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Dijkstra},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:61"}},{"Add":{"text":" journal = {Numerische Mathematik},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:100"}},{"Add":{"text":" title = {A note on two problems in connexion with graphs},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:165"}},{"Add":{"text":" year = {1959},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:187"}},{"Add":{"text":" pages = {269--271},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:213"}},{"Add":{"text":" volume = {1},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:232"}},{"Add":{"text":" bibsource = {dblp computer science bibliography, https://dblp.org},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:302"}},{"Add":{"text":" biburl = {https://dblp.org/rec/journals/nm/Dijkstra59.bib},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:367"}},{"Add":{"text":" doi = {10.1007/BF01386390},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:403"}},{"Add":{"text":" timestamp = {Mon, 29 Jul 2019 15:59:06 +0200},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:452"}},{"Add":{"text":" url = {https://doi.org/10.1007/BF01386390},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:504"}}]}},{"Edit":{"inode":{"change":3,"pos":18034},"path":"literature.bib","line":897,"atom":{"NewVertex":{"up_context":[{"change":3,"pos":2373}],"down_context":[{"change":3,"pos":35999}],"flag":"BLOCK","start":506,"end":1441,"inode":{"change":3,"pos":18034}}},"contents":[{"Add":{"text":"@InProceedings{kremer2016lncs,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:536"}},{"Add":{"text":" author = {Gereon Kremer and Florian Corzilius and Erika {\\'{A}}brah{\\'{a}}m},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:619"}},{"Add":{"text":" booktitle = {Computer Algebra in Scientific Computing - 18th International Workshop, {CASC} 2016, Bucharest, Romania, September 19-23, 2016, Proceedings},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:776"}},{"Add":{"text":" title = {A Generalised Branch-and-Bound Approach and Its Application in {SAT} Modulo Nonlinear Integer Arithmetic},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:898"}},{"Add":{"text":" year = {2016},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:920"}},{"Add":{"text":" editor = {Vladimir P. Gerdt and Wolfram Koepf and Werner M. Seiler and Evgenii V. Vorozhtsov},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1020"}},{"Add":{"text":" pages = {315--335},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1046"}},{"Add":{"text":" publisher = {Springer},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1072"}},{"Add":{"text":" series = {Lecture Notes in Computer Science},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1123"}},{"Add":{"text":" volume = {9890},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1145"}},{"Add":{"text":" bibsource = {dblp computer science bibliography, https://dblp.org},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1215"}},{"Add":{"text":" biburl = {https://dblp.org/rec/conf/casc/KremerCA16.bib},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1278"}},{"Add":{"text":" doi = {10.1007/978-3-319-45641-6\\_21},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1325"}},{"Add":{"text":" timestamp = {Thu, 23 Sep 2021 11:48:04 +0200},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1374"}},{"Add":{"text":" url = {https://doi.org/10.1007/978-3-319-45641-6\\_21},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1437"}},{"Add":{"text":"}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1439"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1440"}}]}},{"Replacement":{"inode":{"change":3,"pos":2412},"path":"figures/ch3_p3_program.tex","line":16,"ins":{"up_context":[{"change":3,"pos":3123}],"down_context":[{"change":3,"pos":3184}],"flag":"BLOCK","start":1442,"end":1506,"inode":{"change":3,"pos":2412}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":3123},"to":{"change":3,"start":3123,"end":3184},"introduced_by":3}],"inode":{"change":3,"pos":2412}},"ins_contents":[{"Add":{"text":" (l1) edge[bend left =30] node {$t_{2}:\\tau_2 = y \\geq 0","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1505"}}],"del_contents":[{"Del":{"text":" (l1) edge[bend left =30] node {$t_{2}:\\tau_2 = y > 0","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3183"}}]}},{"Replacement":{"inode":{"change":3,"pos":2412},"path":"figures/ch3_p3_program.tex","line":18,"ins":{"up_context":[{"change":3,"pos":3211}],"down_context":[{"change":3,"pos":3756}],"flag":"BLOCK","start":1507,"end":1603,"inode":{"change":3,"pos":2412}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":3211},"to":{"change":3,"start":3211,"end":3756},"introduced_by":3}],"inode":{"change":3,"pos":2412}},"ins_contents":[{"Add":{"text":" (l2) edge[bend left=30] node {$t_{3}: \\eta_3(y) = y -","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1568"}},{"Add":{"text":" \\textsc{Bern}(0.5)$} (l1)","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1602"}}],"del_contents":[{"Del":{"text":" (l2) edge[bend left=30] node {$t_{3}:\\begin{aligned} ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3272"}},{"Del":{"text":" p&=0.5 \\\\ \\eta_3(y) &= y - \\textsc{Bern}(0.5)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3330"}},{"Del":{"text":" \\end{aligned}$} (l1)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3359"}},{"Del":{"text":" % (l2) edge[bend left=30,draw=rwth-75] node[swap] {$t_{3a}: p=0.5$} (l1)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3440"}},{"Del":{"text":" % (l1) edge node {$t_4:x \\leq 0$} (l2)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3500"}},{"Del":{"text":" % edge[loop above] node {$t_2:\\begin{aligned} x &> 1 \\\\ x&:=","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3582"}},{"Del":{"text":" % x+1\\end{aligned}$} ()","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3627"}},{"Del":{"text":" % edge[loop below] node {$t_3:\\begin{aligned} x &> 0 \\\\ x&:=","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3710"}},{"Del":{"text":" % x-1\\end{aligned}$} ()","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3755"}}]}},{"Replacement":{"inode":{"change":3,"pos":3838},"path":"figures/ch3_p3_lift.tex","line":16,"ins":{"up_context":[{"change":3,"pos":4665}],"down_context":[{"change":3,"pos":4742}],"flag":"BLOCK","start":1604,"end":1684,"inode":{"change":3,"pos":3838}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":4665},"to":{"change":3,"start":4665,"end":4742},"introduced_by":3}],"inode":{"change":3,"pos":3838}},"ins_contents":[{"Add":{"text":" (l1) edge[bend left =30] node {$t_{2}:\\begin{aligned}\\tau_2 =& y \\geq 0","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1683"}}],"del_contents":[{"Del":{"text":" (l1) edge[bend left =30] node {$t_{2}:\\begin{aligned}\\tau_2 =& y > 0","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4741"}}]}},{"Replacement":{"inode":{"change":3,"pos":3838},"path":"figures/ch3_p3_lift.tex","line":18,"ins":{"up_context":[{"change":3,"pos":4782}],"down_context":[{"change":3,"pos":4931}],"flag":"BLOCK","start":1685,"end":1781,"inode":{"change":3,"pos":3838}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":4782},"to":{"change":3,"start":4782,"end":4931},"introduced_by":3}],"inode":{"change":3,"pos":3838}},"ins_contents":[{"Add":{"text":" (l2) edge[bend left=30] node {$t_{3}: \\eta_3(y) = y -","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1746"}},{"Add":{"text":" \\textsc{Bern}(0.5)$} (l1)","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1780"}}],"del_contents":[{"Del":{"text":" (l2) edge[bend left=30] node {$t_{3}:\\begin{aligned} ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4843"}},{"Del":{"text":" p&=0.5 \\\\ \\eta_3(y) &= y - \\textsc{Bern}(0.5)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4901"}},{"Del":{"text":" \\end{aligned}$} (l1)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4930"}}]}},{"Replacement":{"inode":{"change":3,"pos":7593},"path":"ch6_appendix.tex","line":34,"ins":{"up_context":[{"change":3,"pos":18354}],"down_context":[{"change":3,"pos":18429}],"flag":"BLOCK","start":1782,"end":1857,"inode":{"change":3,"pos":7593}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":18354},"to":{"change":3,"start":18354,"end":18429},"introduced_by":3}],"inode":{"change":3,"pos":7593}},"ins_contents":[{"Add":{"text":" Combined $\\beta \\models \\varphi_1 \\Land\\neg \\varphi_2 \\lightning$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1856"}}],"del_contents":[{"Del":{"text":" Combined $\\beta \\models \\varphi_1 \\Land\\neg \\varphi_2 \\lightning$ ","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:18428"}}]}},{"Edit":{"inode":{"change":3,"pos":7661},"path":"ch5_conclusion.tex","line":2,"atom":{"NewVertex":{"up_context":[{"change":3,"pos":7684}],"down_context":[{"change":3,"pos":7684}],"flag":"BLOCK","start":1858,"end":2612,"inode":{"change":3,"pos":7661}}},"contents":[{"Add":{"text":"\\section{Future Work}\\label{sec:future_work}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1902"}},{"Add":{"text":"In \\Cref{ch:preliminaries} and \\Cref{ch:theory} we have skipped over the concept","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:1983"}},{"Add":{"text":"of costs, which are closely related to runtime complexity. It is common to","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2058"}},{"Add":{"text":"assign costs to a transition which can vary depending on the value of program","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2136"}},{"Add":{"text":"variables. During a run of the program, those costs are added up. The runtime","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2214"}},{"Add":{"text":"runtime complexity of a program is a special case of cost-analysis where every","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2293"}},{"Add":{"text":"transition has costs equal to 1. Because the costs would be equally copied from","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2373"}},{"Add":{"text":"one transition to the unfolded transition, we conjecture that the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2439"}},{"Add":{"text":"partial evaluation preserves worst-case expected costs similarly to the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2511"}},{"Add":{"text":"worst-case expected runtime complexity. However, answering this question needs","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2590"}},{"Add":{"text":"additional research.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2611"}}]}},{"Replacement":{"inode":{"change":3,"pos":7661},"path":"ch5_conclusion.tex","line":14,"ins":{"up_context":[{"change":3,"pos":7685}],"down_context":[],"flag":"BLOCK","start":2613,"end":5588,"inode":{"change":3,"pos":7661}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":7685},"to":{"change":3,"start":7283,"end":7326},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":7326},"to":{"change":3,"start":3029,"end":3074},"introduced_by":3}],"inode":{"change":3,"pos":7661}},"ins_contents":[{"Add":{"text":"In \\Sref{sec:performance}, we have seen that the new implementation is not as","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2690"}},{"Add":{"text":"good as iRankFinder on a small number of programs in the test sets. Since we","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2767"}},{"Add":{"text":"tried to mirror the behaviour of iRankFinder as close as possible, there were","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2845"}},{"Add":{"text":"most probably mistakes made during implementation. For example, our","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2913"}},{"Add":{"text":"implementation of the heuristics deviate from iRankFinders implementation or the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:2994"}},{"Add":{"text":"selecting the feedback set for locations had an unexpected impact on the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3067"}},{"Add":{"text":"analysis. In any case, the worsened analysis results warrant further","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3136"}},{"Add":{"text":"investigation. Until then, \\gls{koat} can keep using iRankFinder for","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3205"}},{"Add":{"text":"non-probabilistic integer programs.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3241"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3242"}},{"Add":{"text":"During implementation, the constraints of versions are converted back and forth","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3322"}},{"Add":{"text":"to Aprons representation of polyhedra. There is probably room for improvement,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3401"}},{"Add":{"text":"by reducing the number of conversions to and from apron. Furthermore, the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3475"}},{"Add":{"text":"satisfiability check for the constraint of a version is done twice. Once before","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3555"}},{"Add":{"text":"the unolding and once after the unfolding during abstraction even though the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3632"}},{"Add":{"text":"satisfiability after the unfolding is already implied. We can probably remove","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3710"}},{"Add":{"text":"one of the satisfiability checks and in turn reduce the number of calls to the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3789"}},{"Add":{"text":"\\gls{smt}-solver.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3807"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3808"}},{"Add":{"text":"The heuristics used for property generation were taken over from","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3873"}},{"Add":{"text":"\\citeauthor{Domenech19}, however in our implementation we found that the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:3946"}},{"Add":{"text":"heuristics resulted in a large number of properties and thus a larger than","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4021"}},{"Add":{"text":"probably necessary evaluation graph. With regard to sub-\\gls{scc} level","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4093"}},{"Add":{"text":"evaluation, one could search for better heuristics which take the context of the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4174"}},{"Add":{"text":"previous analysis better into account.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4213"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4214"}},{"Add":{"text":"A last idea, that came up during the writing of this thesis is some sort of","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4290"}},{"Add":{"text":"online-abstraction. The localised property-based abstraction requires locations","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4370"}},{"Add":{"text":"to be selected for abstraction before the evaluation. Instead, one can imagine","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4449"}},{"Add":{"text":"an evaluation algorithm that abstracts only when traversing a loop on-the-fly,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4528"}},{"Add":{"text":"even ignoring the first $k$ number of iterations and thus only abstracting when","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4608"}},{"Add":{"text":"really necessary. Such an online-abstraction could possibly increase the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4681"}},{"Add":{"text":"precision of the partial evaluation even further. It is not immediately obvious","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4761"}},{"Add":{"text":"if this would improve the sub-\\gls{scc} level analysis of \\gls{koat} but this","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4839"}},{"Add":{"text":"might be interesting for some other class of programs, we haven't considered in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4919"}},{"Add":{"text":"this thesis.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4932"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4933"}},{"Add":{"text":"\\section{Conclusion}\\label{sec:conclusion} ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:4977"}},{"Add":{"text":"In \\Cref{ch:theory} the method of control-flow refinement via partial evaluation","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5058"}},{"Add":{"text":"was expanded to \\gls{pip}. The method is proven to be sound for \\gls{pip} and","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5136"}},{"Add":{"text":"can be used to refine a \\gls{pip} when classical analysis fails at finding tight","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5217"}},{"Add":{"text":"runtime bounds. In \\Cref{ch:implementation} we have presented a new","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5285"}},{"Add":{"text":"implementation of partial evaluation on probabilistic programs and successfully","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5365"}},{"Add":{"text":"inferred tighter expected runtime complexity bounds for an example \\gls{pip}. To","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5446"}},{"Add":{"text":"the best of our knowledge, we applied partial evaluation for the first time in a","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5527"}},{"Add":{"text":"complexity analysis frame work for probabilistic programs.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5586"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5587"}}],"del_contents":[{"Del":{"text":"\\section{Conclusion}\\label{sec:conclusion}","pos":"MQJMCLGFSBSFJ3JPCIHQSBZS4XZPGJI7NGGRASX5UVGJO6HES25AC:7325"}},{"Del":{"text":"\\section{Future Work}\\label{sec:future_work}","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:3073"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":2,"ins":{"up_context":[{"change":3,"pos":7815}],"down_context":[{"change":3,"pos":7815}],"flag":"BLOCK","start":5589,"end":6075,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":7815},"to":{"change":2,"start":21862,"end":22710},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"This chapter will present the implementation of partial evaluation with","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5660"}},{"Add":{"text":"localised property-based abstraction as introduced in \\Cref{ch:theory}. In","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5735"}},{"Add":{"text":"\\Sref{sec:implementation} we describe the implementation using pseudo-code. We","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5814"}},{"Add":{"text":"will mostly focus on the practical aspects of the partial evaluation.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5884"}},{"Add":{"text":"\\Sref{sec:technical_details} will briefly describe the technical details of the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:5964"}},{"Add":{"text":"implementation. Finally, in \\Sref{sec:performance} we will analyse the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6035"}},{"Add":{"text":"performance of the new implementation.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6074"}}],"del_contents":[{"Del":{"text":"This chapter will present the implementation of partial-evaluation with","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:21933"}},{"Del":{"text":"localised property-based abstraction as introduced in the previous","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22000"}},{"Del":{"text":"\\Cref{ch:theory}. In \\Sref{sec:implementation} we describe the implementation","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22078"}},{"Del":{"text":"using pseudo-code. We will mostly on the practical aspects of the partial","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22152"}},{"Del":{"text":"evaluation. ","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22165"}},{"Del":{"text":"% \\Sref{sec:abstract_domain} will briefly discuss the possible","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22228"}},{"Del":{"text":"% abstract domains and the reasons why we decided to use the abstract domain of","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22308"}},{"Del":{"text":"% Polyhedra. \\Sref{sec:loop_heads} will focus on the different possibilities to","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22388"}},{"Del":{"text":"% mark locations for abstraction, which is part of the history-dependent aspect of","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22471"}},{"Del":{"text":"% the abstraction layer. \\Sref{sec:technical_details} describes the technical","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22549"}},{"Del":{"text":"% details of the actual implementation and finally in \\Sref{sec:performance} the","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22630"}},{"Del":{"text":"% correctness and performance of the implementation is evaluated practically. ","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22709"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":11,"ins":{"up_context":[{"change":3,"pos":7378}],"down_context":[{"change":2,"pos":23254}],"flag":"BLOCK","start":6076,"end":6456,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":7378},"to":{"change":2,"start":22711,"end":22786},"introduced_by":2},{"previous":"","flag":"DELETED","from":{"change":2,"pos":22786},"to":{"change":3,"start":1233,"end":1308},"introduced_by":2},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":1233},"to":{"change":3,"start":1233,"end":1308},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":1308},"to":{"change":2,"start":22787,"end":23254},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"First, we present two variants for selecting locations on all the loops such","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6152"}},{"Add":{"text":"that the property from Lemma \\ref{lem:localized_pba_finite} for the localised","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6230"}},{"Add":{"text":"property-based abstraction is satisfied. Second, we will introduce the heuristic","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6311"}},{"Add":{"text":"for property generation, that was put forward by \\citeauthor{Domenech19}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6384"}},{"Add":{"text":"\\cite{Domenech19}. Third, we will describe the technique for selecting","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6455"}}],"del_contents":[{"Del":{"text":"In this section, we present a pseudo-code implementation for the $\\unfold$","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22785"}},{"Del":{"text":"operation, and the partial evaluation of an entire program as described in","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:1307"}},{"Del":{"text":"\\Sref{sec:partialevaluation} using localised property-based abstraction. First,","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22866"}},{"Del":{"text":"we will present two variants for selecting locations on all the loops such that","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:22946"}},{"Del":{"text":"the property from Lemma \\ref{lem:localized_pba_finite} is satisfied. Second, we","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:23026"}},{"Del":{"text":"will introduce the heuristic for property generation, that was put forward by","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:23104"}},{"Del":{"text":"\\citeauthor{Domenech19} \\cite{Domenech19} and give an implementation as a","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:23178"}},{"Del":{"text":"pseudo-code algorithm. Third, we will describe the technique for selecting","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:23253"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":23,"ins":{"up_context":[{"change":2,"pos":23659}],"down_context":[{"change":2,"pos":23729}],"flag":"BLOCK","start":6457,"end":6527,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":23659},"to":{"change":2,"start":23659,"end":23729},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"Recall Lemma \\ref{lem:localized_pba_finite}; as long as the localized","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6526"}}],"del_contents":[{"Del":{"text":"Recall Lemma \\ref{lem:localized_pba_finite}. As long as the localized","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:23728"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":26,"ins":{"up_context":[{"change":2,"pos":23887}],"down_context":[{"change":2,"pos":23967}],"flag":"BLOCK","start":6528,"end":6608,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":23887},"to":{"change":2,"start":23887,"end":23967},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"graph is finite. The naive approach to assert this property, is to use a global","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6607"}}],"del_contents":[{"Del":{"text":"graph is finite. The native approach to assert this property is to use a global","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:23966"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":32,"ins":{"up_context":[{"change":2,"pos":24358}],"down_context":[{"change":3,"pos":3312}],"flag":"BLOCK","start":6609,"end":6630,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":24358},"to":{"change":2,"start":24358,"end":24372},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"unnecessarily slow. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6629"}}],"del_contents":[{"Del":{"text":"even slower. ","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:24371"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":43,"ins":{"up_context":[{"change":2,"pos":24736}],"down_context":[{"change":3,"pos":2315}],"flag":"BLOCK","start":6631,"end":6895,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":24736},"to":{"change":2,"start":24736,"end":24975},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"number of loop-heads by computing the feedback-vertex set of the graph. Reducing","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6711"}},{"Add":{"text":"the number of loop-heads makes the successive partial evaluation faster, and the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6792"}},{"Add":{"text":"partial-evaluation graph more precise in the mostly linear parts, where","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6864"}},{"Add":{"text":"abstraction is not necessary.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6894"}}],"del_contents":[{"Del":{"text":"number of loop-heads by computing the feedback-vertex set. Reducing the number","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:24814"}},{"Del":{"text":"of loop-heads makes successive evaluation faster, and the partial-evaluation","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:24891"}},{"Del":{"text":"graph more precise in the mostly linear parts, where abstraction is not","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:24963"}},{"Del":{"text":"necessary.","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:24974"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":52,"ins":{"up_context":[{"change":3,"pos":3325}],"down_context":[{"change":3,"pos":3403}],"flag":"BLOCK","start":6896,"end":6974,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":3325},"to":{"change":3,"start":3325,"end":3403},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" Given a directed graph $G = (V, E)$, find the minimal subset $R \\subseteq","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:6973"}}],"del_contents":[{"Del":{"text":" Given a directed graph $G = (V, E)$. Find the minimal subset $R \\subseteq","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:3402"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":65,"ins":{"up_context":[{"change":3,"pos":709}],"down_context":[{"change":2,"pos":25272}],"flag":"BLOCK","start":6975,"end":7051,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":709},"to":{"change":2,"start":25198,"end":25272},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"An instance of a hitting set problem can be solved using a \\gls{ilp} and an","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7050"}}],"del_contents":[{"Del":{"text":"An instance of a hitting set problem can be solved using \\gls{ilp} and an","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:25271"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":67,"ins":{"up_context":[{"change":2,"pos":25346}],"down_context":[{"change":3,"pos":36356}],"flag":"BLOCK","start":7052,"end":7172,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":25346},"to":{"change":2,"start":25346,"end":25466},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"\\gls{ilp} instance, where the variables $x_{\\ell_0}, \\dots,x_{\\ell_n}$ are","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7126"}},{"Add":{"text":"variables for every location in the program.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7171"}}],"del_contents":[{"Del":{"text":"\\gls{ilp} instance, where the variables $x_{\\ell_0}, \\dots,x_{\\ell_n}$ are variables","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:25430"}},{"Del":{"text":"for every location in the program.","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:25465"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":78,"ins":{"up_context":[{"change":3,"pos":4696}],"down_context":[{"change":3,"pos":5175}],"flag":"BLOCK","start":7173,"end":7723,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":4696},"to":{"change":2,"start":25778,"end":26342},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"The location variables are constrained to $0 \\leq x_{\\ell_j}\\leq 1$ for all","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7248"}},{"Add":{"text":"$\\ell_j \\in \\Loc_\\Prog$ by \\eqref{eq:ilp-geq0} and \\eqref{eq:ilp-leq1}. For an","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7327"}},{"Add":{"text":"assignment $\\sigma : \\Loc_\\Prog \\rightarrow \\N$, a value $\\sigma(x_{\\ell_j}) = 1$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7409"}},{"Add":{"text":"indicates that the location $\\ell_j$ is part of the hitting set; a value","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7482"}},{"Add":{"text":"$\\sigma(x_{\\ell_j}) = 0$ indicates that it is not. The constraints","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7549"}},{"Add":{"text":"\\eqref{eq:lip1-hit} guarantee that at least one location on every loop is","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7623"}},{"Add":{"text":"selected, and the optimisation goal \\eqref{eq:ilp-vfs} minimises the number of","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7702"}},{"Add":{"text":"selected locations.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7722"}}],"del_contents":[{"Del":{"text":"The value for the location variables are constrained to $0 \\leq x_{\\ell_j}\\leq","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:25856"}},{"Del":{"text":"1$ for all $\\ell_j \\in \\Loc_\\Prog$ by \\eqref{eq:ilp-geq0} and","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:25918"}},{"Del":{"text":"\\eqref{eq:ilp-leq1}. For an assignment $\\sigma : \\Loc_\\Prog \\rightarrow \\N$ a","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:25996"}},{"Del":{"text":"value $\\sigma(x_{\\ell_j}) = 1$ indicates that the location $\\ell_j$ is part of","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:26075"}},{"Del":{"text":"the hitting set; a value $\\sigma(x_{\\ell_j}) = 0$ indicates that it is not. The","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:26155"}},{"Del":{"text":"constraints \\eqref{eq:lip1-hit} guarantees that at least one location on every","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:26234"}},{"Del":{"text":"loop is selected, and the optimization goal \\eqref{eq:ilp-vfs} minimizes the","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:26311"}},{"Del":{"text":"number of selected locations.","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:26341"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":87,"ins":{"up_context":[{"change":3,"pos":5176}],"down_context":[{"change":3,"pos":5615}],"flag":"BLOCK","start":7724,"end":7834,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":5176},"to":{"change":2,"start":26343,"end":26464},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"Finding the feedback-vertex-set and loops is independent of the partial","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7795"}},{"Add":{"text":"evaluation and can be done a-priori. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7833"}}],"del_contents":[{"Del":{"text":"The computation of the feedback-vertex-set and loops is independent of the","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:26417"}},{"Del":{"text":"partial evaluation and can be done a-priori. ","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:26463"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":96,"ins":{"up_context":[{"change":2,"pos":26882}],"down_context":[{"change":3,"pos":5657}],"flag":"BLOCK","start":7835,"end":7957,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":26882},"to":{"change":2,"start":26882,"end":27014},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"\\cite{Domenech19}. We briefly describe this heuristic. For a location $\\ell \\in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7914"}},{"Add":{"text":"L$, the following heuristics are defined.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:7956"}}],"del_contents":[{"Del":{"text":"\\cite{Domenech19}. We will just briefly describe this heuristic. For a location","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:26961"}},{"Del":{"text":"$\\ell \\in L$, the following heuristics are defined.","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:27013"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":100,"ins":{"up_context":[{"change":2,"pos":27030}],"down_context":[{"change":2,"pos":27092}],"flag":"BLOCK","start":7958,"end":8016,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":27030},"to":{"change":2,"start":27030,"end":27092},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" \\PRh(\\ell) &= \\set{\\psi}{(\\ell, \\_, \\tau, \\_, \\_) \\in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8015"}}],"del_contents":[{"Del":{"text":" \\PRh(\\ell) &= \\set{\\psi}{t = (\\ell, \\_, \\tau, \\_, \\_) \\in","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:27091"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":102,"ins":{"up_context":[{"change":2,"pos":27132}],"down_context":[{"change":2,"pos":27203}],"flag":"BLOCK","start":8017,"end":8084,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":27132},"to":{"change":2,"start":27132,"end":27203},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" \\PRhv(\\ell) &= \\set{x \\diamond c}{(\\ell, \\_, \\tau, \\_, \\_) \\in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8083"}}],"del_contents":[{"Del":{"text":" \\PRhv(\\ell) &= \\set{x \\diamond c}{t = (\\ell, \\_, \\tau, \\_, \\_) \\in","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:27202"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":104,"ins":{"up_context":[{"change":2,"pos":27290}],"down_context":[{"change":2,"pos":27354}],"flag":"BLOCK","start":8085,"end":8145,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":27290},"to":{"change":2,"start":27290,"end":27354},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" \\PRc(\\ell) &= \\set{\\psi}{(\\_, \\_, \\tau, \\eta, \\ell) \\in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8144"}}],"del_contents":[{"Del":{"text":" \\PRc(\\ell) &= \\set{\\psi}{t = (\\_, \\_, \\tau, \\eta, \\ell) \\in","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:27353"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":106,"ins":{"up_context":[{"change":2,"pos":27427}],"down_context":[{"change":2,"pos":27496}],"flag":"BLOCK","start":8146,"end":8211,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":27427},"to":{"change":2,"start":27427,"end":27496},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" \\PRcv(\\ell) &= \\set{x \\diamond c}{(\\_, \\_, \\tau, \\eta, \\ell)","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8210"}}],"del_contents":[{"Del":{"text":" \\PRcv(\\ell) &= \\set{x \\diamond c}{t = (\\_, \\_, \\tau, \\eta, \\ell)","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:27495"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":111,"ins":{"up_context":[{"change":2,"pos":27718}],"down_context":[{"change":2,"pos":28484}],"flag":"BLOCK","start":8212,"end":9147,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":27718},"to":{"change":2,"start":27718,"end":28272},"introduced_by":2},{"previous":"","flag":"DELETED","from":{"change":2,"pos":28272},"to":{"change":3,"start":5659,"end":5660},"introduced_by":2},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":3603},"to":{"change":3,"start":5659,"end":5660},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":5660},"to":{"change":2,"start":28273,"end":28484},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"computes bounds for all the program variables that might not be captured","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8284"}},{"Add":{"text":"otherwise, because they are implicitly implied by the guard. $\\PRc$ and $\\PRcv$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8364"}},{"Add":{"text":"do the same for all the incoming transitions. In addition, a heuristic $\\PRd$ is","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8445"}},{"Add":{"text":"defined that propagates properties backwards through a loop, capturing the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8520"}},{"Add":{"text":"conditions that are required to iterate through the loop without leaving the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8597"}},{"Add":{"text":"loop. The loops are first transformed into transition loops $T_{L_1},\\dots,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8673"}},{"Add":{"text":"T_{L_n} \\subseteq \\TP$, as described in \\Sref{sec:graph_operations}. Then for","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8751"}},{"Add":{"text":"every transition loop and loop-head on the loop, the transition loop is rotated","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8831"}},{"Add":{"text":"to have the loop-head first, and the guards are propagated backwards starting at","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8912"}},{"Add":{"text":"the end of the loop. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:8934"}},{"Add":{"text":"For a transition loop $T = \\{t_0, \\dots, t_m\\}$ and a location $\\ell \\in L_j$ on","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9015"}},{"Add":{"text":"that loop, a backwards propagation function $\\back : \\T \\times \\C \\rightarrow","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9093"}},{"Add":{"text":"\\C$ and the heuristic $\\PRd$ are defined as follows:","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9146"}}],"del_contents":[{"Del":{"text":"computes bounds for all program variables that might not be captured otherwise,","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:27797"}},{"Del":{"text":"because they they implicitly implied by the guard.","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:27848"}},{"Del":{"text":"$\\PRc$ and $PRcv$ do the same for all incoming transition. In addition, a","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:27922"}},{"Del":{"text":"heuristic $\\PRd$ is defined that propagates properties backwards through a loop,","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:28003"}},{"Del":{"text":"capturing the conditions that are required to iterate through the loop without","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:28082"}},{"Del":{"text":"leaving the loop. The loops are first transformed into transition loops","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:28154"}},{"Del":{"text":"$T_{L_1},\\dots, T_{L_n} \\subseteq \\TP$, as described in","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:28210"}},{"Del":{"text":"\\Sref{sec:graph_operations}. Then for every transition loop ","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:28271"}},{"Del":{"text":"","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:5659"}},{"Del":{"text":"For a transition loop $T = \\{t_0, \\dots, t_m\\}$ and a location $\\ell \\in L_j$","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:28350"}},{"Del":{"text":"on that loop, a backwards propagation function $\\back : \\T \\times \\C \\rightarrow","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:28431"}},{"Del":{"text":"\\C$ and the heuristic $\\PRd$ is defined as follows:","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:28483"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":136,"ins":{"up_context":[{"change":3,"pos":6036}],"down_context":[{"change":2,"pos":29056}],"flag":"BLOCK","start":9148,"end":9317,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":6036},"to":{"change":2,"start":28920,"end":29056},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"The localised property-based abstraction using those properties $\\Psi_\\ell$ for","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9227"}},{"Add":{"text":"every locations $\\ell \\in L$. For all other $\\ell \\not\\in L$, we set $\\Psi_\\ell","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9307"}},{"Add":{"text":"= \\bot$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9316"}}],"del_contents":[{"Del":{"text":"The localised property-based abstraction using those properties for the","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:28991"}},{"Del":{"text":"locations marked for abstraction is used by our implementation.","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:29055"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":140,"ins":{"up_context":[{"change":2,"pos":29057}],"down_context":[{"change":2,"pos":29096}],"flag":"BLOCK","start":9318,"end":9376,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":29057},"to":{"change":2,"start":29057,"end":29096},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"\\subsection{Selecting transitions for partial evaluation}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9375"}}],"del_contents":[{"Del":{"text":"\\subsection{Selecting Transitions $T$}","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:29095"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":142,"ins":{"up_context":[{"change":2,"pos":29176}],"down_context":[{"change":3,"pos":6711}],"flag":"BLOCK","start":9377,"end":9975,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":29176},"to":{"change":2,"start":29176,"end":29620},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"every transition are computed. Whenever the analysis fails to find linear","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9450"}},{"Add":{"text":"runtime-complexity bounds for a transition, the transition is added to a set of","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9530"}},{"Add":{"text":"transitions $\\T_\\text{cfr}$ that will need further refinement. The set of","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9604"}},{"Add":{"text":"transitions $T$ used for control-flow-refinement is generated by selecting the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9683"}},{"Add":{"text":"smallest loop for all transitions in $\\T_\\text{cfr}$ and expanding it to all","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9760"}},{"Add":{"text":"parallel transitions. The reasoning behind the selection was explained in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9834"}},{"Add":{"text":"detail by \\citeauthor{giesl2022lncs} \\cite{giesl2022lncs} for classical","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9906"}},{"Add":{"text":"program. The algorithm is displayed in Algorithm \\ref{alg:selectT}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9974"}}],"del_contents":[{"Del":{"text":"every transition are computed. Whenever the analysis fails at finding linear","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:29252"}},{"Del":{"text":"runtime-complexity bounds for a transition, this transition is added to a set of","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:29333"}},{"Del":{"text":"transitions $\\T_\\text{nl}$ which need further refinement. The set of transitions","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:29414"}},{"Del":{"text":"$T$ used for control-flow-refinement is generated by selecting the smallest loop","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:29495"}},{"Del":{"text":"for all transitions in $\\T_\\text{nl}$ and expanding it to all parallel","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:29566"}},{"Del":{"text":"transitions. The algorithm is describe in details by","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:29619"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":151,"ins":{"up_context":[{"change":3,"pos":6712}],"down_context":[{"change":2,"pos":29639}],"flag":"BLOCK","start":9976,"end":10939,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":6712},"to":{"change":2,"start":29621,"end":29639},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"\\begin{algorithm}[ht]","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:9997"}},{"Add":{"text":" \\caption{Selecting transitions $T$ for partial evaluation for a set of","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10072"}},{"Add":{"text":" non-trivial transitions $\\T_\\text{cfr}$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10116"}},{"Add":{"text":" \\cite{giesl2022lncs}.\\label{alg:selectT}}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10162"}},{"Add":{"text":" \\KwData{ A PIP $\\Prog = (\\PV, \\Loc_\\Prog, \\ell_0, \\GTP)$, and a non-empty","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10240"}},{"Add":{"text":" subset of transitions $\\T_\\text{cfr} \\subseteq \\TP$.} ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10299"}},{"Add":{"text":" \\KwResult{A subset of transitions $T \\subseteq \\TP$.}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10357"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10358"}},{"Add":{"text":" \\SetKwFunction{FSelect}{select}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10394"}},{"Add":{"text":" \\SetKwProg{Fn}{Function}{:}{end}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10431"}},{"Add":{"text":" \\Fn{\\FSelect{$\\T_\\text{cfr}$}}{","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10467"}},{"Add":{"text":" $T = \\emptyset$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10493"}},{"Add":{"text":" \\For{$t = (\\ell, \\_, \\_, \\_, \\ell') \\in \\T_\\text{cfr}$}{","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10558"}},{"Add":{"text":" $T_t \\leftarrow \\text{a shortest path from } \\ell \\text{ to }","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10632"}},{"Add":{"text":" \\ell'$\\label{alg:line:shortest_path}\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10683"}},{"Add":{"text":" $T_t \\leftarrow T \\union \\{t\\}$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10729"}},{"Add":{"text":" $T_t \\leftarrow T \\union \\set{(\\ell, \\_, \\_, \\_, \\ell') \\in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10801"}},{"Add":{"text":" \\T_\\Prog}{(\\ell, \\_, \\_, \\_, \\ell') \\in T}$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10859"}},{"Add":{"text":" $T \\leftarrow T \\union T_t$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10901"}},{"Add":{"text":" }","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10911"}},{"Add":{"text":" \\Return{$T$}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10932"}},{"Add":{"text":" }","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10938"}}],"del_contents":[{"Del":{"text":"\\begin{algorithm}","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:29638"}}]}},{"Edit":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":174,"atom":{"NewVertex":{"up_context":[{"change":2,"pos":29655}],"down_context":[{"change":2,"pos":29655}],"flag":"BLOCK","start":10940,"end":11260,"inode":{"change":3,"pos":7792}}},"contents":[{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:10940"}},{"Add":{"text":"The shortest path on line \\ref{alg:line:shortest_path} of Algorithm","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11008"}},{"Add":{"text":"\\ref{alg:selectT} can be computed using one of many variants of Djikstra's","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11083"}},{"Add":{"text":"algorithm \\cite{dijkstra1959}, which is readily available in most graph","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11155"}},{"Add":{"text":"libraries. The shortest path is closed into a transition loop and then all","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11230"}},{"Add":{"text":"parallel edges are selected.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11259"}}]}},{"Edit":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":181,"atom":{"NewVertex":{"up_context":[{"change":2,"pos":29656}],"down_context":[{"change":3,"pos":7258}],"flag":"BLOCK","start":11261,"end":11988,"inode":{"change":3,"pos":7792}}},"contents":[{"Add":{"text":"\\subsection{Partial evaluation}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11292"}},{"Add":{"text":"The unfolding is implemented using the abstract domain of convex polyhedra. In","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11371"}},{"Add":{"text":"practice, any other abstract domain can be used, but since the set of transitions","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11453"}},{"Add":{"text":"is kept small in general and precision is favoured over speed, polyhedra allow","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11532"}},{"Add":{"text":"exact projections of linear constraints. Let $\\texttt{proj}: 2^\\V \\times \\Poly","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11611"}},{"Add":{"text":"\\rightarrow \\Poly$ be the projection, and $\\texttt{update}: (\\PV \\rightarrow","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11688"}},{"Add":{"text":"\\Z[\\V]) \\times \\Poly \\rightarrow \\Poly$ be an update in the abstract domain. We","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11768"}},{"Add":{"text":"can convert to and from polyhedra with the $\\texttt{polyh} : \\C \\rightarrow","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11844"}},{"Add":{"text":"\\Poly$ and $\\texttt{constr}: \\Poly \\rightarrow \\C$ function. An implementation","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11923"}},{"Add":{"text":"of the unfold operation is shown in Algorithm \\ref{alg:unfold}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:11987"}}]}},{"Edit":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":192,"atom":{"EdgeMap":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":7259},"to":{"change":2,"start":29657,"end":29696},"introduced_by":2}],"inode":{"change":3,"pos":7792}}},"contents":[{"Del":{"text":"\\subsection{Partial Evaluation on $T$}","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:29695"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":194,"ins":{"up_context":[{"change":3,"pos":7302}],"down_context":[{"change":3,"pos":7332}],"flag":"BLOCK","start":11989,"end":12059,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":7302},"to":{"change":3,"start":7302,"end":7332},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" \\KwData{ A \\gls{pip} $\\Prog = (\\PV, \\Loc_\\Prog, \\ell_0, \\GTP)$ } ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12058"}}],"del_contents":[{"Del":{"text":" \\KwData{ A PIP $\\Prog$ } ","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:7331"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":200,"ins":{"up_context":[{"change":3,"pos":7563}],"down_context":[{"change":3,"pos":7959}],"flag":"BLOCK","start":12060,"end":12306,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":7563},"to":{"change":3,"start":7563,"end":7959},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" $P \\leftarrow \\texttt{polyh}(\\varphi \\land \\tau \\Land \\lfloor\\eta\\rceil)$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12143"}},{"Add":{"text":" $P \\leftarrow \\texttt{update}(\\tilde{\\eta}, P)$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12201"}},{"Add":{"text":" $P \\leftarrow \\texttt{proj}(\\PV_\\Prog, P)$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12254"}},{"Add":{"text":" $\\varphi' \\leftarrow \\texttt{constr}(P)$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12305"}}],"del_contents":[{"Del":{"text":" \\uIf{$\\varphi \\Land \\tau$ is satisfiable} {","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:7618"}},{"Del":{"text":" $P \\leftarrow \\texttt{polyh}(\\varphi \\land \\tau \\Land \\lfloor\\eta\\rceil)$\\;","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:7706"}},{"Del":{"text":" $P \\leftarrow \\texttt{update}(\\tilde{\\eta}, P)$\\;","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:7768"}},{"Del":{"text":" $P \\leftarrow \\texttt{proj}(\\PV_\\Prog, P)$\\;","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:7825"}},{"Del":{"text":" $\\varphi' \\leftarrow \\texttt{constr}(P)$\\;","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:7880"}},{"Del":{"text":" } \\Else{","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:7897"}},{"Del":{"text":" $\\varphi' \\leftarrow \\texttt{false}$\\;","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:7948"}},{"Del":{"text":" }","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:7958"}}]}},{"Edit":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":207,"atom":{"NewVertex":{"up_context":[{"change":3,"pos":8108}],"down_context":[{"change":3,"pos":8108}],"flag":"BLOCK","start":12307,"end":15501,"inode":{"change":3,"pos":7792}}},"contents":[{"Add":{"text":" }","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12312"}},{"Add":{"text":"\\end{algorithm}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12328"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12329"}},{"Add":{"text":"Finally, we present the partial evaluation on subsets of transitions $T \\subseteq","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12411"}},{"Add":{"text":"\\TP$ that were selected as shown in the previous section using a localised","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12486"}},{"Add":{"text":"abstraction function $S$. The full algorithm is displayed in Algorithm","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12557"}},{"Add":{"text":"\\ref{alg:pe}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12571"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12572"}},{"Add":{"text":"First, the program is converted into its graph $G_\\Prog$, then the transitions","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12651"}},{"Add":{"text":"from $T$ are removed and the graph is lifted to a partial evaluation graph. An","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12730"}},{"Add":{"text":"empty set of versions $V$ is created that remembers the versions that have","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12805"}},{"Add":{"text":"already been evaluated in order to avoid unnecessary computations. Additionally","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12885"}},{"Add":{"text":"a new set of general transitions $\\GTPn$ is initialised, where all newly found","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:12964"}},{"Add":{"text":"general transitions are added. A single evaluation starts from the target","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13038"}},{"Add":{"text":"location $\\ell \\in \\Loc_\\Prog$ of an entry transition $t = (\\_, \\_, \\_, \\_,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13114"}},{"Add":{"text":"\\ell) \\in \\TPin(T)$ of the subset of transitions $T$. It evaluates recursively","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13193"}},{"Add":{"text":"all the newly encountered versions until the evaluation would leave $T$ or the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13272"}},{"Add":{"text":"version has already been encountered. The evaluation is repeated for every entry","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13353"}},{"Add":{"text":"transition, expanding the partial evaluation every time. We just have to","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13426"}},{"Add":{"text":"consider the special case when the subset of transitions contains the starting","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13505"}},{"Add":{"text":"location, in which case an evaluation is also started from $\\ell_0$. The","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13578"}},{"Add":{"text":"satisfiability check on line \\ref{alg:line:check_sat} of Algorithm \\ref{alg:pe}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13658"}},{"Add":{"text":"cuts the evaluation short and avoids unnecessary computations for unreachable","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13736"}},{"Add":{"text":"versions. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13747"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13748"}},{"Add":{"text":"\\begin{algorithm}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13766"}},{"Add":{"text":" \\caption{Partial evaluation of a subset of transitions $T$ of a \\gls{pip}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13844"}},{"Add":{"text":" $\\Prog$.\\label{alg:pe}}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13872"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13873"}},{"Add":{"text":" \\KwData{ A \\gls{pip} $\\Prog = (\\PV, \\Loc_\\Prog, \\ell_0, \\GTP)$, localised","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:13951"}},{"Add":{"text":" abstraction function $S$ and a subset of transitions $T \\subseteq \\TP$.}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14028"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14029"}},{"Add":{"text":" \\KwResult{A PIP $\\Prog'$ with equal runtime-complexity bounds.}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14097"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14098"}},{"Add":{"text":" \\SetKw{Continue}{continue}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14129"}},{"Add":{"text":" \\SetKwFunction{FEvaluate}{evaluate}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14169"}},{"Add":{"text":" \\SetKwProg{Fn}{Function}{:}{end}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14206"}},{"Add":{"text":" $G \\leftarrow (G_\\Prog - T)\\lift$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14246"}},{"Add":{"text":" $V \\leftarrow \\emptyset$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14277"}},{"Add":{"text":" % $\\GTPn \\leftarrow$ \\set{\\{\\}}{\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14316"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14317"}},{"Add":{"text":" \\Fn{\\FEvaluate{$G$, $v$}}{","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14348"}},{"Add":{"text":" \\lIf{$v \\in V$}{\\Return $G$}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14385"}},{"Add":{"text":" $V \\leftarrow V \\uplus \\{v\\}$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14425"}},{"Add":{"text":" $\\langle \\ell, \\varphi\\rangle \\leftarrow v$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14479"}},{"Add":{"text":" \\For{$g \\in \\GTPout(\\ell)$}{","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14516"}},{"Add":{"text":" \\lIf{$\\tau_g \\land \\varphi$ is satisfiable\\label{alg:line:check_sat}}{\\Continue}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14609"}},{"Add":{"text":" $g' \\leftarrow \\emptyset$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14647"}},{"Add":{"text":" \\For{$t \\in g$}{","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14676"}},{"Add":{"text":" \\uIf{$t \\in T$}{","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14709"}},{"Add":{"text":" $(\\langle\\ell,\\varphi\\rangle, p, \\tau, \\eta, \\langle\\ell',","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14788"}},{"Add":{"text":" \\varphi'\\rangle) \\leftarrow \\unfold(\\varphi, t)$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14859"}},{"Add":{"text":" $t' \\leftarrow (\\langle\\ell,\\varphi\\rangle, p, \\tau, \\eta,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14938"}},{"Add":{"text":" S(\\langle\\ell',\\varphi'\\rangle))$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:14994"}},{"Add":{"text":" $g' \\leftarrow g \\union \\{t'\\}$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15048"}},{"Add":{"text":" $G \\leftarrow \\FEvaluate{$G$, $\\langle\\ell',\\varphi\\rangle$}$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15132"}},{"Add":{"text":" }\\uElse{","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15157"}},{"Add":{"text":" $t' \\leftarrow (\\langle\\ell,\\varphi\\rangle, p, \\tau, \\eta,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15236"}},{"Add":{"text":" \\langle\\ell',\\texttt{true}\\rangle)$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15294"}},{"Add":{"text":" $g' \\leftarrow g' \\union {t'}$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15347"}},{"Add":{"text":" }","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15365"}},{"Add":{"text":" }","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15379"}},{"Add":{"text":" $G \\leftarrow G + g'$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15415"}},{"Add":{"text":" $\\GTPn \\leftarrow \\GTPn \\union \\{g'\\}$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15468"}},{"Add":{"text":" }","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15478"}},{"Add":{"text":" \\Return {$G$}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15500"}}]}},{"Edit":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":274,"atom":{"NewVertex":{"up_context":[{"change":3,"pos":8114}],"down_context":[{"change":3,"pos":8114}],"flag":"BLOCK","start":15502,"end":15914,"inode":{"change":3,"pos":7792}}},"contents":[{"Add":{"text":" \\lIf{$(\\ell_0, \\_, \\_, \\_,\\_) \\in T$} {","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15545"}},{"Add":{"text":" $G \\leftarrow \\FEvaluate{$G$, $\\langle\\ell_0,\\texttt{true}\\rangle$}$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15622"}},{"Add":{"text":" }","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15628"}},{"Add":{"text":" \\For{$(\\_, \\_, \\_, \\_, \\ell) \\in \\TPin(T)$} {","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15678"}},{"Add":{"text":" $G \\leftarrow \\FEvaluate{$G$, $\\langle\\ell,\\texttt{true}\\rangle$}$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15755"}},{"Add":{"text":" }","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15761"}},{"Add":{"text":" $\\GTPn \\leftarrow \\text{general transitions of }E(G)$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15821"}},{"Add":{"text":" $\\Prog'\\leftarrow (\\PV,V(G),\\langle\\ell,\\texttt{true},\\GTPn)$\\;","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15889"}},{"Add":{"text":" \\Return $\\Prog'$\\; ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15913"}}]}},{"Edit":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":285,"atom":{"NewVertex":{"up_context":[{"change":3,"pos":8378}],"down_context":[{"change":2,"pos":29697}],"flag":"BLOCK","start":15915,"end":16670,"inode":{"change":3,"pos":7792}}},"contents":[{"Add":{"text":"Consider the case where a transition a of general transition $g \\in \\GTP$ is","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:15991"}},{"Add":{"text":"removed because it is part of the subset $T$, making $g$ an illegal general","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16067"}},{"Add":{"text":"transition. Still the general transition is lifted into the initial partial","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16143"}},{"Add":{"text":"evaluation graph. However, if there exists an entry of $T$ to the location","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16218"}},{"Add":{"text":"$\\ell_g$, then the version $\\langle \\ell_g, \\texttt{true}$ and all transitions","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16297"}},{"Add":{"text":"in $g$ are unfolded by the first evaluation step for this entry transition","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16372"}},{"Add":{"text":"making $g$ whole again in the partial evaluation graph. If the $\\langle \\ell_g,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16452"}},{"Add":{"text":"\\texttt{true}$ is reachable trough the partial evaluation the version is","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16525"}},{"Add":{"text":"evaluated then at its corresponding step. If it is not reachable the invalid","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16602"}},{"Add":{"text":"general transition is removed at the end as part of the clean-up.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16668"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16669"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":299,"ins":{"up_context":[{"change":3,"pos":8653}],"down_context":[{"change":3,"pos":8986}],"flag":"BLOCK","start":16671,"end":16980,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":8653},"to":{"change":3,"start":8653,"end":8986},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"\\gls{ocaml}. For solving satisfiability of constraints Z3\\cite{z3} was used as","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16749"}},{"Add":{"text":"it is already used in other parts of \\gls{koat}. For projections of constraints","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16829"}},{"Add":{"text":"and handling polyhedra we recurred \\gls{apron}\\cite{apron} and the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16896"}},{"Add":{"text":"\\gls{ppl}\\cite{bagnara2008socp}, which can be used as an abstract domain in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16972"}},{"Add":{"text":"apron.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:16979"}}],"del_contents":[{"Del":{"text":"\\gls{ocaml}. For solving satisfiability of constraints and formulas in general","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:8731"}},{"Del":{"text":"Z3\\cite{z3} was used as it is already used in other parts of \\gls{koat}. For","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:8808"}},{"Del":{"text":"projections of constraints and handling polyhedra we recurred","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:8870"}},{"Del":{"text":"\\gls{apron}\\cite{apron} and the \\gls{ppl}\\cite{bagnara2008socp}, which can be","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:8948"}},{"Del":{"text":"used as an abstract domain in apron.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:8985"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":305,"ins":{"up_context":[{"change":3,"pos":8987}],"down_context":[{"change":3,"pos":9066}],"flag":"BLOCK","start":16981,"end":17055,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":8987},"to":{"change":3,"start":8987,"end":9066},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"Both the Algorithm \\ref{alg:pe} which applies control-flow-refinement via","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17054"}}],"del_contents":[{"Del":{"text":"Both the Algorithm \\ref{alg:cfr_pip} which applies control-flow-refinement via","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:9065"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":320,"ins":{"up_context":[{"change":3,"pos":10015}],"down_context":[{"change":3,"pos":10083}],"flag":"BLOCK","start":17056,"end":17101,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10015},"to":{"change":2,"start":29755,"end":29777},"introduced_by":2}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"\\section{Performance}\\label{sec:performance}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17100"}}],"del_contents":[{"Del":{"text":"\\section{Performance}","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:29776"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":323,"ins":{"up_context":[{"change":3,"pos":6873}],"down_context":[{"change":3,"pos":10154}],"flag":"BLOCK","start":17102,"end":17521,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":6873},"to":{"change":3,"start":6873,"end":7310},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"implementation. The performance is analysed for classical programs using","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17174"}},{"Add":{"text":"benchmarks on test sets from the \\gls{tpdb}\\cite{tpdb}. The results are compared","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17255"}},{"Add":{"text":"to benchmarks on the same data sets using the latest version of \\gls{koat}. In","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17334"}},{"Add":{"text":"addition to the performance comparison, the correctness is asserted by comparing","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17415"}},{"Add":{"text":"the upper bounds found by the new implementation to lower bounds proven by the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17494"}},{"Add":{"text":"analysis tool \\gls{loat}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17520"}}],"del_contents":[{"Del":{"text":"implementation. The performance is analysed for classical and probabilistic","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:6948"}},{"Del":{"text":"programs using benchmarks on test sets from the \\gls{tpdb}\\cite{tpdb}. The","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:7023"}},{"Del":{"text":"results are compared to benchmarks on the same data sets using the latest","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:7097"}},{"Del":{"text":"version of \\gls{koat}. In addition to the performance comparison, the","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:7167"}},{"Del":{"text":"correctness is asserted by comparing the upper bounds found by the new","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:7238"}},{"Del":{"text":"implementation to lower bounds proven by the analysis tool \\gls{loat}.","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:7309"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":330,"ins":{"up_context":[{"change":3,"pos":10155}],"down_context":[{"change":3,"pos":7910}],"flag":"BLOCK","start":17522,"end":18693,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10155},"to":{"change":3,"start":7311,"end":7910},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"All of the code was compiled using OCaml 4.14.0. Among the more important","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17595"}},{"Add":{"text":"dependencies is the SMT library Z3 in version 4.11.0.0, Apron in version 0.9.13,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17676"}},{"Add":{"text":"and the \\gls{ppl} in version 1.2. \\gls{koat} and all dependencies were built","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17753"}},{"Add":{"text":"inside a docker container. All the benchmarks were done on an AMD Ryzen 7 5800X","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17833"}},{"Add":{"text":"8-Core Processor with 32 GB RAM. Eight benchmarks are executed in parallel, each","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17914"}},{"Add":{"text":"in their own Ubuntu 20.04 docker container. A single analysis timeouts after 240","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:17995"}},{"Add":{"text":"seconds. The measured computation time for each test includes the upstart and","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:18073"}},{"Add":{"text":"take down time of the docker container, but the timeout is measured internally","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:18152"}},{"Add":{"text":"in \\gls{koat}. For each benchmark configuration, we obtain a number of programs","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:18232"}},{"Add":{"text":"for which the analysis found a constant, linear, quadratic, higher-order","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:18305"}},{"Add":{"text":"polynomial and exponential runtime-complexity bounds, as well as the average","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:18382"}},{"Add":{"text":"time $AVG$ of analysis across all programs. Since the timeout is not very","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:18456"}},{"Add":{"text":"representative, we also include the average analysis time $AVG^+$ only those","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:18533"}},{"Add":{"text":"programs for which \\gls{koat} found finite complexity bounds. Besides we do not","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:18613"}},{"Add":{"text":"include the number for linear bounds in the number for quadratic bounds, etc. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:18692"}}],"del_contents":[{"Del":{"text":"All code was compiled using OCaml 4.14.0. Among the more important dependencies","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:7390"}},{"Del":{"text":"is the SMT library \\textit{Z3} in version 4.11.0.0, \\textit{Apron} in version","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:7468"}},{"Del":{"text":"0.9.13, and the \\gls{ppl} in version 1.2. \\gls{koat} and all dependencies are","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:7546"}},{"Del":{"text":"build inside a docker container. All benchmarks are done on a AMD Ryzen 7 5800X","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:7626"}},{"Del":{"text":"8-Core Processor with 32 GB RAM. 8 benchmarks are executed in parallel each in","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:7705"}},{"Del":{"text":"their own Ubuntu 20.04 docker container. The measured computation time for each","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:7785"}},{"Del":{"text":"test includes the upstart and take down time of the docker container, but the","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:7863"}},{"Del":{"text":"timeout is measured internally in \\gls{koat}.","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:7909"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":348,"ins":{"up_context":[{"change":3,"pos":8062}],"down_context":[{"change":3,"pos":8375}],"flag":"BLOCK","start":18694,"end":18962,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":8062},"to":{"change":3,"start":3594,"end":3674},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":3674},"to":{"change":3,"start":8143,"end":8375},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":8143},"to":{"change":3,"start":8143,"end":8375},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"measure \\gls{koat}'s performance.\\cite{giesl2022lncs} At the time of writing,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:18771"}},{"Add":{"text":"the latest version of the master branch is used.\\footnote{commit","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:18836"}},{"Add":{"text":"\\texttt{a2eb156ac31db0f4daaa89822786a413632e637f}} The old version of \\gls{koat}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:18917"}},{"Add":{"text":"is evaluated with four different settings: ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:18961"}}],"del_contents":[{"Del":{"text":"measure \\gls{koat}s performance.\\cite{giesl2022lncs} The at the time of writing","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:3673"}},{"Del":{"text":"latest version on the master branch is used. ","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:8188"}},{"Del":{"text":"\\gls{koat}\\footnote{commit \\texttt{a2eb156ac31db0f4daaa89822786a413632e637f}} is","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:8269"}},{"Del":{"text":"evaluated with four different settings: ","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:8310"}},{"Del":{"text":"The new implementation is tested with four different settings: ","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:8374"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":370,"ins":{"up_context":[{"change":3,"pos":9414}],"down_context":[{"change":3,"pos":9491}],"flag":"BLOCK","start":18963,"end":19040,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":9414},"to":{"change":3,"start":9414,"end":9491},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"The new implementation is evaluated on classical integer programs using two ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19039"}}],"del_contents":[{"Del":{"text":"The new implementation is evaluated on classical integer programs using four","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:9490"}}]}},{"Edit":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":373,"atom":{"EdgeMap":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":9523},"to":{"change":3,"start":9523,"end":9883},"introduced_by":3}],"inode":{"change":3,"pos":7792}}},"contents":[{"Del":{"text":" \\item \\enquote{new-\\acrshort{koat}} as a correctness check, this","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:9591"}},{"Del":{"text":" configuration uses the same settings as \\enquote{\\acrshort{koat}} and","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:9669"}},{"Del":{"text":" is expected to have similar result. ","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:9714"}},{"Del":{"text":" \\item \\enquote{new-\\acrshort{koat} + \\acrshort{mprf}5} as is this configuration that has the","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:9811"}},{"Del":{"text":" same settings as \\enquote{\\acrshort{koat} + \\acrshort{mprf}5}.","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:9882"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":374,"ins":{"up_context":[{"change":3,"pos":9957}],"down_context":[{"change":3,"pos":10057}],"flag":"BLOCK","start":19041,"end":19152,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":9957},"to":{"change":3,"start":9957,"end":10057},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" control-flow-refinement described in Algorithm \\ref{alg:pe} with","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19113"}},{"Add":{"text":" computation of the \\gls{fvs}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19151"}}],"del_contents":[{"Del":{"text":" sub-\\gls{scc} control-flow-refinement described in Algorithm","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:10025"}},{"Del":{"text":" \\ref{alg:sub_scc_cfr}.","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:10056"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":377,"ins":{"up_context":[{"change":3,"pos":10132}],"down_context":[{"change":3,"pos":10267}],"flag":"BLOCK","start":19153,"end":19299,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10132},"to":{"change":3,"start":10132,"end":10267},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" control-flow-refinement described in Algorithm \\ref{alg:pe} with","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19225"}},{"Add":{"text":" computation of the \\gls{fvs} and \\gls{mprf} with a maximum depth","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19298"}}],"del_contents":[{"Del":{"text":" sub-\\gls{scc} control-flow-refinement described in Algorithm","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:10200"}},{"Del":{"text":" \\ref{alg:sub_scc_cfr} and \\gls{mprf} with a maximum depth","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:10266"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":383,"ins":{"up_context":[{"change":3,"pos":10389}],"down_context":[{"change":3,"pos":10547}],"flag":"BLOCK","start":19300,"end":19451,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10389},"to":{"change":3,"start":10389,"end":10547},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"the \\gls{tpdb}. The first test set it called \\enquote{Complexity\\_ITS} and it","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19377"}},{"Add":{"text":"comprises 801 integer programs from various studies. The results of this","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19450"}}],"del_contents":[{"Del":{"text":"the \\gls{tpdb}. The first test set it called \\enquote{Complexity\\_ITS} that is","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:10467"}},{"Del":{"text":"comprised of 801 integer programs from various literature. The results of this","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:10546"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":394,"ins":{"up_context":[{"change":3,"pos":10885}],"down_context":[{"change":3,"pos":11158}],"flag":"BLOCK","start":19452,"end":19710,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10885},"to":{"change":3,"start":10885,"end":11158},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" \\acrshort{koat} & 128 & 218 & 95 & 10 & 6 & 457 & 3.17 & 8.84 \\\\ \\hline","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19531"}},{"Add":{"text":" \\acrshort{koat} + \\acrshort{mprf}5 & 128 & 235 & 98 & 14 & 6 & 481 &","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19608"}},{"Add":{"text":" 2.74 & 12.36 \\\\ \\hline","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19639"}},{"Add":{"text":" \\acrshort{koat} + iRF & 133 & 244 & 98 & 9 & 6 & 490 & 4.48 &","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19709"}}],"del_contents":[{"Del":{"text":" \\acrshort{koat} & 128 & 218 & 95 & 10 & 6 & 457 & 4.69 & 8.84 \\\\ \\hline","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:10964"}},{"Del":{"text":" \\acrshort{koat} + \\acrshort{mprf}5 & 128 & 235 & 98 & 14 & 6 & 481 & 5.58 &","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:11048"}},{"Del":{"text":" 12.36 \\\\ \\hline","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:11072"}},{"Del":{"text":" \\acrshort{koat} + old-\\acrshort{cfr} & 133 & 244 & 98 & 9 & 6 & 490 & 6.12 &","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:11157"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":400,"ins":{"up_context":[{"change":3,"pos":11253}],"down_context":[{"change":3,"pos":11296}],"flag":"BLOCK","start":19711,"end":19754,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":11253},"to":{"change":3,"start":11253,"end":11296},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" & 6 & 505 & 4.25 & 25.05 \\\\ \\hline","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19753"}}],"del_contents":[{"Del":{"text":" & 6 & 505 & 7.66 & 25.05 \\\\ \\hline","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:11295"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":402,"ins":{"up_context":[{"change":3,"pos":11311}],"down_context":[{"change":3,"pos":11599}],"flag":"BLOCK","start":19755,"end":20130,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":11311},"to":{"change":3,"start":11311,"end":11599},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" % new-\\acrshort{koat} & & & & & & & \\\\ \\hline","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19808"}},{"Add":{"text":" % new-\\acrshort{koat} + \\acrshort{mprf}5 & & & & & & & \\\\ \\hline","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19881"}},{"Add":{"text":" new-\\acrshort{koat} + nat.-\\acrshort{cfr} & 134 & 239 & 97 & 8 & 6 & 484","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19962"}},{"Add":{"text":" & 3.22 & 21.67 \\\\ \\hline","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:19995"}},{"Add":{"text":" new-\\acrshort{koat} + \\acrshort{mprf}5 + nat.-\\acrshort{cfr} & 134 &","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20072"}},{"Add":{"text":" 250 & 98 & 12 & 6 & 500 & 3.02 & 26.28 \\\\ \\hline","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20129"}}],"del_contents":[{"Del":{"text":" new-\\acrshort{koat} & & & & & & & \\\\ \\hline","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:11362"}},{"Del":{"text":" new-\\acrshort{koat} + \\acrshort{mprf}5 & & & & & & & \\\\ \\hline","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:11433"}},{"Del":{"text":" new-\\acrshort{koat} + nat.-\\acrshort{cfr} & & & & & & &\\\\ \\hline","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:11506"}},{"Del":{"text":" new-\\acrshort{koat} + \\acrshort{mprf}5 + nat.-\\acrshort{cfr} & & & & & & &\\\\ \\hline","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:11598"}}]}},{"Edit":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":412,"atom":{"NewVertex":{"up_context":[{"change":3,"pos":11756}],"down_context":[{"change":3,"pos":11784}],"flag":"BLOCK","start":20131,"end":20867,"inode":{"change":3,"pos":7792}}},"contents":[{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20131"}},{"Add":{"text":"We observe a noticeable improvement in computational performance when using the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20211"}},{"Add":{"text":"native implementation over iRankFinder. Without \\gls{mprf} the average analysis","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20291"}},{"Add":{"text":"of programs where a finite complexity bound was found shortens by 1.26 seconds","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20370"}},{"Add":{"text":"from 4.48 seconds to 3.22 seconds; and with \\gls{mprf} by 1.23 seconds from 4.25","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20451"}},{"Add":{"text":"seconds to 3.02 seconds. Unfortunately overall, fewer finite bounds can be found","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20532"}},{"Add":{"text":"for the given test set. Where \\gls{koat} found finite runtime bounds for 490","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20609"}},{"Add":{"text":"programs with iRankFinder, it manages to find finite complexity bounds for only","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20689"}},{"Add":{"text":"484 programs with the new implementation. Similarly, the old implementation with","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20770"}},{"Add":{"text":"\\gls{mprf} and iRankFinder found finite complexity bounds for only 500 program","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20849"}},{"Add":{"text":"in the test set.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20866"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":426,"ins":{"up_context":[{"change":3,"pos":11937}],"down_context":[{"change":3,"pos":12080}],"flag":"BLOCK","start":20868,"end":21016,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":11937},"to":{"change":3,"start":11937,"end":12080},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":"programs, C programs are transformed into integer programs using the tool","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:20941"}},{"Add":{"text":"\\texttt{llvm2kittel}\\cite{llvm2kittel}. The results of this benchmark are","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21015"}}],"del_contents":[{"Del":{"text":"programs, C programs are transformed integer programs using the tool","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:12005"}},{"Del":{"text":"\\textit{llvm2kittel}\\cite{llvm2kittel}. The results of this benchmark are","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:12079"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":437,"ins":{"up_context":[{"change":3,"pos":12410}],"down_context":[{"change":3,"pos":12562}],"flag":"BLOCK","start":21017,"end":21169,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":12410},"to":{"change":3,"start":12410,"end":12562},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" \\gls{koat} & 23 & 175 & 68 & 10 & 0 & 276 & 3.40 & 4.69 \\\\ \\hline","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21090"}},{"Add":{"text":" \\gls{koat} + \\acrshort{mprf}5 & 23 & 203 & 96 & 10 & 0 & 305 & 3.40 &","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21168"}}],"del_contents":[{"Del":{"text":" \\gls{koat} & 23 & 175 & 68 & 10 & 0 & 276 & 3.74 & 4.69 \\\\ \\hline","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:12483"}},{"Del":{"text":" \\gls{koat} + \\acrshort{mprf}5 & 23 & 203 & 96 & 10 & 0 & 305 & 4.43 &","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:12561"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":440,"ins":{"up_context":[{"change":3,"pos":12585}],"down_context":[{"change":3,"pos":12664}],"flag":"BLOCK","start":21170,"end":21249,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":12585},"to":{"change":3,"start":12585,"end":12664},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" \\gls{koat} + old-\\acrshort{cfr} & 25 & 215 & 68 & 9 & 0 & 317 & 5.10 &","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21248"}}],"del_contents":[{"Del":{"text":" \\gls{koat} + old-\\acrshort{cfr} & 25 & 215 & 68 & 9 & 0 & 317 & 5.76 &","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:12663"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":443,"ins":{"up_context":[{"change":3,"pos":12769}],"down_context":[{"change":3,"pos":12810}],"flag":"BLOCK","start":21250,"end":21291,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":12769},"to":{"change":3,"start":12769,"end":12810},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" 0 & 329 & 5.74 & 17.24 \\\\ \\hline","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21290"}}],"del_contents":[{"Del":{"text":" 0 & 329 & 7.10 & 17.24 \\\\ \\hline","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:12809"}}]}},{"Replacement":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":445,"ins":{"up_context":[{"change":3,"pos":12825}],"down_context":[{"change":3,"pos":13113}],"flag":"BLOCK","start":21292,"end":21537,"inode":{"change":3,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":12825},"to":{"change":3,"start":12825,"end":13113},"introduced_by":3}],"inode":{"change":3,"pos":7792}},"ins_contents":[{"Add":{"text":" new-\\acrshort{koat} + nat.-\\acrshort{cfr} & 25 & 202 & 72 & 6 & 0 & 305","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21371"}},{"Add":{"text":" & 4.14 & 10.06 \\\\ \\hline","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21404"}},{"Add":{"text":" new-\\acrshort{koat} + \\acrshort{mprf}5 + nat.-\\acrshort{cfr} & 24 & 222","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21484"}},{"Add":{"text":" & 71 & 6 & 0 & 323 & 3.81 & 13.66 \\\\ \\hline","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21536"}}],"del_contents":[{"Del":{"text":" new-\\acrshort{koat} & & & & & & & \\\\ \\hline","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:12876"}},{"Del":{"text":" new-\\acrshort{koat} + \\acrshort{mprf}5 & & & & & & & \\\\ \\hline","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:12947"}},{"Del":{"text":" new-\\acrshort{koat} + nat.-\\acrshort{cfr} & & & & & & &\\\\ \\hline","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:13020"}},{"Del":{"text":" new-\\acrshort{koat} + \\acrshort{mprf}5 + nat.-\\acrshort{cfr} & & & & & & &\\\\ \\hline","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:13112"}}]}},{"Edit":{"inode":{"change":3,"pos":7792},"path":"ch4_implementation.tex","line":453,"atom":{"NewVertex":{"up_context":[{"change":3,"pos":13283}],"down_context":[],"flag":"BLOCK","start":21538,"end":22306,"inode":{"change":3,"pos":7792}}},"contents":[{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21538"}},{"Add":{"text":"For the second test set, we observe a similar improvement in analysis time as","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21616"}},{"Add":{"text":"well. The average analysis time of programs where a finite runtime-bound is","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21692"}},{"Add":{"text":"found shortens by 0.96 seconds from 5.10 seconds to 4.14 seconds without","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21765"}},{"Add":{"text":"\\glspl{mprf}; and by 1.93 seconds from 5.74 seconds to 3.81 seconds with","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21838"}},{"Add":{"text":"\\glspl{mprf}. However, the new implementation can only find complexity bounds","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21916"}},{"Add":{"text":"for 305 instead of 317 programs without \\glspl{mprf} and 323 instead of 329 when","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:21997"}},{"Add":{"text":"using \\glspl{mprf}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22017"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22018"}},{"Add":{"text":"This worsened behaviour needs further investigation. We suspect there to be","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22094"}},{"Add":{"text":"either a bug in the implementation or that the properties are not optimally","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22170"}},{"Add":{"text":"chosen. Besides, there is certainly room for performance optimisation in the new","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22251"}},{"Add":{"text":"implementation.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22267"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22268"}},{"Add":{"text":"\\subsection{Probabilistic programs}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22304"}},{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22305"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":2,"ins":{"up_context":[{"change":3,"pos":7972}],"down_context":[{"change":3,"pos":71}],"flag":"BLOCK","start":22307,"end":22726,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":7972},"to":{"change":3,"start":14463,"end":14859},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":14859},"to":{"change":3,"start":26198,"end":26207},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":26198},"to":{"change":3,"start":26198,"end":26207},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"The goal of \\gls{koat}s analysis is to find tight upper bounds for \\glspl{pip}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22386"}},{"Add":{"text":"The analysis techniques used, such as \\acrfullpl{mprf} and \\cite{twn}, have","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22462"}},{"Add":{"text":"their limits and cannot always find tight bounds for every program. This","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22535"}},{"Add":{"text":"motivates control-flow refinement in order to find a program that is equivalent","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22615"}},{"Add":{"text":"to the original one, which opens the possibility for the analysis to find","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22689"}},{"Add":{"text":"tighter runtime complexity bounds. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22725"}}],"del_contents":[{"Del":{"text":"The goal of \\gls{koat}'s analysis is to find tight upper bounds for \\glspl{pip}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14543"}},{"Del":{"text":"The analysis techniques used, such as \\acrfullpl{mprf}, have their limits and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14621"}},{"Del":{"text":"cannot always find tight bounds for every program. This motivates control-flow","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14700"}},{"Del":{"text":"refinement in order to find a program that is equivalent to the original one,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14778"}},{"Del":{"text":"which opens the possibility for the analysis to find tighter runtime-complexity","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14858"}},{"Del":{"text":"bounds. ","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:26206"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":9,"ins":{"up_context":[{"change":3,"pos":72}],"down_context":[{"change":3,"pos":4049}],"flag":"BLOCK","start":22727,"end":22807,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":72},"to":{"change":3,"start":14860,"end":14938},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"Recall Definition \\ref{def:runtime_complexity} -- the newly constructed refined","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22806"}}],"del_contents":[{"Del":{"text":"Recall Definition \\ref{def:runtime_complexity}; the newly constructed refined","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14937"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":11,"ins":{"up_context":[{"change":3,"pos":4123}],"down_context":[{"change":3,"pos":4199}],"flag":"BLOCK","start":22808,"end":22888,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":4123},"to":{"change":3,"start":14939,"end":15020},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"original program. This implies that the runtime complexity bounds found for the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22887"}}],"del_contents":[{"Del":{"text":"original program, which implies that the runtime-complexity bounds found for the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15019"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":21,"ins":{"up_context":[{"change":3,"pos":15438}],"down_context":[{"change":3,"pos":15515}],"flag":"BLOCK","start":22889,"end":22967,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":15438},"to":{"change":3,"start":15438,"end":15515},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"\\glspl{pip}. The partial evaluation will be constructed in a similar way, but","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:22966"}}],"del_contents":[{"Del":{"text":"\\glspl{pip}. The partial evaluation will be constructed in a similar way but","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15514"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":24,"ins":{"up_context":[{"change":3,"pos":1498}],"down_context":[{"change":3,"pos":15659}],"flag":"BLOCK","start":22968,"end":23049,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":1498},"to":{"change":3,"start":15585,"end":15659},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"The construction is carried out in three steps. First, an unfolding operation is","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23048"}}],"del_contents":[{"Del":{"text":"The construction is done in three steps. First, an unfolding operation is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15658"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":27,"ins":{"up_context":[{"change":3,"pos":15818}],"down_context":[{"change":3,"pos":27336}],"flag":"BLOCK","start":23050,"end":23203,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":15818},"to":{"change":3,"start":15818,"end":15974},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"and evaluation will be adapted multiple times in order to find finite partial","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23127"}},{"Add":{"text":"evaluation graphs, and to evaluate only the relevant parts of the program.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23202"}}],"del_contents":[{"Del":{"text":"and evaluation will be adapted multiple times in order to 1. find finite","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15890"}},{"Del":{"text":"partial evaluation graphs, and 2. evaluate only the relevant parts of the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15964"}},{"Del":{"text":"program.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15973"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":32,"ins":{"up_context":[{"change":3,"pos":16131}],"down_context":[{"change":3,"pos":27551}],"flag":"BLOCK","start":23204,"end":23265,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":3,"pos":16131},"to":{"change":3,"start":27490,"end":27551},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":27490},"to":{"change":3,"start":27490,"end":27551},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"worst-case runtime complexity of non-probabilistic programs.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23264"}}],"del_contents":[{"Del":{"text":"worst-case runtime-complexity of non-probabilistic programs.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:27550"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":43,"ins":{"up_context":[{"change":3,"pos":27924}],"down_context":[{"change":3,"pos":4774}],"flag":"BLOCK","start":23266,"end":23370,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":27924},"to":{"change":3,"start":27924,"end":28003},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" probabilistic program $\\Prog_2$ from the introduction (\\fref{fig:prob_c} on","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23345"}},{"Add":{"text":" \\pref{fig:prob_c}),","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23369"}}],"del_contents":[{"Del":{"text":" probabilistic program $\\Prog_2$ from the introduction (\\fref{fig:prob_c}),","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:28002"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":47,"ins":{"up_context":[{"change":3,"pos":5395}],"down_context":[{"change":3,"pos":7972}],"flag":"BLOCK","start":23371,"end":23617,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":3,"pos":5395},"to":{"change":3,"start":28157,"end":28237},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":28157},"to":{"change":3,"start":28157,"end":28237},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":28237},"to":{"change":3,"start":4854,"end":5009},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" \\{x,y,u,w\\}$ and has the program variables $\\PV_{\\Prog_3} = \\{x,y\\}$. $u$,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23449"}},{"Add":{"text":" and $w$ are temporary variables whose values are reassigned by the scheduler","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23530"}},{"Add":{"text":" on every transition and whose values do not propagate throughout the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23603"}},{"Add":{"text":" program.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23616"}}],"del_contents":[{"Del":{"text":" \\{x,y,u,w\\}$ and has the program variables $\\PV_{\\Prog_3} = \\{x,y\\}$. $u,w$","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:28236"}},{"Del":{"text":" are temporary variables whose values are reassigned by the scheduler on","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:4929"}},{"Del":{"text":" every transition and whose values do not propagate throughout the program.","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:5008"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":58,"ins":{"up_context":[{"change":3,"pos":16476}],"down_context":[{"change":3,"pos":29286}],"flag":"BLOCK","start":23618,"end":24336,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":16476},"to":{"change":3,"start":16476,"end":16787},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":16787},"to":{"change":3,"start":29165,"end":29286},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":29165},"to":{"change":3,"start":29165,"end":29286},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" not change the expected runtime complexity of the program. On the general","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23695"}},{"Add":{"text":" transition $g_1 = \\{t_{1a}, t_{1b}\\}$ at location $\\ell_1$ the program","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23770"}},{"Add":{"text":" branches probabilistically, with a probability of $p=0.5$. On the transition","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23851"}},{"Add":{"text":" $t_3$, the value with which $y$ is decremented is sampled probabilistically","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:23931"}},{"Add":{"text":" from a Bernoulli distribution, where both possible outcomes have the same","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24009"}},{"Add":{"text":" probability $p=0.5$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24034"}},{"Add":{"text":" ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24039"}},{"Add":{"text":" When entering any of the two loops at $t_{1a}$ or $t_{1b}$ the value of $x$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24119"}},{"Add":{"text":" is unbounded due to the non-deterministic sampling of $x$ in $t_0$. Hence","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24197"}},{"Add":{"text":" \\gls{koat} cannot find \\gls{mprf} bounding the value of $x$ throughout the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24276"}},{"Add":{"text":" execution and compute runtime bounds for those loops. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24335"}}],"del_contents":[{"Del":{"text":" not change anything about the expected runtime complexity of the program. On","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16556"}},{"Del":{"text":" the general transition $g_1 = \\{t_{1a}, t_{1b}\\}$ at location $\\ell_1$ the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16635"}},{"Del":{"text":" program branches probabilistically, with a probability of $p=0.5$. On the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16713"}},{"Del":{"text":" transition $t_3$, the value with which $y$ is decremented is sampled","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16786"}},{"Del":{"text":" probabilistically from a Bernoulli distribution, where both possible","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:29237"}},{"Del":{"text":" outcomes have the same probability $p=0.5$.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:29285"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":77,"ins":{"up_context":[{"change":3,"pos":29607}],"down_context":[{"change":3,"pos":17045}],"flag":"BLOCK","start":24337,"end":24414,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":29607},"to":{"change":3,"start":16968,"end":17045},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"impossible: first, the number of starting states is infinite, and second the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24413"}}],"del_contents":[{"Del":{"text":"impossible: First, the number of starting states is infinite, and second the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17044"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":81,"ins":{"up_context":[{"change":3,"pos":17232}],"down_context":[{"change":3,"pos":17544}],"flag":"BLOCK","start":24415,"end":24808,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":17232},"to":{"change":3,"start":17232,"end":17309},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":17309},"to":{"change":3,"start":5146,"end":5220},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":5146},"to":{"change":3,"start":5146,"end":5220},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":5220},"to":{"change":3,"start":17310,"end":17544},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"complex, since the transitions taken do not only depend on the current state but","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24495"}},{"Add":{"text":"also on the answer of the scheduler and some random event. Instead of simulating","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24576"}},{"Add":{"text":"every single possible run, it would be more efficient to treat \\emph{similar}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24654"}},{"Add":{"text":"states as one. Instead of simulating a single assignment, the set of all","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24727"}},{"Add":{"text":"possible assignments after a step is over-approximated by a set of constraints.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24807"}}],"del_contents":[{"Del":{"text":"complex, since the transitions taken do not only depend on the current state","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17308"}},{"Del":{"text":"but also on the answer of the scheduler and some random event. Instead of","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:5219"}},{"Del":{"text":"simulating every single possible run, it would be more efficient to treat","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17383"}},{"Del":{"text":"\\emph{similar} states as one. Instead of simulating a single assignment, the set","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17464"}},{"Del":{"text":"of all possible assignments after a step is described by a set of constraints.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17543"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":87,"ins":{"up_context":[{"change":3,"pos":17616}],"down_context":[{"change":3,"pos":1532}],"flag":"BLOCK","start":24809,"end":25004,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":17616},"to":{"change":3,"start":17616,"end":17811},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"formalised for \\glspl{chc} by \\citeauthor{gallagher2019eptcs}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24870"}},{"Add":{"text":"\\cite{gallagher2019eptcs}. In this thesis, the definitions are adapted to better","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:24951"}},{"Add":{"text":"fit the algorithm and the formalism of \\Glspl{pip}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25003"}}],"del_contents":[{"Del":{"text":"formalised for \\glspl{chc} by","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17645"}},{"Del":{"text":"\\citeauthor{gallagher2019eptcs} \\cite{gallagher2019eptcs}. In this thesis, the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17724"}},{"Del":{"text":"definitions are adapted to better fit the algorithm and the formalism of","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17797"}},{"Del":{"text":"\\Glspl{pip}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17810"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":99,"ins":{"up_context":[{"change":3,"pos":3431}],"down_context":[{"change":3,"pos":5544}],"flag":"BLOCK","start":25005,"end":25236,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":3376},"to":{"change":3,"start":31921,"end":32001},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":32001},"to":{"change":3,"start":5396,"end":5544},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"The versions are the vertices/locations of the partial evaluation graph. The","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25081"}},{"Add":{"text":"edges of the partial evaluation graph are transitions $\\VTP$ (\\enquote{$\\VT$}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25159"}},{"Add":{"text":"stands for \\enquote{version transitions}) analogously defined to Definition","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25235"}}],"del_contents":[{"Del":{"text":"The versions are the edges/locations of the partial evaluation graph. The edges","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:32000"}},{"Del":{"text":"of the partial evaluation graph are transitions $\\VTP$ (\\enquote{$\\VT$} stands","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:5474"}},{"Del":{"text":"for \\enquote{version transitions}) analogously defined to Definition","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:5543"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":154,"ins":{"up_context":[{"change":3,"pos":6402}],"down_context":[{"change":3,"pos":4374}],"flag":"BLOCK","start":25237,"end":25481,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":3,"pos":6402},"to":{"change":3,"start":32980,"end":33203},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":4169},"to":{"change":3,"start":32980,"end":33203},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" An unfolding operation $\\unfold : 2^{\\C_\\PV} \\times (\\PV \\rightarrow","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25309"}},{"Add":{"text":" \\Z[\\V\\union\\D)]) \\rightarrow 2^{\\C_\\PV}$ of a set of constraints $\\varphi","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25387"}},{"Add":{"text":" \\in 2^{\\C_\\PV}$ with an update $\\eta: \\PV\\rightarrow\\Z[\\V \\union \\D]$ is","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25464"}},{"Add":{"text":" defined as ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25480"}}],"del_contents":[{"Del":{"text":" An unfolding operation $\\unfold : \\Phi_\\PV \\times (\\PV \\rightarrow","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:33050"}},{"Del":{"text":" \\Z[\\V\\union\\D)]) \\rightarrow \\Phi_\\PV$ of a formula $\\varphi \\in \\Phi_\\PV$","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:33129"}},{"Del":{"text":" with an update $\\eta: \\PV\\rightarrow\\Z[\\V \\union \\D]$ is defined as ","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:33202"}}]}},{"Edit":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":167,"atom":{"EdgeMap":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":6786},"to":{"change":3,"start":0,"end":11},"introduced_by":3}],"inode":{"change":3,"pos":7949}}},"contents":[{"Del":{"text":"% \\newpage","pos":"45HZPHVHYJ2BXATQ4MKCMPMMPZFBNJECY7JV3WX63BSA4CSQN6HAC:10"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":168,"ins":{"up_context":[{"change":3,"pos":784}],"down_context":[{"change":3,"pos":5819}],"flag":"BLOCK","start":25482,"end":25555,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":784},"to":{"change":3,"start":5749,"end":5819},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" For any $\\varphi\\in 2^{\\C_\\PV}$ and probabilistic update $\\eta : \\PV","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25554"}}],"del_contents":[{"Del":{"text":" For any $\\varphi\\in\\Phi_\\PV$ and probabilistic update $\\eta : \\PV","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:5818"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":177,"ins":{"up_context":[{"change":3,"pos":464}],"down_context":[{"change":3,"pos":542}],"flag":"BLOCK","start":25556,"end":25636,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":464},"to":{"change":3,"start":33669,"end":33751},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" Let $\\varphi\\in\\C_\\PV$, $s, s' \\in \\Sigma$, $s \\models \\varphi$ and $s'","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25635"}}],"del_contents":[{"Del":{"text":" Let $\\varphi\\in\\Phi_\\PV$, $s, s' \\in \\Sigma$, $s \\models \\varphi$ and $s'","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:33750"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":211,"ins":{"up_context":[{"change":3,"pos":34727}],"down_context":[{"change":3,"pos":34864}],"flag":"BLOCK","start":25637,"end":25820,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":34727},"to":{"change":3,"start":34727,"end":34864},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" &\\begin{aligned}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25661"}},{"Add":{"text":" \\;=(&y > 0 \\Land x' = x \\Land y' = y - d \\Land 0 \\leq d \\leq 1 \\\\&\\Land x'' =","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25751"}},{"Add":{"text":" x' \\Land y'' = y')|_{\\PV''}[x''/x]_{x\\in\\PV}\\end{aligned}\\\\","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25819"}}],"del_contents":[{"Del":{"text":" &= (y > 0 \\Land x' = x \\Land y' = y - d \\Land 0 \\leq d \\leq 1 \\Land x'' =","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:34808"}},{"Del":{"text":" x' \\Land y'' = y')|_{\\PV''}[x''/x]_{x\\in\\PV}\\\\","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:34863"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":227,"ins":{"up_context":[{"change":3,"pos":7206}],"down_context":[{"change":3,"pos":7280}],"flag":"BLOCK","start":25821,"end":25899,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":7206},"to":{"change":3,"start":7206,"end":7280},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" that constructs a new transition connecting both versions. The update and","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25898"}}],"del_contents":[{"Del":{"text":" that constructs a new transition between the versions. The update and","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:7279"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":266,"ins":{"up_context":[{"change":3,"pos":9574}],"down_context":[{"change":3,"pos":9260}],"flag":"BLOCK","start":25900,"end":26112,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":9574},"to":{"change":3,"start":8852,"end":8932},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":8932},"to":{"change":3,"start":19789,"end":19870},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":19870},"to":{"change":3,"start":9012,"end":9055},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":9012},"to":{"change":3,"start":9012,"end":9055},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"For any $V(G_n) \\subseteq V(G_{n+1})$ and $E(G_n) \\subseteq E(G_{n+1})$ where $n","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:25980"}},{"Add":{"text":"\\in \\N$. Moreover, let $G^* = \\evaluate_\\Prog^*$ be the limit the evaluation","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26057"}},{"Add":{"text":"approaches with an increasing number of repetitions. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26111"}}],"del_contents":[{"Del":{"text":"For any $V(G_n) \\subseteq V(G_{n+1})$ and $E(G_n) \\subseteq E(G_{n+1})$ for any","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:8931"}},{"Del":{"text":"$n \\in \\N$. Let $G^* = \\evaluate_\\Prog^*$ be the limit the evaluation approaches","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19869"}},{"Del":{"text":"with an increasing number of repetitions. ","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:9054"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":306,"ins":{"up_context":[{"change":3,"pos":6419}],"down_context":[{"change":3,"pos":6498}],"flag":"BLOCK","start":26113,"end":26191,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":6419},"to":{"change":3,"start":20450,"end":20529},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" evaluation step unfolds $\\langle \\ell_1, \\textit{true}\\rangle$ over $t_3$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26190"}}],"del_contents":[{"Del":{"text":" evaluation step, unfolds $\\langle \\ell_1, \\textit{true}\\rangle$ over $t_3$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20528"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":315,"ins":{"up_context":[{"change":3,"pos":2690}],"down_context":[{"change":3,"pos":10909}],"flag":"BLOCK","start":26192,"end":26349,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":2690},"to":{"change":3,"start":10751,"end":10909},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"As one can see in Example \\ref{ex:infinite_evaluation}, the evaluation graph of","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26271"}},{"Add":{"text":"a program can be infinite, especially when the program does not terminate on","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26348"}}],"del_contents":[{"Del":{"text":"As one can see in example \\ref{ex:infinite_evaluation}, the evaluation graph of","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:10830"}},{"Del":{"text":"a program can be infinite, especially, when the program does not terminate on","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:10908"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":322,"ins":{"up_context":[{"change":3,"pos":11120}],"down_context":[{"change":3,"pos":2993}],"flag":"BLOCK","start":26350,"end":26606,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":11120},"to":{"change":3,"start":11120,"end":11362},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" \\textit{abstraction space}. A function $\\alpha: 2^{\\C_\\PV} \\rightarrow \\A$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26428"}},{"Add":{"text":" is called an \\textit{abstraction function} if for every $\\varphi\\subset","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26504"}},{"Add":{"text":" 2^{\\C_\\PV}$, $\\llbracket \\varphi \\rrbracket \\subseteq \\llbracket","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26573"}},{"Add":{"text":" \\alpha(\\varphi)\\rrbracket$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26605"}}],"del_contents":[{"Del":{"text":" \\textit{abstraction space}. A function $\\alpha: \\Phi \\rightarrow \\A$ is","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:11195"}},{"Del":{"text":" called an \\textit{abstraction function} if for every $\\varphi\\subset\\Phi$,","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:11274"}},{"Del":{"text":" $\\llbracket \\varphi \\rrbracket \\subseteq \\llbracket","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:11330"}},{"Del":{"text":" \\alpha(\\varphi)\\rrbracket$","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:11361"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":329,"ins":{"up_context":[{"change":3,"pos":10074}],"down_context":[{"change":3,"pos":3700}],"flag":"BLOCK","start":26607,"end":26843,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10074},"to":{"change":3,"start":10074,"end":10226},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10226},"to":{"change":3,"start":20691,"end":20762},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":20762},"to":{"change":3,"start":10301,"end":10317},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10301},"to":{"change":3,"start":10301,"end":10317},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"above are satisfied. One common abstraction method is the so-called","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26674"}},{"Add":{"text":"property-based abstraction \\cite{Domenech19} which selects a subset of","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26745"}},{"Add":{"text":"properties from a finite set of properties that are all entailed by the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26817"}},{"Add":{"text":"original constraint set.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26842"}}],"del_contents":[{"Del":{"text":"above are satisfied. One common abstraction method is is the so called","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:10144"}},{"Del":{"text":"property-based abstraction\\cite{Domenech19} which selects a subset of properties","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:10225"}},{"Del":{"text":"from a finite set of properties, that are all entailed by the original","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20761"}},{"Del":{"text":"constraint set.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:10316"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":337,"ins":{"up_context":[{"change":3,"pos":10511}],"down_context":[{"change":3,"pos":20763}],"flag":"BLOCK","start":26844,"end":26925,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10511},"to":{"change":3,"start":10511,"end":10589},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" abstraction space of the property-based abstraction is defined as $\\A_\\Psi =","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26924"}}],"del_contents":[{"Del":{"text":" abstraction space of the property-based abstraction is defined $\\A_\\Psi =","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:10588"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":340,"ins":{"up_context":[{"change":3,"pos":20885}],"down_context":[{"change":3,"pos":10726}],"flag":"BLOCK","start":26926,"end":26991,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":3,"pos":20885},"to":{"change":3,"start":10667,"end":10726},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10667},"to":{"change":3,"start":10667,"end":10726},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" $\\alpha_\\Psi : 2^{\\C_\\PV} \\rightarrow \\A_\\Psi$ is defined as","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:26990"}}],"del_contents":[{"Del":{"text":" $\\alpha_\\Psi : \\Phi \\rightarrow \\A_\\Psi$ is defined as","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:10725"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":348,"ins":{"up_context":[{"change":3,"pos":10956}],"down_context":[{"change":3,"pos":11128}],"flag":"BLOCK","start":26992,"end":27177,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10956},"to":{"change":3,"start":10956,"end":11128},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" A property based abstraction function is a valid abstraction function. For","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:27070"}},{"Add":{"text":" all $\\varphi \\in 2^{\\C_\\PV}$ and finite set of properties $\\Psi \\subset \\C$,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:27151"}},{"Add":{"text":" the following holds.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:27176"}}],"del_contents":[{"Del":{"text":" The property based abstraction is a valid abstraction function. For all","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:11031"}},{"Del":{"text":" $\\varphi \\in \\Phi$ and finite set of properties $\\Psi \\subset \\C$, the","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:11106"}},{"Del":{"text":" following holds.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:11127"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":354,"ins":{"up_context":[{"change":3,"pos":11249}],"down_context":[{"change":3,"pos":11297}],"flag":"BLOCK","start":27178,"end":27227,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":11249},"to":{"change":3,"start":11249,"end":11297},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" \\proof{see Appendix on \\pref{apdx:lem:pba}.}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:27226"}}],"del_contents":[{"Del":{"text":" \\proof{see Appendix on \\pref{apdx:lem:pba}}","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:11296"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":357,"ins":{"up_context":[{"change":3,"pos":3937}],"down_context":[{"change":3,"pos":11714}],"flag":"BLOCK","start":27228,"end":27302,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":3937},"to":{"change":3,"start":11644,"end":11714},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"The unfolding and evaluation are modified slightly to accommodate for the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:27301"}}],"del_contents":[{"Del":{"text":"The unfolding and evaluation are modified slightly to accommodate the","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:11713"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":379,"ins":{"up_context":[{"change":3,"pos":4778}],"down_context":[{"change":3,"pos":5666}],"flag":"BLOCK","start":27303,"end":27599,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":4778},"to":{"change":3,"start":20886,"end":21092},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":12666},"to":{"change":3,"start":5170,"end":5171},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":21092},"to":{"change":3,"start":5170,"end":5171},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":23520},"to":{"change":3,"start":5170,"end":5171},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":5170},"to":{"change":3,"start":5170,"end":5171},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":5171},"to":{"change":3,"start":21093,"end":21912},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"For better readability, we denote abstracted constraints with the set notation","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:27381"}},{"Add":{"text":"using curly braces. We write $\\langle \\ell, \\{\\psi_1, \\dots, \\psi_m\\}\\rangle$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:27459"}},{"Add":{"text":"instead of $\\langle \\ell, \\psi_1 \\land \\dots \\land \\psi_m\\rangle$, in order to","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:27538"}},{"Add":{"text":"highlight the condensing characteristic of the abstraction.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:27598"}}],"del_contents":[{"Del":{"text":"% \\begin{lemma}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20901"}},{"Del":{"text":"% Let $\\alpha: \\Phi \\rightarrow \\A$ be an abstraction function over a finite","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20982"}},{"Del":{"text":"% abstraction space $\\A\\subset\\C$. The partial evaluation graph $G_\\alpha^*$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21063"}},{"Del":{"text":"% exists and is finite.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21091"}},{"Del":{"text":"","pos":"M3WK5HF3XU5MN2JKJMRJVDMC74F4MZAQ23FTGJRW4CGU5ZL72MHAC:5170"}},{"Del":{"text":"% \\begin{proof}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21112"}},{"Del":{"text":"% The abstraction $\\alpha$ can map to only finitely many values in the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21191"}},{"Del":{"text":"% finite abstraction space $\\A$. Every evaluation step adds only versions","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21273"}},{"Del":{"text":"% from a reduced and finite set of version $\\Loc_\\Prog \\times \\A \\subset","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21354"}},{"Del":{"text":"% \\vers_\\Prog$. Hence the set of vertices of partial evaluation graph","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21432"}},{"Del":{"text":"% equally finite. A transition is added after a version $\\langle \\ell,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21511"}},{"Del":{"text":"% \\_\\rangle$ for every outgoing transitions $t \\in \\TPout(\\ell)$ only. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21591"}},{"Del":{"text":"% Since $\\TPout(\\ell) \\subseteq \\T_\\Prog$ is finite, only finitely many","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21671"}},{"Del":{"text":"% transitions can exist between any two versions.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21729"}},{"Del":{"text":"% The graph grows with every evaluation step, hence the limit exists and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21810"}},{"Del":{"text":"% it must be reached in a finite number of evaluation steps.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21879"}},{"Del":{"text":"% \\end{proof}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21897"}},{"Del":{"text":"% \\end{lemma}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21911"}}]}},{"Edit":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":384,"atom":{"EdgeMap":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":5667},"to":{"change":3,"start":13459,"end":13595},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":13595},"to":{"change":3,"start":6112,"end":6113},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":6112},"to":{"change":3,"start":6112,"end":6113},"introduced_by":3}],"inode":{"change":3,"pos":7949}}},"contents":[{"Del":{"text":"For better readability, we write denote abstracted constraints with the set","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:13534"}},{"Del":{"text":"notation. $\\langle \\ell, \\{\\psi_1, \\dots, \\psi_m\\}\\rangle$ ","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:13594"}},{"Del":{"text":"","pos":"M3WK5HF3XU5MN2JKJMRJVDMC74F4MZAQ23FTGJRW4CGU5ZL72MHAC:6112"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":385,"ins":{"up_context":[{"change":3,"pos":11658}],"down_context":[{"change":3,"pos":13755}],"flag":"BLOCK","start":27600,"end":27774,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":3,"pos":11658},"to":{"change":3,"start":13612,"end":13684},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":13612},"to":{"change":3,"start":13612,"end":13684},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":13684},"to":{"change":3,"start":21913,"end":21989},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" Consider program $\\Prog_3$ from \\fref{fig:ex_prob} on \\pref{fig:ex_prob}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:27676"}},{"Add":{"text":" again. We define $\\psi_1,\\psi_2,\\psi_3 \\in \\C$ and an abstraction space","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:27752"}},{"Add":{"text":" $\\A_\\Psi$ where ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:27773"}}],"del_contents":[{"Del":{"text":" Consider program $\\Prog_3$ from \\fref{fig:ex_prob} again. We define","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:13683"}},{"Del":{"text":" $\\psi_1,\\psi_2,\\psi_3 \\in \\C$ and an abstraction space $\\A_\\Psi$ where ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21988"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":402,"ins":{"up_context":[{"change":3,"pos":14268}],"down_context":[{"change":3,"pos":22631}],"flag":"BLOCK","start":27775,"end":27851,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":14268},"to":{"change":3,"start":22555,"end":22631},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" The property-based abstraction $\\alpha_\\Psi$ uses the abstraction space","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:27850"}}],"del_contents":[{"Del":{"text":" The property based abstraction $\\alpha_\\Psi$ uses the abstraction 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For any version $\\langle","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:28718"}},{"Add":{"text":" \\ell,\\varphi\\rangle \\in \\vers_\\Prog$ the abstraction is required to be of","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:28796"}},{"Add":{"text":" the form $S (\\langle \\ell, \\varphi' \\rangle)$ with","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:28851"}},{"Add":{"text":" $\\llbracket\\varphi\\rrbracket \\subseteq \\llbracket \\varphi'\\rrbracket$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:28926"}}],"del_contents":[{"Del":{"text":"\\begin{definition}[Localized abstraction]","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:13757"}},{"Del":{"text":" A \\emph{localized} abstraction is a function $S : \\vers_\\Prog","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:13823"}},{"Del":{"text":" \\rightarrow \\vers_\\Prog$ is a function that \\emph{can} abstract the","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:13895"}},{"Del":{"text":" constraint of a version, but is not required to do so. 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Location $\\ell_0$ and $\\ell_2$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:29429"}}],"del_contents":[{"Del":{"text":" Lets return to our example program $\\Prog_3$. Location $\\ell_0$ and $\\ell_2$","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:16248"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":548,"ins":{"up_context":[{"change":3,"pos":16503}],"down_context":[{"change":3,"pos":16557}],"flag":"BLOCK","start":29431,"end":29485,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":16503},"to":{"change":3,"start":16503,"end":16557},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" The localised property-based abstraction function","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:29484"}}],"del_contents":[{"Del":{"text":" The localized property-based abstraction function","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:16556"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":554,"ins":{"up_context":[{"change":3,"pos":23644}],"down_context":[{"change":3,"pos":23838}],"flag":"BLOCK","start":29486,"end":29689,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":23644},"to":{"change":3,"start":23644,"end":23838},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" \\fref{fig:ex_localized_pba}. It looks very similar to before, but we saved","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:29564"}},{"Add":{"text":" some entailment checks on location $\\ell_2$, in return for which the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:29637"}},{"Add":{"text":" versions at $\\ell_2$ are no longer collapsed. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:29688"}}],"del_contents":[{"Del":{"text":" \\fref{fig:ex_localized_pba} It looks very similar to before, but we saved","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23721"}},{"Del":{"text":" some entailment checks on location $\\ell_2$, in return the versions at","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23796"}},{"Del":{"text":" $\\ell_2$ are not collapsed anymore. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23837"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":561,"ins":{"up_context":[{"change":3,"pos":23930}],"down_context":[{"change":3,"pos":24011}],"flag":"BLOCK","start":29690,"end":29771,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":23930},"to":{"change":3,"start":23930,"end":24011},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" \\caption{Partial evaluation of $\\Prog_3$ with a localised property based","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:29770"}}],"del_contents":[{"Del":{"text":" \\caption{Partial evaluation of $\\Prog_3$ with a localized property based","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24010"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":566,"ins":{"up_context":[{"change":3,"pos":17078}],"down_context":[{"change":3,"pos":17806}],"flag":"BLOCK","start":29772,"end":30006,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":17078},"to":{"change":3,"start":17078,"end":17301},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" practice the abstraction is expensive and abstracting fewer locations can","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:29849"}},{"Add":{"text":" speed up the partial evaluation. The real strength of localised abstraction","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:29929"}},{"Add":{"text":" will be seen when only evaluating a subset of transitions of $\\Prog_3$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:30005"}}],"del_contents":[{"Del":{"text":" practice the abstraction is expensive and abstracting fewer locations, can","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17156"}},{"Del":{"text":" speed up the partial evaluation. The real strength of localized abstraction","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17236"}},{"Del":{"text":" will be seen when only evaluating a component of $\\Prog_3$.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17300"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":571,"ins":{"up_context":[{"change":3,"pos":7589}],"down_context":[{"change":3,"pos":24182}],"flag":"BLOCK","start":30007,"end":30080,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":7589},"to":{"change":3,"start":24109,"end":24182},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"Clearly, a global abstraction function is a special case of a localised ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:30079"}}],"del_contents":[{"Del":{"text":"Clearly, a global abstraction function is a special case of a localized ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24181"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":573,"ins":{"up_context":[{"change":3,"pos":24253}],"down_context":[{"change":3,"pos":24333}],"flag":"BLOCK","start":30081,"end":30161,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":24253},"to":{"change":3,"start":24253,"end":24333},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"abstraction function. How to select properties for the localised property-based","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:30160"}}],"del_contents":[{"Del":{"text":"abstraction function. 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In fact, we will see","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:30318"}},{"Add":{"text":"that the localised abstraction function must abstract on at least one location","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:30397"}},{"Add":{"text":"per loop in the original program. This is formalized by the Lemma","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:30463"}},{"Add":{"text":"\\ref{lem:localized_pba_finite}. But before that, we will adapt the evaluation one","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:30545"}}],"del_contents":[{"Del":{"text":"The property based abstraction crafted in Example \\ref{ex:localized_pba} was","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24464"}},{"Del":{"text":"carefully crafted to guarantee a finite abstraction graph. 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","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:31282"}}],"del_contents":[{"Del":{"text":"The component-level evaluation is then repeated on $G_0$ until a fix-point is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25860"}},{"Del":{"text":"found. 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\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29428"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":682,"ins":{"up_context":[{"change":3,"pos":29523}],"down_context":[{"change":3,"pos":29675}],"flag":"BLOCK","start":32501,"end":32660,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":29523},"to":{"change":3,"start":29523,"end":29675},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" Note that the transitions $t_{2,1}$ and $t_{2,2}$ are new transitions but","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:32578"}},{"Add":{"text":" they transition to the trivial version instead of being unfolded due to $t_2","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:32659"}}],"del_contents":[{"Del":{"text":" Note, the transition $t_{2,1}$ and $t_{2,2}$ are new transitions but they","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29600"}},{"Del":{"text":" transition to the trivial version instead beeing unfolded due to $t_2","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29674"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":687,"ins":{"up_context":[{"change":3,"pos":29772}],"down_context":[{"change":3,"pos":29847}],"flag":"BLOCK","start":32661,"end":32738,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":29772},"to":{"change":3,"start":29772,"end":29846},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" step. We obtain $G_3 = G_2 = G^*_{\\Prog_3,S_{\\bot,\\Psi_{\\ell_1},\\bot}}$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:32737"}}],"del_contents":[{"Del":{"text":" step. We get $G_3 = G_2 = G^*_{\\Prog_3,S_{\\bot,\\Psi_{\\ell_1},\\bot}}$.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29845"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":693,"ins":{"up_context":[{"change":3,"pos":30008}],"down_context":[{"change":3,"pos":30206}],"flag":"BLOCK","start":32739,"end":32933,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":30008},"to":{"change":3,"start":30008,"end":30206},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" \\caption{Partial evaluation graph of $\\Prog_3$ from \\fref{fig:ex_prob}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:32817"}},{"Add":{"text":" on a subset of transitions $T = \\{t_{1a},t_{1b}\\}$ and with a localised","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:32897"}},{"Add":{"text":" property-based abstraction","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:32932"}}],"del_contents":[{"Del":{"text":" \\caption{Component-level partial evaluation graph of $\\Prog_3$ from","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30083"}},{"Del":{"text":" \\fref{fig:ex_prob} on a component $T = \\{t_{1a},t_{1b}\\}$ and with a","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30160"}},{"Del":{"text":" localized property-based abstraction","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30205"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":701,"ins":{"up_context":[{"change":2,"pos":30017}],"down_context":[{"change":2,"pos":30018}],"flag":"BLOCK","start":32934,"end":33216,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":2,"pos":30017},"to":{"change":3,"start":30311,"end":30592},"introduced_by":2},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":30311},"to":{"change":3,"start":30311,"end":30592},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" In practice, versions that have no incoming transitions in $G_0$ are","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:33006"}},{"Add":{"text":" removed, before the evaluation to avoid unnecessary unfoldings. 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","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30591"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":707,"ins":{"up_context":[{"change":3,"pos":30595}],"down_context":[{"change":3,"pos":30848}],"flag":"BLOCK","start":33217,"end":33463,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":30595},"to":{"change":3,"start":30595,"end":30848},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"The partial evaluation over the entire program is a special case of the partial","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:33296"}},{"Add":{"text":"evaluation with a subset $T = \\T_\\Prog$. 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All following lemmas","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30747"}},{"Del":{"text":"and theorems hold equally for a (global) abstraction and partial evaluations of","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30827"}},{"Del":{"text":"the entire program.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30847"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":714,"ins":{"up_context":[{"change":3,"pos":30929}],"down_context":[{"change":3,"pos":31004}],"flag":"BLOCK","start":33464,"end":33539,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":30929},"to":{"change":3,"start":30929,"end":31004},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" localised abstraction function $S$. For every transition $t = (\\langle","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:33538"}}],"del_contents":[{"Del":{"text":" localized abstraction function $S$. For every transition $t = (\\langle","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31003"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":719,"ins":{"up_context":[{"change":3,"pos":18393}],"down_context":[{"change":3,"pos":18459}],"flag":"BLOCK","start":33540,"end":33608,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":18393},"to":{"change":3,"start":18393,"end":18459},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" {\\bar{t}} = (\\ell_1, p, \\tau', \\eta, \\ell_2)\\textrm{, and }","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:33607"}}],"del_contents":[{"Del":{"text":" {\\bar{t}} = (\\ell, p, \\tau', \\eta, \\ell_2)\\textrm{, and }","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:18458"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":723,"ins":{"up_context":[{"change":3,"pos":25176}],"down_context":[{"change":3,"pos":18615}],"flag":"BLOCK","start":33609,"end":33701,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":25176},"to":{"change":3,"start":31159,"end":31246},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" \\proof{Follows directly from the construction of the unfolding and the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:33683"}},{"Add":{"text":" evaluation.}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:33700"}}],"del_contents":[{"Del":{"text":" \\proof{Follows directly from construction of the unfolding and the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31229"}},{"Del":{"text":" evaluation}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31245"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":729,"ins":{"up_context":[{"change":3,"pos":31326}],"down_context":[{"change":3,"pos":31664}],"flag":"BLOCK","start":33702,"end":34036,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":31326},"to":{"change":3,"start":31326,"end":31664},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"\\ref{ex:component_pe}, the localised property-based abstraction was carefully","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:33779"}},{"Add":{"text":"crafted so that the evaluation graph is finite. 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As was already foreshadowed the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31482"}},{"Del":{"text":"partial evaluation graph is for a localized abstraction function is finite when at least one location","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31584"}},{"Del":{"text":"per loop in the original program is abstracted to a finite abstraction space. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31663"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":736,"ins":{"up_context":[{"change":3,"pos":31711}],"down_context":[{"change":3,"pos":31789}],"flag":"BLOCK","start":34037,"end":34115,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":31711},"to":{"change":3,"start":31711,"end":31789},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" Let $S$ be a localised abstraction function. The partial evaluation graph","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:34114"}}],"del_contents":[{"Del":{"text":" Let $S$ be a localized abstraction function. The partial evaluation graph","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31788"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":739,"ins":{"up_context":[{"change":3,"pos":31944}],"down_context":[{"change":3,"pos":31966}],"flag":"BLOCK","start":34116,"end":34144,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":31944},"to":{"change":3,"start":31944,"end":31966},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" 2^{\\C_\\PV}}$ is finite.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:34143"}}],"del_contents":[{"Del":{"text":" \\Phi}$ is 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We can construct an infinite path $p = v_0v_1\\dots = \\langle\\ell_0,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:34407"}}],"del_contents":[{"Del":{"text":" We can construct an infinite path $p = v_0v_1\\dots = \\langle\\ell_0","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32438"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":753,"ins":{"up_context":[{"change":3,"pos":32596}],"down_context":[{"change":3,"pos":32674}],"flag":"BLOCK","start":34409,"end":34489,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":32596},"to":{"change":3,"start":32596,"end":32674},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" $\\evaluate_\\Prog^*$ does not converge, there must exist a fresh version","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:34488"}}],"del_contents":[{"Del":{"text":" $\\evaluate_\\Prog^*$ doesn't 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","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:34626"}}],"del_contents":[{"Del":{"text":" that is appears an infinite number of times on $p'$. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33780"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":771,"ins":{"up_context":[{"change":3,"pos":33782}],"down_context":[{"change":3,"pos":34174}],"flag":"BLOCK","start":34628,"end":35025,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":33782},"to":{"change":3,"start":33782,"end":34174},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" By assumption there must exist a location $\\ell^* \\in L$ for which","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:34702"}},{"Add":{"text":" $\\set{S(\\langle\\ell,\\varphi\\rangle)}{\\varphi \\in 2^{\\C_\\PV}}$ is finite. At","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:34786"}},{"Add":{"text":" the same time, versions $\\langle\\ell^*,\\_\\rangle$ must appear an infinite","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:34868"}},{"Add":{"text":" number of times on $p$, and they can only be abstracted to finitely","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:34944"}},{"Add":{"text":" many possible versions, which is a contradiction of the versions on $p$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:35024"}}],"del_contents":[{"Del":{"text":" By assumption there must exists a location $\\ell^* \\in L$ for which","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33857"}},{"Del":{"text":" $\\set{S(\\langle\\ell,\\varphi\\rangle)}{\\varphi \\in \\Phi}$ is finite. At","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33935"}},{"Del":{"text":" the same time versions $\\langle\\ell^*,\\_\\rangle$ must appear an infinite","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34016"}},{"Del":{"text":" number of times on $p$, and there can be only abstracted to finitely","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34093"}},{"Del":{"text":" many possible versions, which is a contradiction to the versions on $p$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34173"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":782,"ins":{"up_context":[{"change":3,"pos":34282}],"down_context":[{"change":3,"pos":34672}],"flag":"BLOCK","start":35026,"end":35422,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":34282},"to":{"change":3,"start":34282,"end":34672},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"We will proceed with proving that the partial evaluation graph restricted to a","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:35104"}},{"Add":{"text":"subset of transitions can be used for constructing an new program $\\Prog'$ with","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:35184"}},{"Add":{"text":"equal expected runtime complexity. All the results hold equally for partial","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:35260"}},{"Add":{"text":"evaluation of the entire program and independently of the choice of the (global","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:35340"}},{"Add":{"text":"or localised) abstraction function, as long as the localised evaluation function","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:35421"}}],"del_contents":[{"Del":{"text":"We will proceed with proving, that the partial evaluation graph of a component","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34360"}},{"Del":{"text":"level partial evaluation can be used for constructing an new program $\\Prog'$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34438"}},{"Del":{"text":"with equal expected runtime-complexity. All results equally hold for partial","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34515"}},{"Del":{"text":"evaluation of entire program and independently of the choice of the (global or","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34594"}},{"Del":{"text":"localized) abstraction function as long as the localized evaluation function","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34671"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":791,"ins":{"up_context":[{"change":3,"pos":34836}],"down_context":[{"change":3,"pos":34913}],"flag":"BLOCK","start":35423,"end":35500,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":34836},"to":{"change":3,"start":34836,"end":34913},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" localised abstraction function $S$ on a component $T \\subseteq \\TP$. The","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:35499"}}],"del_contents":[{"Del":{"text":" localized abstraction function $S$ on a component $T \\subseteq \\TP$. The","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34912"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":796,"ins":{"up_context":[{"change":3,"pos":25430}],"down_context":[{"change":3,"pos":35143}],"flag":"BLOCK","start":35501,"end":35663,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":25430},"to":{"change":3,"start":34988,"end":35143},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" \\item there exists a unique general transition $\\bar{g}_i \\in \\GTP$ and","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:35580"}},{"Add":{"text":" some $\\varphi \\in 2^{\\C_\\PV}$ for which the following property holds:","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:35662"}}],"del_contents":[{"Del":{"text":" \\item there exist a unique general transition $\\bar{g}_i \\in \\GTP$ and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35066"}},{"Del":{"text":" some $\\varphi \\in \\Phi$ for which the following property holds:","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35142"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":803,"ins":{"up_context":[{"change":2,"pos":30109}],"down_context":[{"change":3,"pos":35476}],"flag":"BLOCK","start":35664,"end":35700,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":30109},"to":{"change":2,"start":30109,"end":30139},"introduced_by":2}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" 2^{\\C_\\PV}}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:35699"}}],"del_contents":[{"Del":{"text":" 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abstraction","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36672"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":845,"ins":{"up_context":[{"change":3,"pos":20446}],"down_context":[{"change":3,"pos":20522}],"flag":"BLOCK","start":35854,"end":35930,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":20446},"to":{"change":3,"start":20446,"end":20522},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"$\\Prog$ has the same worst-case expected runtime complexity as the original","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:35929"}}],"del_contents":[{"Del":{"text":"$\\Prog$ has the same worst-case expected runtime-complexity as the 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Lemma","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:36152"}},{"Add":{"text":"\\ref{lem:constrfgeq} and Theorem \\ref{thm:thightness} show that the worst-case","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:36231"}},{"Add":{"text":"expected runtime-complexity does not worsen with partial evaluation either.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:36307"}}],"del_contents":[{"Del":{"text":"Theorem \\ref{thm:correctness} which will shows that any runtime-complexity","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:37063"}},{"Del":{"text":"bounds for the partial evaluation are runtime-complexity bounds of the original","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:37143"}},{"Del":{"text":"program, which demonstrates the correctness of the construction. Lemma","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:37214"}},{"Del":{"text":"\\ref{lem:constrfgeq} and Theorem \\ref{thm:thightness} will show that worst-case","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:37294"}},{"Del":{"text":"expected runtime-complexity doesn't get worse with partial evaluation.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:37365"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":859,"ins":{"up_context":[{"change":3,"pos":21442}],"down_context":[{"change":3,"pos":9198}],"flag":"BLOCK","start":36309,"end":36447,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":21442},"to":{"change":3,"start":21442,"end":21578},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"admissible finite prefixes, since non-admissible ones do not count in the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:36382"}},{"Add":{"text":"expected runtime-complexity as they have a probability of zero.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:36446"}}],"del_contents":[{"Del":{"text":"admissible finite prefixes, since non-admissible ones, do not count in the","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:21516"}},{"Del":{"text":"expected runtime-complexity as they have a probability zero.","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:21577"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":917,"ins":{"up_context":[{"change":3,"pos":23187}],"down_context":[{"change":3,"pos":23627}],"flag":"BLOCK","start":36448,"end":37011,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":23187},"to":{"change":3,"start":23187,"end":23349},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":23349},"to":{"change":2,"start":31568,"end":31920},"introduced_by":2}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" Let $\\Prog$ be a \\gls{pip}, $s_0 \\in \\Sigma$ an arbitrary initial state,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:36524"}},{"Add":{"text":" $\\scheduler \\in \\MDS_\\Prog$ a Markovian scheduler, and $\\Prog'$ the partial","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:36604"}},{"Add":{"text":" evaluation as described in Definition \\ref{def:partial_evaluation} with a","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:36682"}},{"Add":{"text":" localised abstraction function $S$ and the subset of transitions $T","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:36754"}},{"Add":{"text":" \\subseteq \\TP$. There exists a Markovian scheduler $\\scheduler' \\in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:36826"}},{"Add":{"text":" \\MDS_{\\Prog'}$, such that for all admissible finite prefixes","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:36891"}},{"Add":{"text":" $f\\in\\fpath_\\Prog$ with $\\prSs(\\varf) > 0$ there exists a finite prefix","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:36967"}},{"Add":{"text":" $\\varf' =","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:36981"}},{"Add":{"text":" \\simu_{\\Prog,S,T}(\\varf)","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37010"}}],"del_contents":[{"Del":{"text":" For a \\gls{pip} $\\Prog$, and any initial state $s_0 \\in \\Sigma$, a Markovian","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:23267"}},{"Del":{"text":" scheduler $\\scheduler \\in \\MDS_\\Prog$ and the partial evaluation $\\Prog'$ as","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:23348"}},{"Del":{"text":" described in Definition \\ref{def:partial_evaluation} with localised","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:31639"}},{"Del":{"text":" abstraction function $S$ and component $T$. There exists a Markovian","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:31712"}},{"Del":{"text":" scheduler $\\scheduler' \\in \\MDS_{\\Prog'}$, such that for all admissible","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:31788"}},{"Del":{"text":" finite prefixes $f\\in\\fpath_\\Prog$ with $\\prSs(\\varf) > 0$ there exists a","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:31866"}},{"Del":{"text":" finite prefix $\\varf' = \\simu_{\\Prog,S,T}(\\varf)","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:31919"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":949,"ins":{"up_context":[{"change":2,"pos":32183}],"down_context":[{"change":2,"pos":32457}],"flag":"BLOCK","start":37012,"end":37291,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":32183},"to":{"change":2,"start":32183,"end":32457},"introduced_by":2}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" \\in g, \\varphi' \\in 2^{\\C_\\PV} \\text{ and } t \\not\\in T}$. $g'$ is the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37086"}},{"Add":{"text":" general transition that contains all the unfolded and copied transitions for","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37167"}},{"Add":{"text":" $t \\in g$. For better readability we define a function","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37226"}},{"Add":{"text":" $\\texttt{eval}_{\\Prog,S,T} : \\GTP \\rightarrow \\GTPn$ where ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37290"}}],"del_contents":[{"Del":{"text":" \\in g, \\varphi' \\in \\Phi, \\text{ but } t \\not\\in T}$. $g'$ is the general","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:32260"}},{"Del":{"text":" transition that contains all the unfolded and copied transitions for $t \\in","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:32340"}},{"Del":{"text":" g$. For better readability we define a function $\\texttt{eval}_{\\Prog,S,T} : \\GTP","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:32426"}},{"Del":{"text":" \\rightarrow \\GTPn$ where ","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:32456"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":958,"ins":{"up_context":[{"change":2,"pos":32767}],"down_context":[{"change":2,"pos":32828}],"flag":"BLOCK","start":37292,"end":37359,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":32767},"to":{"change":2,"start":32767,"end":32828},"introduced_by":2}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" \\eta, \\ell') \\in g \\backslash T, \\varphi' \\in 2^{\\C_\\PV}}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37358"}}],"del_contents":[{"Del":{"text":" \\eta, \\ell') \\in g \\backslash T, \\varphi' \\in \\Phi}.","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:32827"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":963,"ins":{"up_context":[{"change":3,"pos":24563}],"down_context":[{"change":3,"pos":24716}],"flag":"BLOCK","start":37360,"end":37516,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":24563},"to":{"change":3,"start":24563,"end":24716},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" 2^{\\C_\\PV}$. $s' \\models \\varphi$ is not guaranteed. $s' \\not\\models\\varphi","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37439"}},{"Add":{"text":" \\Land \\tau_g$ if and only if $s \\not\\models \\varphi$. In that case, the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37515"}}],"del_contents":[{"Del":{"text":" \\Phi_\\PV$. $s' \\models \\varphi$ is not guaranteed. $s' \\not\\models\\varphi","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:24640"}},{"Del":{"text":" \\Land \\tau_g$ if and only if $s \\not\\models \\varphi$. In that case the","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:24715"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":993,"ins":{"up_context":[{"change":3,"pos":14578}],"down_context":[{"change":3,"pos":14011}],"flag":"BLOCK","start":37517,"end":38089,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":14578},"to":{"change":3,"start":25841,"end":26065},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":26065},"to":{"change":3,"start":18641,"end":18881},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":18881},"to":{"change":3,"start":26301,"end":26359},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":26301},"to":{"change":3,"start":26301,"end":26359},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" $\\scheduler'$ is indeed a valid scheduler: Case 1, let $s' \\models \\varphi$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37597"}},{"Add":{"text":" Case 1.1, let $\\scheduler(\\ell, \\bar{t'},s')$ selects $(g_\\bot, s')$. In","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37674"}},{"Add":{"text":" that case, $\\scheduler(\\ell', t', s') = (g_\\bot, s')$ as well. This is","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37749"}},{"Add":{"text":" correct because if no $(g, \\tilde{s}) \\in \\GTPnout(\\ell) \\times \\Sigma$ with","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37830"}},{"Add":{"text":" $\\tilde{s}|_\\PV = s'$ and $\\tilde{s} \\models \\tau_g$ exist, then does no","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37907"}},{"Add":{"text":" $(g', \\tilde{s'}) \\in \\GTPout(\\ell') \\times \\Sigma$, because all general","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:37984"}},{"Add":{"text":" transitions in $\\GTPout(\\ell')$ have more restrictive guards than their","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38060"}},{"Add":{"text":" original counterparts. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38088"}}],"del_contents":[{"Del":{"text":" Case 1, $s' \\models \\varphi$. Case 1.1, $\\scheduler(\\ell, \\bar{t'},s')$","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:25916"}},{"Del":{"text":" selects $(g_\\bot, s')$. In that case, $\\scheduler(\\ell', t', s') =","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:25987"}},{"Del":{"text":" (g_\\bot, s')$ as well. This is correct because if no $(g, \\tilde{s}) \\in","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:26064"}},{"Del":{"text":" \\GTPnout(\\ell) \\times \\Sigma$ with $\\tilde{s}|_\\PV = s'$ and $\\tilde{s}","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:18716"}},{"Del":{"text":" \\models \\tau_g$ exist, then so can't any $(g', \\tilde{s'}) \\in \\GTPout(\\ell')","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:18798"}},{"Del":{"text":" \\times \\Sigma$, because all general transitions in $\\GTPout(\\ell')$ have more","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:18880"}},{"Del":{"text":" restrictive guards than their original counterparts. ","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:26358"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1008,"ins":{"up_context":[{"change":3,"pos":14238}],"down_context":[{"change":3,"pos":14395}],"flag":"BLOCK","start":38090,"end":38167,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":14238},"to":{"change":3,"start":14322,"end":14395},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" Case 2, let $s' \\not\\models \\varphi$. The properties of a scheduler (see","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38166"}}],"del_contents":[{"Del":{"text":" Case 2, $s' \\not\\models \\varphi$. The properties of a scheduler (see","pos":"45HZPHVHYJ2BXATQ4MKCMPMMPZFBNJECY7JV3WX63BSA4CSQN6HAC:14394"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1014,"ins":{"up_context":[{"change":3,"pos":14806}],"down_context":[{"change":3,"pos":26829}],"flag":"BLOCK","start":38168,"end":38317,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":14806},"to":{"change":3,"start":26674,"end":26829},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" \\rem{If unreachable transitions and versions are removed from the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38237"}},{"Add":{"text":" program, the original scheduler might select a general transition that has","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38316"}}],"del_contents":[{"Del":{"text":" \\rem{In case unreachable transitions and versions are removed from the","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:26748"}},{"Del":{"text":" program, the original scheduler might select a general transition which has","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:26828"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1033,"ins":{"up_context":[{"change":3,"pos":27649}],"down_context":[{"change":3,"pos":15684}],"flag":"BLOCK","start":38318,"end":38397,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":27649},"to":{"change":3,"start":27649,"end":27722},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" similar general transition but it is not important for the final result. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38396"}}],"del_contents":[{"Del":{"text":" similar general transition but not important for the final result. ","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:27721"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1097,"ins":{"up_context":[{"change":3,"pos":30485}],"down_context":[{"change":3,"pos":30642}],"flag":"BLOCK","start":38398,"end":38554,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":30485},"to":{"change":3,"start":30485,"end":30642},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" At this point it becomes clear why filtering unreachable transitions","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38474"}},{"Add":{"text":" and versions can be done in practice. The newly constructed prefix can","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38553"}}],"del_contents":[{"Del":{"text":" At this point is becomes clear, why filtering unreachable transitions","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:30562"}},{"Del":{"text":" ans versions can be done in practice. The newly constructed prefix can","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:30641"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1100,"ins":{"up_context":[{"change":3,"pos":30722}],"down_context":[{"change":3,"pos":30800}],"flag":"BLOCK","start":38555,"end":38635,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":30722},"to":{"change":3,"start":30722,"end":30800},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" the presence of non-reachable transitions and versions does not change","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38634"}}],"del_contents":[{"Del":{"text":" the presence of non-reachable transitions and versions doesn't change","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:30799"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1111,"ins":{"up_context":[{"change":3,"pos":28611}],"down_context":[{"change":3,"pos":28781}],"flag":"BLOCK","start":38636,"end":38831,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":28611},"to":{"change":3,"start":30985,"end":31133},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":31133},"to":{"change":3,"start":28745,"end":28781},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":28745},"to":{"change":3,"start":28745,"end":28781},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" For a \\gls{pip} $\\Prog$, the partial evaluation $\\Prog'$ of $\\Prog$ as","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38710"}},{"Add":{"text":" described in Definition \\ref{def:partial_evaluation}, and an initial state","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38789"}},{"Add":{"text":" $s_0 \\in\\Sigma$ the following holds:","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38830"}}],"del_contents":[{"Del":{"text":" For a \\gls{pip} $\\Prog$, the partial evaluation $\\Prog'$ as described in","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:31061"}},{"Del":{"text":" Definition \\ref{def:partial_evaluation}, and an initial state $s_0","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:31132"}},{"Del":{"text":" \\in\\Sigma$ the following holds:","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:28780"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1145,"ins":{"up_context":[{"change":3,"pos":32129}],"down_context":[{"change":3,"pos":30840}],"flag":"BLOCK","start":38832,"end":38891,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":32129},"to":{"change":3,"start":32129,"end":32182},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" on the right side of Equation \\eqref{eq:sumsleq}. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38890"}}],"del_contents":[{"Del":{"text":" on the right side of of (\\ref{eq:sumsleq}). ","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:32181"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1147,"ins":{"up_context":[{"change":3,"pos":30841}],"down_context":[{"change":3,"pos":30984}],"flag":"BLOCK","start":38892,"end":39045,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":30841},"to":{"change":3,"start":30841,"end":30984},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" Taking the supremum over all schedulers on both sides of Equation","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:38965"}},{"Add":{"text":" \\eqref{eq:esleq} does not change the relation, since we just found for","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39044"}}],"del_contents":[{"Del":{"text":" Taking the supremum over all schedulers on both sides of","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:30905"}},{"Del":{"text":" (\\ref{eq:esleq}) doesn't change the relation, since we just found for","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:30983"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1158,"ins":{"up_context":[{"change":3,"pos":31298}],"down_context":[{"change":3,"pos":32627}],"flag":"BLOCK","start":39046,"end":39549,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":31298},"to":{"change":3,"start":32183,"end":32627},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"The following Theorem \\ref{thm:thightness} shows, that the partial evaluation","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39123"}},{"Add":{"text":"does not worsen the worst-case expected runtime complexity either. Note, this","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39201"}},{"Add":{"text":"does not imply that a perfectly tight bound for the worst-case expected runtime","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39281"}},{"Add":{"text":"can be found. Similar to Lemma \\ref{lem:constrfleq} we first construct a new","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39358"}},{"Add":{"text":"scheduler $\\scheduler \\in \\HDS_\\Prog$ and show that when using this scheduler, a","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39439"}},{"Add":{"text":"finite path of equal probability and runtime-complexity exists in $\\Prog$ for","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39517"}},{"Add":{"text":"every finite path in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39538"}},{"Add":{"text":"$\\Prog'$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39548"}}],"del_contents":[{"Del":{"text":"The following theorem shows, that the partial evaluation hasn't gotten worse","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:32259"}},{"Del":{"text":"either. Note, this doesn't imply that a perfectly tight bound for the worst-case","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:32340"}},{"Del":{"text":"expected runtime can be found. Similarly to Lemma \\ref{lem:constrfleq} we first","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:32420"}},{"Del":{"text":"construct a new scheduler $\\scheduler \\in \\HDS_\\Prog$ and show that when using","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:32499"}},{"Del":{"text":"this scheduler, a finite path of equal probability and runtime-complexity exists","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:32580"}},{"Del":{"text":"in $\\Prog$ for every finite path in $\\Prog'$.","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:32626"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1181,"ins":{"up_context":[{"change":3,"pos":33222}],"down_context":[{"change":2,"pos":34452}],"flag":"BLOCK","start":39550,"end":39629,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":33222},"to":{"change":3,"start":33222,"end":33299},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" We require a history dependent scheduler instead of a Markovian scheduler,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39628"}}],"del_contents":[{"Del":{"text":" We require a history dependent scheduler instead of Markovian scheduler,","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:33298"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1250,"ins":{"up_context":[{"change":2,"pos":35757}],"down_context":[{"change":2,"pos":35831}],"flag":"BLOCK","start":39630,"end":39703,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":35757},"to":{"change":2,"start":35757,"end":35831},"introduced_by":2}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" selects $\\scheduler'(c'_{n+1}) = (g_bot,s)$ as well. This is correct","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39702"}}],"del_contents":[{"Del":{"text":" selects $\\scheduler'(c'_{n+1}) = (g_bot,s)$ as well. Which is correct","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:35830"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1253,"ins":{"up_context":[{"change":2,"pos":35991}],"down_context":[{"change":2,"pos":36052}],"flag":"BLOCK","start":39704,"end":39764,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":35991},"to":{"change":2,"start":35991,"end":36052},"introduced_by":2}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" Hence, no general transition in $\\Prog$ is satisfiable.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39763"}}],"del_contents":[{"Del":{"text":" Hence, no general transition in $\\Prog$","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:36034"}},{"Del":{"text":" satisfiable ","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:36051"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1255,"ins":{"up_context":[{"change":2,"pos":36053}],"down_context":[{"change":2,"pos":36099}],"flag":"BLOCK","start":39765,"end":39857,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":36053},"to":{"change":2,"start":36053,"end":36099},"introduced_by":2}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" By using the induction hypothesis we arrive at the following, concluding the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39845"}},{"Add":{"text":" proof.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39856"}}],"del_contents":[{"Del":{"text":" By using the induction hypothesis we get ","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:36098"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1284,"ins":{"up_context":[{"change":3,"pos":32441}],"down_context":[{"change":2,"pos":36428}],"flag":"BLOCK","start":39858,"end":40014,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":32441},"to":{"change":3,"start":8989,"end":9067},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":9067},"to":{"change":2,"start":36354,"end":36428},"introduced_by":2}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":" By Lemma \\ref{lem:hdsvsmds} the fact that we used a history-dependent","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:39935"}},{"Add":{"text":" scheduler does not matter in the supremum and we optain the following","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:40013"}}],"del_contents":[{"Del":{"text":" By Lemma \\ref{lem:hdsvsmds} the fact that we used a history dependent","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:9066"}},{"Del":{"text":" scheduler doesn't matter in the supremum and we get the following","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:36427"}}]}},{"Replacement":{"inode":{"change":3,"pos":7949},"path":"ch3_theory.tex","line":1307,"ins":{"up_context":[{"change":3,"pos":9504}],"down_context":[],"flag":"BLOCK","start":40015,"end":40084,"inode":{"change":3,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":9504},"to":{"change":3,"start":9504,"end":9585},"introduced_by":3}],"inode":{"change":3,"pos":7949}},"ins_contents":[{"Add":{"text":"worst-case expected runtime do not worsen with a partial 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For example","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:37445"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":50,"ins":{"up_context":[{"change":3,"pos":37667}],"down_context":[{"change":3,"pos":37734}],"flag":"BLOCK","start":40257,"end":40332,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":37667},"to":{"change":3,"start":37667,"end":37734},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"$[x_i/y_i]_{x_i \\in A} = (\\dots([x_1/y_i]) \\dots) [x_n/y_n]$. For example,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:40331"}}],"del_contents":[{"Del":{"text":"$[x_i/y_i]_{x_i \\in A} = [x_1/y_i] \\dots [x_n/y(n)]$. For example,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:37733"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":62,"ins":{"up_context":[{"change":3,"pos":10603}],"down_context":[{"change":3,"pos":9801}],"flag":"BLOCK","start":40333,"end":40501,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":3,"pos":9329},"to":{"change":3,"start":10768,"end":10840},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":10603},"to":{"change":3,"start":10768,"end":10840},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":11738},"to":{"change":3,"start":10768,"end":10840},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10768},"to":{"change":3,"start":10768,"end":10840},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | 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\\V}.","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3068"}}]}},{"Edit":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":68,"atom":{"EdgeMap":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":9912},"to":{"change":3,"start":3070,"end":3285},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":3285},"to":{"change":3,"start":10041,"end":10042},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10041},"to":{"change":3,"start":10041,"end":10042},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10042},"to":{"change":3,"start":3286,"end":3462},"introduced_by":3}],"inode":{"change":3,"pos":8115}}},"contents":[{"Del":{"text":" % Analogously, the set of \\textit{linear strict inequality} constraints is","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3148"}},{"Del":{"text":" % defined as the set","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3173"}},{"Del":{"text":" % \\begin{align*}","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3194"}},{"Del":{"text":" % \\Lin^{<}=\\set{a_0 + \\sum_{i=1}^{n} a_i v_i < 0}{a_i \\in \\R}.","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3265"}},{"Del":{"text":" % \\end{align*}","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3284"}},{"Del":{"text":"","pos":"A5NVU6DZYOG6GRXJU6DZLCUQTKA7LPUWOGQ4NCJSY6FJB6GSP3OAC:10041"}},{"Del":{"text":" % The union of strict and non-strict linear constraints is called","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3355"}},{"Del":{"text":" % \\textit{mixed linear inequality} constraints denoted as $\\Lin = \\Lin^{\\leq}","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3437"}},{"Del":{"text":" % \\union \\Lin^{<}$.","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3461"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":69,"ins":{"up_context":[{"change":3,"pos":10213}],"down_context":[{"change":3,"pos":10309}],"flag":"BLOCK","start":40582,"end":40697,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10213},"to":{"change":3,"start":3463,"end":3543},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":3543},"to":{"change":3,"start":10293,"end":10309},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10293},"to":{"change":3,"start":10293,"end":10309},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" The set of \\textit{non-strict polynomial inequality constraints} with","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:40655"}},{"Add":{"text":" variables $A$ is defined as the set ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:40696"}}],"del_contents":[{"Del":{"text":" The set of \\textit{non-strict polynomial inequality constraints} is defined","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3542"}},{"Del":{"text":" as the set ","pos":"A5NVU6DZYOG6GRXJU6DZLCUQTKA7LPUWOGQ4NCJSY6FJB6GSP3OAC:10308"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":72,"ins":{"up_context":[{"change":3,"pos":10331}],"down_context":[{"change":3,"pos":10381}],"flag":"BLOCK","start":40698,"end":40743,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10331},"to":{"change":3,"start":3544,"end":3587},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" \\PC_A = \\set{e \\leq 0}{e \\in \\Z[A]}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:40742"}}],"del_contents":[{"Del":{"text":" \\C = \\set{e \\leq 0}{e \\in \\Z[\\V]}.","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3586"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":74,"ins":{"up_context":[{"change":3,"pos":10401}],"down_context":[{"change":3,"pos":1406}],"flag":"BLOCK","start":40744,"end":40842,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10401},"to":{"change":3,"start":3588,"end":3902},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:40744"}},{"Add":{"text":" Let $\\C$ and $\\PV$ denote the linear and polynomial constraints with","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:40817"}},{"Add":{"text":" variables $A = \\V$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:40841"}}],"del_contents":[{"Del":{"text":" % whereas the set of \\textit{strict polynomial inequality} constraints is","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3665"}},{"Del":{"text":" % defined as the set ","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3691"}},{"Del":{"text":" % \\begin{equation*}","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3715"}},{"Del":{"text":" % \\C^< = \\set{e < 0}{e \\in \\Z[\\V]},","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3759"}},{"Del":{"text":" % \\end{equation*}","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3781"}},{"Del":{"text":" % and the set of \\textit{mixed polynomial inequality} constraints is the set","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3862"}},{"Del":{"text":" % $\\C = \\C^{\\leq} \\union \\C^{<}$. ","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3901"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":80,"ins":{"up_context":[{"change":3,"pos":10715}],"down_context":[{"change":3,"pos":10926}],"flag":"BLOCK","start":40843,"end":41128,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10715},"to":{"change":3,"start":3903,"end":4142},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" The set of non-strict linear constraints is a subset of the set of","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:40913"}},{"Add":{"text":" non-strict polynomial constraints. For the most part we will just refer to","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:40992"}},{"Add":{"text":" \\enquote{constraints} and mean linear constraints. Whenever we encounter","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41069"}},{"Add":{"text":" polynomial constraints we will mention it explicitly.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41127"}}],"del_contents":[{"Del":{"text":" Non-strict linear constraints are a subset of non-strict polynomial","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:3974"}},{"Del":{"text":" constraints. For the most part we will just refer to \\enquote{constraints}","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:4053"}},{"Del":{"text":" when the distinction between linear and polynomial constraints is","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:4123"}},{"Del":{"text":" unimportant. ","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:4141"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":86,"ins":{"up_context":[{"change":3,"pos":1087}],"down_context":[{"change":3,"pos":38306}],"flag":"BLOCK","start":41129,"end":41206,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":1087},"to":{"change":3,"start":42942,"end":43093},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":43093},"to":{"change":3,"start":38233,"end":38306},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"For foundations on \\gls{fo}-logic we refer the reader to \\cite{barwise1977}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41205"}}],"del_contents":[{"Del":{"text":"The set of quantifier-free formulas of first-order logic over a variable set","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:43018"}},{"Del":{"text":"$A\\subset\\V$ is denoted with $\\Phi_A$, or simply $\\Phi$, when $A=\\V$. For","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:43092"}},{"Del":{"text":"foundations on \\gls{fo}-logic we refer the reader to \\cite{barwise1977}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:38305"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":89,"ins":{"up_context":[{"change":3,"pos":43321}],"down_context":[{"change":3,"pos":1249}],"flag":"BLOCK","start":41207,"end":41641,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":43321},"to":{"change":3,"start":38385,"end":38537},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":38537},"to":{"change":3,"start":10761,"end":10919},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10761},"to":{"change":3,"start":10761,"end":10919},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10919},"to":{"change":3,"start":38538,"end":38659},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"\\gls{fo} logic \\cite{barwise1977}. In this thesis, we will only consider integer","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41287"}},{"Add":{"text":"arithmetic. Let $\\Zs := (\\Z, +, \\cdot, 0, 1, \\leq)$ be the structure of standard","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41368"}},{"Add":{"text":"integer arithmetic. For an assignment $\\beta : \\V \\rightarrow \\Z$ and a formula","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41448"}},{"Add":{"text":"$\\varphi \\in \\Phi$ we write $\\beta \\models \\varphi$ instead of $\\Zs, \\beta","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41523"}},{"Add":{"text":"\\models \\varphi$. Finally, we define two trivial formulas $\\texttt{true} = (0 =","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41603"}},{"Add":{"text":"0)$ and $\\texttt{false} = (1 < 0)$. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41640"}}],"del_contents":[{"Del":{"text":"\\gls{fo} logic.\\cite{barwise1977} In this thesis, we will only consider integer","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:38464"}},{"Del":{"text":"arithmetic. Let $\\Zs := (\\Z, +, \\cdot, 0, 1, \\leq)$ be the structure of","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:38536"}},{"Del":{"text":"standard integer arithmetic. For an assignment $\\beta : \\V \\rightarrow \\Z$ and a","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:10841"}},{"Del":{"text":"formula $\\varphi \\in \\Phi$ we write $\\beta \\models \\varphi$ instead of $\\Zs,","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:10918"}},{"Del":{"text":"\\beta \\models \\varphi$. Finally, we define two trivial formulas $\\texttt{true} =","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:38618"}},{"Del":{"text":"(0 = 0)$ and $\\texttt{false} = (1 < 0)$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:38658"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":116,"ins":{"up_context":[{"change":3,"pos":38870}],"down_context":[{"change":3,"pos":2234}],"flag":"BLOCK","start":41642,"end":41872,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":3,"pos":38870},"to":{"change":3,"start":4872,"end":5111},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10424},"to":{"change":3,"start":4872,"end":5111},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" Two constraints, or set of constraints $\\varphi_1$ and $\\varphi_2$ are","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41716"}},{"Add":{"text":" considered \\textit{equivalent}, denoted as $\\varphi_1 \\equiv \\varphi_2$, when","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41798"}},{"Add":{"text":" $\\llbracket \\varphi_1 \\rrbracket = \\llbracket \\varphi_2 \\rrbracket$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41871"}}],"del_contents":[{"Del":{"text":" Two formulas, constraints, or set of constraints $\\varphi_1$ and $\\varphi_2$","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:4952"}},{"Del":{"text":" are considered \\textit{equivalent}, denoted as $\\varphi_1 \\equiv \\varphi_2$","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:5032"}},{"Del":{"text":" when $\\llbracket \\varphi_1 \\rrbracket = \\llbracket \\varphi_2 \\rrbracket$.","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:5110"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":126,"ins":{"up_context":[{"change":3,"pos":12762}],"down_context":[{"change":3,"pos":11870}],"flag":"BLOCK","start":41873,"end":41926,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":12762},"to":{"change":3,"start":38951,"end":39012},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"$e_1, e_2 \\in \\Z[\\V]$ can be transformed as follows:","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41925"}}],"del_contents":[{"Del":{"text":"$e_1, e_2 \\in \\mathbb{P}[\\V]$ can be transformed as follows:","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:39011"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":173,"ins":{"up_context":[{"change":3,"pos":39715}],"down_context":[{"change":3,"pos":7879}],"flag":"BLOCK","start":41927,"end":42252,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":39715},"to":{"change":3,"start":39715,"end":39835},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"The satisfiability of sets polynomial constraints is known to be","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:41991"}},{"Add":{"text":"undecidable\\cite{kremer2016lncs}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42025"}},{"Add":{"text":"Thus whenever we encounter polynomial constraints the need to be","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42090"}},{"Add":{"text":"over-approximated by linear constraints. The theory relevant for this thesis is","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42170"}},{"Add":{"text":"the quantifier-free linear integer arithmetic (usually called \\texttt{QF\\_LIA}).","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42251"}}],"del_contents":[{"Del":{"text":"The theory relevant for this thesis is the quantifier-free linear integer","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:39788"}},{"Del":{"text":"arithmetic (usually called \\texttt{QF\\_LIA}).","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:39834"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":189,"ins":{"up_context":[{"change":3,"pos":4097}],"down_context":[{"change":3,"pos":11459}],"flag":"BLOCK","start":42253,"end":42478,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":4097},"to":{"change":3,"start":11220,"end":11380},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":11380},"to":{"change":3,"start":11388,"end":11467},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" Let $A \\subseteq V$ and $\\varphi \\in 2^\\C$. The projection of $\\varphi$ onto","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42333"}},{"Add":{"text":" $A$ denoted as $\\varphi|_A$ is a constraint $\\varphi|_A\\in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42396"}},{"Add":{"text":" 2^{\\C_A}$, with $\\llbracket \\varphi|_A \\rrbracket = \\set{\\beta|_A}{\\beta \\in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42477"}}],"del_contents":[{"Del":{"text":" Let $A \\subseteq V$ and $\\varphi \\in \\Phi$. The projection of $\\varphi$ onto","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:11300"}},{"Del":{"text":" $A$ denoted as $\\varphi|_A$ is a quantifier-free FO-formula $\\varphi|_A\\in","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:11379"}},{"Del":{"text":" \\Phi_A$, with $\\llbracket \\varphi|_A \\rrbracket = \\set{\\beta|_A}{\\beta \\in","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:11466"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":222,"ins":{"up_context":[{"change":3,"pos":41483}],"down_context":[{"change":3,"pos":13595}],"flag":"BLOCK","start":42479,"end":42598,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":41483},"to":{"change":3,"start":41483,"end":41564},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":41564},"to":{"change":3,"start":45357,"end":45391},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":45357},"to":{"change":3,"start":45357,"end":45391},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"\\Sigma_A$. It computes the assignment after an update $u$ with a given an","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42552"}},{"Add":{"text":"assignment $s \\in \\Sigma$ before the update:","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42597"}}],"del_contents":[{"Del":{"text":"\\Sigma_A$. It computes the assignment after an update $u$ is given an assignment","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:41563"}},{"Del":{"text":"$s \\in \\Sigma$ before the update:","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:45390"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":232,"ins":{"up_context":[{"change":3,"pos":45668}],"down_context":[{"change":3,"pos":45726}],"flag":"BLOCK","start":42599,"end":42664,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":45668},"to":{"change":3,"start":45668,"end":45726},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" \\textit{before} the update. \\textit{After} the update, holds","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42663"}}],"del_contents":[{"Del":{"text":" \\textit{before} the update. \\textit{After} the update","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:45725"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":237,"ins":{"up_context":[{"change":3,"pos":41607}],"down_context":[{"change":3,"pos":41687}],"flag":"BLOCK","start":42665,"end":42990,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":3,"pos":41607},"to":{"change":3,"start":45832,"end":46063},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":45832},"to":{"change":3,"start":45832,"end":46063},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":46063},"to":{"change":3,"start":41608,"end":41687},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" Let $A, B \\subseteq \\V$. Given an assignment $s\\in\\Sigma$, a set of","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42736"}},{"Add":{"text":" constraints $\\varphi \\in 2^\\C$ and an update $u: A \\rightarrow \\Z[B]$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42811"}},{"Add":{"text":" Without loss of generality $A' = \\set{x'}{x \\in A} \\subset \\V$ is a set of","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42890"}},{"Add":{"text":" fresh variables with $A' \\cap (A \\union B) = \\emptyset$. We write under","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42966"}},{"Add":{"text":" misuse of notation","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:42989"}}],"del_contents":[{"Del":{"text":" Let $A, B \\subseteq \\V$. Given an assignment $s\\in\\Sigma$, a formula","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:45904"}},{"Del":{"text":" $\\varphi \\in \\Phi$ and an update $u: A \\rightarrow \\Z[B]$. Without loss of","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:45983"}},{"Del":{"text":" generality $A' = \\set{x'}{x \\in A} \\subset \\V$ is a set of fresh variables","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:46062"}},{"Del":{"text":" with $A' \\cap (A \\union B) = \\emptyset$. We write under misuse of notation","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:41686"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":243,"ins":{"up_context":[{"change":3,"pos":41709}],"down_context":[{"change":3,"pos":41767}],"flag":"BLOCK","start":42991,"end":43048,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":41709},"to":{"change":3,"start":41709,"end":41767},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" u(\\varphi) = ((\\varphi \\Land \\LAnd_{x\\in A} x' =","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43047"}}],"del_contents":[{"Del":{"text":" u(\\varphi) := ((\\varphi \\Land \\LAnd_{x\\in A} x' =","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:41766"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":250,"ins":{"up_context":[{"change":3,"pos":41955}],"down_context":[{"change":3,"pos":46219}],"flag":"BLOCK","start":43049,"end":43128,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":41955},"to":{"change":3,"start":41955,"end":42034},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" $\\varphi \\in 2^\\C$ and an update $u: A \\rightarrow \\Z[B]$. For any partial","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43127"}}],"del_contents":[{"Del":{"text":" $\\varphi \\in \\Phi$ and an update $u: A \\rightarrow \\Z[B]$. For any partial","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:42033"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":268,"ins":{"up_context":[{"change":3,"pos":12782}],"down_context":[{"change":3,"pos":14927}],"flag":"BLOCK","start":43129,"end":43309,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":12782},"to":{"change":3,"start":42512,"end":42696},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"used abstract domains are Polyhedra \\cite{bagnara2005convexpolyhedron},","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43200"}},{"Add":{"text":"Octagons \\cite{mine2001wcre, mine2007arxiv, mine2006hosc,truchet2010synacs} and","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43280"}},{"Add":{"text":"Boxes \\cite{gurfinkel2010}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43308"}}],"del_contents":[{"Del":{"text":"used abstract domains are Polyhedra \\cite{bagnara2005convexpolyhedron}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:42582"}},{"Del":{"text":"Octagons \\cite{mine2001wcre, mine2007arxiv, mine2006hosc,truchet2010synacs}, and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:42663"}},{"Del":{"text":"the Boxes \\cite{gurfinkel2010}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:42695"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":303,"ins":{"up_context":[{"change":3,"pos":16585}],"down_context":[{"change":3,"pos":16660}],"flag":"BLOCK","start":43310,"end":43389,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":16585},"to":{"change":3,"start":16585,"end":16660},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" Consider the following two constraints over the variables $\\V = \\{x, y\\}$:","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43388"}}],"del_contents":[{"Del":{"text":" Consider the following constraints over the variables $\\V = \\{x, y\\}$:","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:16659"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":311,"ins":{"up_context":[{"change":3,"pos":16990}],"down_context":[{"change":3,"pos":17010}],"flag":"BLOCK","start":43390,"end":43462,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":16990},"to":{"change":3,"start":16990,"end":17010},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" which is a polynomial constraint can be represented as a parabola. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43461"}}],"del_contents":[{"Del":{"text":" as a parabola. ","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:17009"}}]}},{"Edit":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":321,"atom":{"EdgeMap":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":17432},"to":{"change":3,"start":46458,"end":46531},"introduced_by":3}],"inode":{"change":3,"pos":8115}}},"contents":[{"Del":{"text":"The projection operation computes the projection of an \\gls{fo} formula.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:46530"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":323,"ins":{"up_context":[{"change":3,"pos":42941}],"down_context":[{"change":3,"pos":17671}],"flag":"BLOCK","start":43463,"end":43506,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":42941},"to":{"change":3,"start":42941,"end":42983},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"respect to the canonical basis of $\\R^m$. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43505"}}],"del_contents":[{"Del":{"text":"respect to the canonical basis of $\\R^m$.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:42982"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":396,"ins":{"up_context":[{"change":3,"pos":19423}],"down_context":[{"change":3,"pos":15604}],"flag":"BLOCK","start":43507,"end":43767,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":19423},"to":{"change":3,"start":14789,"end":14859},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":14859},"to":{"change":3,"start":19495,"end":19583},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":19495},"to":{"change":3,"start":19495,"end":19583},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"special case of polyhedra. Octagons are used in \\gls{koat} during the generation","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43587"}},{"Add":{"text":"of invariants. They are faster to work with than polyhedra but less precise,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43664"}},{"Add":{"text":"since they can not represent all linear constraints and are forced to","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43734"}},{"Add":{"text":"over-approximated some of them.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43766"}}],"del_contents":[{"Del":{"text":"special case of polyhedra. Octagons are used in \\gls{koat} during the","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:14858"}},{"Del":{"text":"generation of invariants. They are faster to work with than polyhedra but less","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:19573"}},{"Del":{"text":"precise.","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:19582"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":433,"ins":{"up_context":[{"change":3,"pos":44648}],"down_context":[{"change":3,"pos":44879}],"flag":"BLOCK","start":43768,"end":43996,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":44648},"to":{"change":3,"start":44648,"end":44879},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"probability two subsets of outcomes (e.g. $A = \\{\\omega \\in \\Omega\\mid\\text{","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43844"}},{"Add":{"text":" coin shows head}\\}$ and $B = \\{\\omega \\in \\Omega\\mid\\text{coin shows","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43917"}},{"Add":{"text":"tail}\\}$) so you can measure the probability of the union of both subsets ($A","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:43995"}}],"del_contents":[{"Del":{"text":"probability two subsets of outcomes (e.g. $A = \\set{ \\omega \\in \\Omega} {\\text{","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:44727"}},{"Del":{"text":" coin shows head}}$ and $B = \\set{\\omega \\in \\Omega} {\\text{coin shows","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:44801"}},{"Del":{"text":"tail}}$) so you can measure the probability of the union of both subsets ($A","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:44878"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":533,"ins":{"up_context":[{"change":3,"pos":47176}],"down_context":[{"change":3,"pos":35277}],"flag":"BLOCK","start":43997,"end":44045,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":47176},"to":{"change":3,"start":47176,"end":47224},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" = \\sum_{r \\in \\overline{N}} P(X \\geq r)","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44044"}}],"del_contents":[{"Del":{"text":" = \\sum_{r \\in \\overline{N}} P(X \\leq r)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47223"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":670,"ins":{"up_context":[{"change":3,"pos":30977}],"down_context":[{"change":3,"pos":50775}],"flag":"BLOCK","start":44046,"end":44117,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":30977},"to":{"change":3,"start":47811,"end":47883},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"Let $\\D$ be the set of probability distributions. $\\Z[\\V\\union \\D]$ is","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44116"}}],"del_contents":[{"Del":{"text":"Let $\\D$ be the set of probability distributions. $\\Z[\\V\\union \\D]$, is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47882"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":732,"ins":{"up_context":[{"change":3,"pos":52274}],"down_context":[{"change":3,"pos":52418}],"flag":"BLOCK","start":44118,"end":44312,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":52274},"to":{"change":3,"start":52274,"end":52344},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":52344},"to":{"change":3,"start":48432,"end":48509},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" \\min (\\support(D)) \\leq d \\leq \\max (\\support(D)) & \\begin{aligned}&\\text{if}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44203"}},{"Add":{"text":" \\min(\\support(D))\\text{, and } \\\\&\\max (\\support(D)) \\text{ are","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44279"}},{"Add":{"text":" known},\\end{aligned} \\\\","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44311"}}],"del_contents":[{"Del":{"text":" \\min (\\support(D)) \\leq d \\leq \\max (\\support(D)) & \\text{if}","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:52343"}},{"Del":{"text":" \\min(\\support(D)) \\text{and} \\max (\\support(D)) \\text{ is known}, \\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48508"}}]}},{"Edit":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":741,"atom":{"NewVertex":{"up_context":[{"change":3,"pos":52665}],"down_context":[{"change":3,"pos":52665}],"flag":"BLOCK","start":44313,"end":44535,"inode":{"change":3,"pos":8115}}},"contents":[{"Add":{"text":"","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44313"}},{"Add":{"text":"\\begin{rem}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44325"}},{"Add":{"text":" \\gls{koat} supports only the distributions presented in the previous","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44398"}},{"Add":{"text":" section. Hence, those constraints are known and easy to compute for all","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44474"}},{"Add":{"text":" distributions which we encounter in practice.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44524"}},{"Add":{"text":"\\end{rem}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44534"}}]}},{"Edit":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":860,"atom":{"EdgeMap":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":36011},"to":{"change":3,"start":36011,"end":36154},"introduced_by":3}],"inode":{"change":3,"pos":8115}}},"contents":[{"Del":{"text":"% \\begin{lemma}","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36026"}},{"Del":{"text":"","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36027"}},{"Del":{"text":"% $s'\\models\\eta(varphi)$","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36057"}},{"Del":{"text":"% $s'(x_i) = k_i \\in","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36082"}},{"Del":{"text":"% \\text{supp}([\\eta(x_i)](s))$ implies $s'\\models \\tilde{\\eta}(\\)$","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36153"}}]}},{"Edit":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":861,"atom":{"EdgeMap":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":36155},"to":{"change":3,"start":36155,"end":36369},"introduced_by":3}],"inode":{"change":3,"pos":8115}}},"contents":[{"Del":{"text":" % and values $k_i \\in","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36180"}},{"Del":{"text":" % \\text{supp}([\\eta(x_i)](s))$, the assignment $s'(x_i)=k_i$ satisfies the","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36259"}},{"Del":{"text":" % non-probabilistic over-approximation \\todo{todo}","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36314"}},{"Del":{"text":"","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36315"}},{"Del":{"text":" % \\begin{proof}","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36335"}},{"Del":{"text":" % \\end{proof}","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36353"}},{"Del":{"text":"% \\end{lemma}","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36367"}},{"Del":{"text":"","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36368"}}]}},{"Edit":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":862,"atom":{"EdgeMap":{"edges":[{"previous":"","flag":"DELETED","from":{"change":3,"pos":5198},"to":{"change":3,"start":4443,"end":4731},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":54816},"to":{"change":3,"start":4443,"end":4731},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":9345},"to":{"change":3,"start":4443,"end":4731},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":4731},"to":{"change":3,"start":23013,"end":23014},"introduced_by":3}],"inode":{"change":3,"pos":8115}}},"contents":[{"Del":{"text":"\\begin{comment}","pos":"KUEZAMEH2M3K6EDHWMYXQMHZ2QTTH4KBM5CO7A7EB64ABEWRE6WQC:4458"}},{"Del":{"text":" The syntax and semantics of probabilistic programs has been extensively","pos":"KUEZAMEH2M3K6EDHWMYXQMHZ2QTTH4KBM5CO7A7EB64ABEWRE6WQC:4534"}},{"Del":{"text":" studied:","pos":"KUEZAMEH2M3K6EDHWMYXQMHZ2QTTH4KBM5CO7A7EB64ABEWRE6WQC:4547"}},{"Del":{"text":" - [Supermartingales,","pos":"KUEZAMEH2M3K6EDHWMYXQMHZ2QTTH4KBM5CO7A7EB64ABEWRE6WQC:4572"}},{"Del":{"text":" 2017](https://arxiv.org/pdf/1709.04037.pdf#cite.Billingsley%3Abook)","pos":"KUEZAMEH2M3K6EDHWMYXQMHZ2QTTH4KBM5CO7A7EB64ABEWRE6WQC:4644"}},{"Del":{"text":" - [Probabilistic Koat2, 2021](https://arxiv.org/pdf/2010.06367.pdf)","pos":"KUEZAMEH2M3K6EDHWMYXQMHZ2QTTH4KBM5CO7A7EB64ABEWRE6WQC:4716"}},{"Del":{"text":"\\end{comment}","pos":"KUEZAMEH2M3K6EDHWMYXQMHZ2QTTH4KBM5CO7A7EB64ABEWRE6WQC:4730"}},{"Del":{"text":"","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:23013"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":1062,"ins":{"up_context":[{"change":3,"pos":53117}],"down_context":[{"change":3,"pos":36703}],"flag":"BLOCK","start":44536,"end":44583,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":3,"pos":53117},"to":{"change":3,"start":36682,"end":36703},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":36682},"to":{"change":3,"start":36682,"end":36703},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"contains five parts \\cite{puterman1994markov}:","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44582"}}],"del_contents":[{"Del":{"text":"contains five parts:","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36702"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":1071,"ins":{"up_context":[{"change":3,"pos":23021}],"down_context":[{"change":3,"pos":17928}],"flag":"BLOCK","start":44584,"end":44626,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":23021},"to":{"change":3,"start":53203,"end":53256},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":53256},"to":{"change":3,"start":37252,"end":37268},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":37252},"to":{"change":3,"start":37252,"end":37268},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" with every action.\\end{enumerate}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44625"}}],"del_contents":[{"Del":{"text":" with every action. \\cite{puterman1994markov}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53255"}},{"Del":{"text":"\\end{enumerate}","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:37267"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":1080,"ins":{"up_context":[{"change":3,"pos":37673}],"down_context":[{"change":3,"pos":23255}],"flag":"BLOCK","start":44627,"end":44705,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":37673},"to":{"change":3,"start":53257,"end":53336},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"The highest reward is the optimisation goal when taking decisions in a Markov","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44704"}}],"del_contents":[{"Del":{"text":"The highest reward is the optimisation goal, when taking decisions in a Markov","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53335"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":1082,"ins":{"up_context":[{"change":3,"pos":23330}],"down_context":[{"change":3,"pos":53534}],"flag":"BLOCK","start":44706,"end":44919,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":23330},"to":{"change":3,"start":53337,"end":53534},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"accumulated rewards over a whole process. In the literature","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44765"}},{"Add":{"text":"\\cite{puterman1994markov, puterman1990markov}, the properties of a given Markov","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44845"}},{"Add":{"text":"decision process with regard to the highest achievable rewards have been","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44918"}}],"del_contents":[{"Del":{"text":"accumulated rewards over a whole process. In the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53385"}},{"Del":{"text":"literature \\cite{puterman1994markov, puterman1990markov}, the properties of a","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53463"}},{"Del":{"text":"given process with regard to the highest achievable rewards have been","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53533"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":1102,"ins":{"up_context":[{"change":3,"pos":23965}],"down_context":[{"change":3,"pos":38847}],"flag":"BLOCK","start":44920,"end":45105,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":23965},"to":{"change":3,"start":53803,"end":53865},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":53865},"to":{"change":3,"start":23966,"end":24004},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":38747},"to":{"change":3,"start":23966,"end":24004},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":24004},"to":{"change":3,"start":53866,"end":53930},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" \\item The transition probability, which will be defined later.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:44986"}},{"Add":{"text":" \\item The reward is 1 for every non-terminating transition and 0 for the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:45063"}},{"Add":{"text":" terminating transition $t_\\bot$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:45104"}}],"del_contents":[{"Del":{"text":" \\item The transition probability will be defined later in","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53864"}},{"Del":{"text":" \\Sref{ssec:probabilityspace}.","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:24003"}},{"Del":{"text":" \\item The reward is 1 for every non-terminating transition.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53929"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":1357,"ins":{"up_context":[{"change":3,"pos":20927}],"down_context":[{"change":3,"pos":21020}],"flag":"BLOCK","start":45106,"end":45424,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":20927},"to":{"change":3,"start":57219,"end":57427},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" $\\scheduler_\\text{opt} \\in \\MDS_{F_i} \\subset \\MDS$ exists that","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:45177"}},{"Add":{"text":" maximises the expected reward when limited to the finite set of","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:45249"}},{"Add":{"text":" configurations. For all remaining configurations $\\scheduler_\\text{opt}$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:45330"}},{"Add":{"text":" may behave arbitrarily. Let $\\nu_i$ be the reward of this optimal","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:45404"}},{"Add":{"text":" scheduler.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:45423"}}],"del_contents":[{"Del":{"text":" $\\scheduler_\\text{opt} \\in \\MDS_{F_i}$ exists that maximises the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57291"}},{"Del":{"text":" expected reward when limited to the finite set of configurations. Let","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57369"}},{"Del":{"text":" $\\nu_i$ be the reward of this optimal scheduler.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57426"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":1364,"ins":{"up_context":[{"change":3,"pos":57506}],"down_context":[{"change":3,"pos":57577}],"flag":"BLOCK","start":45425,"end":45503,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":57506},"to":{"change":3,"start":57506,"end":57577},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" yield an expected reward no less than the history dependent scheduler","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:45502"}}],"del_contents":[{"Del":{"text":" yield an expected no less than the history dependent scheduler","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57576"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":1371,"ins":{"up_context":[{"change":3,"pos":21374}],"down_context":[{"change":3,"pos":21449}],"flag":"BLOCK","start":45504,"end":45578,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":21374},"to":{"change":3,"start":21374,"end":21449},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":" Finally, we let $N$ grow to infinity where $\\nu_{i,N}$ is now the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:45577"}}],"del_contents":[{"Del":{"text":" Finally, we let $N$ grow to infinity. 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All","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:23631"}},{"Del":{"text":"locations in this component are denoted by a function $\\Loc : 2^{\\TP}","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:23701"}},{"Del":{"text":"\\rightarrow 2^{\\Loc_\\Prog}$ with ","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:23735"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":1459,"ins":{"up_context":[{"change":3,"pos":36539}],"down_context":[{"change":3,"pos":37223}],"flag":"BLOCK","start":45831,"end":46066,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":36539},"to":{"change":3,"start":23836,"end":24059},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"For a subset of transitions $T \\subset \\T_\\Prog$ the entry transitions are","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:45905"}},{"Add":{"text":"described by a function $\\TPin : 2^{\\TP} \\rightarrow 2^{\\TP}$ and the exit","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:45980"}},{"Add":{"text":"transitions are described by a function $\\TPout : 2^{\\TP} \\rightarrow 2^{\\TP}$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46059"}},{"Add":{"text":"with ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46065"}}],"del_contents":[{"Del":{"text":"For a component $T \\subset \\T_\\Prog$ the entry transitions are described by a","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:23913"}},{"Del":{"text":"function $\\TPin : 2^{\\TP} \\rightarrow 2^{\\TP}$ and the exit transitions are","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:23989"}},{"Del":{"text":"described by a function $\\TPout : 2^{\\TP} \\rightarrow 2^{\\TP}$ with ","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:24058"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":1469,"ins":{"up_context":[{"change":3,"pos":37532}],"down_context":[{"change":3,"pos":24312}],"flag":"BLOCK","start":46067,"end":46148,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":37532},"to":{"change":3,"start":24231,"end":24312},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"For a \\gls{pip} $\\Prog$ a loop (or cycle) is a finite sequence of locations $L =","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46147"}}],"del_contents":[{"Del":{"text":"For a \\gls{pip} $\\Prog$ A loop (or cycle) is a finite sequence of locations $L =","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:24311"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":1476,"ins":{"up_context":[{"change":3,"pos":24768}],"down_context":[{"change":3,"pos":25041}],"flag":"BLOCK","start":46149,"end":46453,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":24768},"to":{"change":3,"start":24768,"end":25041},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"of subsets of transitions $T_L$ which we call the \\emph{transition loops} of $L$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46229"}},{"Add":{"text":"with","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46234"}},{"Add":{"text":"\\begin{equation*}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46252"}},{"Add":{"text":" \\begin{gathered}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46273"}},{"Add":{"text":" T_L = \\set{\\{t_0,\\dots,t_n\\}}{t_i = (\\ell_i, \\_, \\_, \\_, \\ell_{i+1}) \\in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46350"}},{"Add":{"text":" \\T_\\Prog, 0 \\leq i < n, t_n = (\\ell_i, \\_, \\_, \\_, \\ell_0) \\in \\T_\\Prog}. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46433"}},{"Add":{"text":" \\end{gathered}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46452"}}],"del_contents":[{"Del":{"text":"of components $T_L$ which we call the \\emph{transition loops} of $L$ with","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:24841"}},{"Del":{"text":"\\begin{equation*} ","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:24860"}},{"Del":{"text":" T_L = \\set{\\{t_0,\\dots,t_n\\}}{t_i = (\\ell_i, \\_, \\_, \\_, \\ell_{i+1}) \\in \\T_\\Prog \\text{ for } 0 \\leq i < n \\text{ and } t_n =","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:24991"}},{"Del":{"text":" (\\ell_i, \\_, \\_, \\_, \\ell_0) \\in \\T_\\Prog}. ","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:25040"}}]}},{"Replacement":{"inode":{"change":3,"pos":8115},"path":"ch2_preliminaries.tex","line":1485,"ins":{"up_context":[{"change":3,"pos":38264}],"down_context":[{"change":3,"pos":13792}],"flag":"BLOCK","start":46454,"end":46674,"inode":{"change":3,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":38264},"to":{"change":3,"start":25058,"end":25247},"introduced_by":3}],"inode":{"change":3,"pos":8115}},"ins_contents":[{"Add":{"text":"A subset of transitions $T \\subseteq \\T_\\Prog$ is called a \\emph{strongly","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46527"}},{"Add":{"text":"connected} component when for every two locations $\\ell, \\ell' \\in V(T)$ there","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46606"}},{"Add":{"text":"exists a path between the two using only the transitions from $T$.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46673"}}],"del_contents":[{"Del":{"text":"A component $T \\subseteq \\T_\\Prog$ is \\emph{strongly connected} when for every","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:25136"}},{"Del":{"text":"two locations $\\ell, \\ell' \\in V(T)$ there exists a path between the two using","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:25215"}},{"Del":{"text":"only the transitions from $T$.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:25246"}}]}},{"Replacement":{"inode":{"change":3,"pos":14309},"path":"ch1_introduction.tex","line":10,"ins":{"up_context":[{"change":3,"pos":59023}],"down_context":[{"change":3,"pos":59104}],"flag":"BLOCK","start":46675,"end":46758,"inode":{"change":3,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":59023},"to":{"change":3,"start":59023,"end":59104},"introduced_by":3}],"inode":{"change":3,"pos":14309}},"ins_contents":[{"Add":{"text":"their program in critical scenarios. In an ever more complex world, with ever more","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46757"}}],"del_contents":[{"Del":{"text":"his program in critical scenarios. In an ever more complex world, with ever more","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59103"}}]}},{"Replacement":{"inode":{"change":3,"pos":14309},"path":"ch1_introduction.tex","line":14,"ins":{"up_context":[{"change":3,"pos":59327}],"down_context":[{"change":3,"pos":8564}],"flag":"BLOCK","start":46759,"end":46895,"inode":{"change":3,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":59327},"to":{"change":3,"start":59327,"end":59408},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":59408},"to":{"change":3,"start":15856,"end":15911},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":15856},"to":{"change":3,"start":15856,"end":15911},"introduced_by":3}],"inode":{"change":3,"pos":14309}},"ins_contents":[{"Add":{"text":"programming paradigms as tightly and quickly as possible \\cite{giesl2017aprove,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46838"}},{"Add":{"text":"montoya2014aplas,alarcon2017muterm,irankfinder2018wst}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46894"}}],"del_contents":[{"Del":{"text":"programming paradigms as tightly and quickly as possible. \\cite{giesl2017aprove,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59407"}},{"Del":{"text":"montoya2014aplas,alarcon2017muterm,irankfinder2018wst}","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:15910"}}]}},{"Replacement":{"inode":{"change":3,"pos":14309},"path":"ch1_introduction.tex","line":24,"ins":{"up_context":[{"change":3,"pos":59729}],"down_context":[{"change":3,"pos":59882}],"flag":"BLOCK","start":46896,"end":47051,"inode":{"change":3,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":59729},"to":{"change":3,"start":59729,"end":59882},"introduced_by":3}],"inode":{"change":3,"pos":14309}},"ins_contents":[{"Add":{"text":"two program variables $x$ and $y$, and one temporary variable $u$, which is used","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:46976"}},{"Add":{"text":"to non-deterministically sample $x$ on the first line of the program. The","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:47050"}}],"del_contents":[{"Del":{"text":"two program variables $x$, and $y$, and one temporary variable $u$, which is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59805"}},{"Del":{"text":"used to non-deterministically sample $x$ in the first transition $t_0$. The","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59881"}}]}},{"Replacement":{"inode":{"change":3,"pos":14309},"path":"ch1_introduction.tex","line":30,"ins":{"up_context":[{"change":3,"pos":60190}],"down_context":[{"change":3,"pos":60402}],"flag":"BLOCK","start":47052,"end":47332,"inode":{"change":3,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":60190},"to":{"change":3,"start":60190,"end":60402},"introduced_by":3}],"inode":{"change":3,"pos":14309}},"ins_contents":[{"Add":{"text":"\\gls{cfr} on sub-\\gls{scc} level, the program can be transformed into an","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:47124"}},{"Add":{"text":"equivalent program (see \\fref{fig:classic_pe_c}) where the analysis with","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:47197"}},{"Add":{"text":"\\gls{mprf} succeeds at finding a finite runtime-complexity bound, because now","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:47275"}},{"Add":{"text":"the variable $x$ is constrained when entering the loop.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:47331"}}],"del_contents":[{"Del":{"text":"\\gls{cfr} on sub-\\gls{scc} level, the program (see \\fref{fig:classic_pe_c}) can","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60269"}},{"Del":{"text":"be transformed into an equivalent program where the analysis with \\gls{mprf}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60346"}},{"Del":{"text":"succeeds at finding a finite runtime-complexity bound.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60401"}}]}},{"Replacement":{"inode":{"change":3,"pos":14309},"path":"ch1_introduction.tex","line":66,"ins":{"up_context":[{"change":3,"pos":31381}],"down_context":[{"change":3,"pos":31461}],"flag":"BLOCK","start":47333,"end":47401,"inode":{"change":3,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":31381},"to":{"change":3,"start":16197,"end":16264},"introduced_by":3}],"inode":{"change":3,"pos":14309}},"ins_contents":[{"Add":{"text":"programs \\cite{meyer2021tacas}. Unfortunately, by using iRankfinder","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:47400"}}],"del_contents":[{"Del":{"text":"programs\\cite{meyer2021tacas}. Unfortunately, by using iRankfinder","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:16263"}}]}},{"Replacement":{"inode":{"change":3,"pos":14309},"path":"ch1_introduction.tex","line":86,"ins":{"up_context":[{"change":3,"pos":9618}],"down_context":[{"change":3,"pos":32042}],"flag":"BLOCK","start":47402,"end":47482,"inode":{"change":3,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":9618},"to":{"change":3,"start":9618,"end":9677},"introduced_by":3}],"inode":{"change":3,"pos":14309}},"ins_contents":[{"Add":{"text":" find runtime complexity bounds using only probabilistic ranking functions.}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:47481"}}],"del_contents":[{"Del":{"text":" find runtime-complexity bounds using only \\gls{mprf}.}","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:9676"}}]}},{"Replacement":{"inode":{"change":3,"pos":14309},"path":"ch1_introduction.tex","line":89,"ins":{"up_context":[{"change":3,"pos":32056}],"down_context":[{"change":3,"pos":32702}],"flag":"BLOCK","start":47483,"end":48154,"inode":{"change":3,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":32056},"to":{"change":3,"start":32056,"end":32135},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":32135},"to":{"change":3,"start":65487,"end":65636},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":65636},"to":{"change":3,"start":60916,"end":61350},"introduced_by":3}],"inode":{"change":3,"pos":14309}},"ins_contents":[{"Add":{"text":"Consider the very similar probabilistic integer program $\\Prog_2$ which is","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:47557"}},{"Add":{"text":"displayed in \\fref{fig:prob_c}. The variables $x$ and $y$ are now only","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:47628"}},{"Add":{"text":"in/-decremented with 50\\% probability in every iteration. The program will","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:47703"}},{"Add":{"text":"probably always terminate, since a longer runtime becomes increasingly unlikely.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:47784"}},{"Add":{"text":"The current version of \\gls{koat} fails to find a finite runtime and expected","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:47862"}},{"Add":{"text":"complexity bound for this program. It would be useful to transform $\\Prog_2$","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:47939"}},{"Add":{"text":"into an equivalent program like the one shown in \\fref{fig:prob_pe_c}, similar","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48018"}},{"Add":{"text":"to the \\gls{cfr} technique presented by \\citeauthor{giesl2022lncs}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48085"}},{"Add":{"text":"\\cite{giesl2022lncs} and \\citeauthor{Domenech19} \\cite{Domenech19}.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48153"}}],"del_contents":[{"Del":{"text":"Consider the very similar probabilistic integer program $\\Prog_2$ displayed in","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:32134"}},{"Del":{"text":"\\fref{fig:prob_c}. The variables $x$ and $y$ are now only in/-decremented with","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:65565"}},{"Del":{"text":"50\\% probability in every iteration. The program will probably always","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:65635"}},{"Del":{"text":"terminate, since a longer runtime becomes increasingly unlikely. The current","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60992"}},{"Del":{"text":"version of \\gls{koat} fails to find a finite runtime and expected complexity","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61069"}},{"Del":{"text":"bound for this program. It would be useful to transform $\\Prog_2$ into an","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61143"}},{"Del":{"text":"equivalent program like the one shown in \\fref{fig:prob_pe_c}, similar to the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61221"}},{"Del":{"text":"\\gls{cfr} technique presented by \\citeauthor{giesl2022lncs} \\cite{giesl2022lncs}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61302"}},{"Del":{"text":"and \\citeauthor{Domenech19} \\cite{Domenech19}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61349"}}]}},{"Replacement":{"inode":{"change":3,"pos":14309},"path":"ch1_introduction.tex","line":115,"ins":{"up_context":[{"change":3,"pos":10688}],"down_context":[{"change":3,"pos":40668}],"flag":"BLOCK","start":48155,"end":48233,"inode":{"change":3,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":10688},"to":{"change":3,"start":40591,"end":40668},"introduced_by":3}],"inode":{"change":3,"pos":14309}},"ins_contents":[{"Add":{"text":"The tool iRankFinder \\cite{irankfinder2018wst} is another tool taking part in","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48232"}}],"del_contents":[{"Del":{"text":"The tool iRankFinder\\cite{irankfinder2018wst} is another tool taking part in","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:40667"}}]}},{"Replacement":{"inode":{"change":3,"pos":14309},"path":"ch1_introduction.tex","line":129,"ins":{"up_context":[{"change":3,"pos":62541}],"down_context":[{"change":3,"pos":62616}],"flag":"BLOCK","start":48234,"end":48310,"inode":{"change":3,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":62541},"to":{"change":3,"start":62541,"end":62616},"introduced_by":3}],"inode":{"change":3,"pos":14309}},"ins_contents":[{"Add":{"text":"with a number of limitations \\cite{giesl2022lncs}. Besides code-quality and","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48309"}}],"del_contents":[{"Del":{"text":"with a number of limitations.\\cite{giesl2022lncs} Besides code-quality and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62615"}}]}},{"Replacement":{"inode":{"change":3,"pos":14309},"path":"ch1_introduction.tex","line":132,"ins":{"up_context":[{"change":3,"pos":41905}],"down_context":[{"change":3,"pos":42129}],"flag":"BLOCK","start":48311,"end":48758,"inode":{"change":3,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":41905},"to":{"change":3,"start":41905,"end":42129},"introduced_by":3}],"inode":{"change":3,"pos":14309}},"ins_contents":[{"Add":{"text":"\\gls{koat} with regard to non-probabilistic integer programs by reimplementing","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48389"}},{"Add":{"text":"the control flow refinement technique of iRankFinder \\cite{irankfinder2018wst}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48468"}},{"Add":{"text":"natively in KoAT instead of calling the external library. By adapting the method","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48549"}},{"Add":{"text":"of partial evaluation to the paradigm of probabilistic programs and proving its","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48629"}},{"Add":{"text":"correctness, partial evaluation is made available for probabilistic programs for","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48710"}},{"Add":{"text":"the first time (to the best of our knowledge).","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48757"}}],"del_contents":[{"Del":{"text":"\\gls{koat} by reimplementing the control flow refinement technique of","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:41974"}},{"Del":{"text":"iRankFinder\\cite{irankfinder2018wst} natively in KoAT instead of calling the","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:42051"}},{"Del":{"text":"external library and adapting it to the paradigm of probabilistic programs. ","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:42128"}}]}},{"Replacement":{"inode":{"change":3,"pos":14309},"path":"ch1_introduction.tex","line":140,"ins":{"up_context":[{"change":3,"pos":42208}],"down_context":[{"change":3,"pos":40329}],"flag":"BLOCK","start":48759,"end":49420,"inode":{"change":3,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":42208},"to":{"change":3,"start":66925,"end":67122},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":67122},"to":{"change":3,"start":62698,"end":62769},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":62769},"to":{"change":3,"start":67198,"end":67286},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":67198},"to":{"change":3,"start":67198,"end":67286},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":67286},"to":{"change":3,"start":62770,"end":62846},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":62846},"to":{"change":3,"start":67365,"end":67432},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":67365},"to":{"change":3,"start":67365,"end":67432},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":67432},"to":{"change":3,"start":62847,"end":62923},"introduced_by":3},{"previous":"","flag":"DELETED","from":{"change":3,"pos":62923},"to":{"change":3,"start":67513,"end":67544},"introduced_by":3},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":3,"pos":67513},"to":{"change":3,"start":67513,"end":67544},"introduced_by":3}],"inode":{"change":3,"pos":14309}},"ins_contents":[{"Add":{"text":"\\Cref{ch:preliminaries}, focusing especially on constraints, probability theory,","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48839"}},{"Add":{"text":"and the formalism behind (probabilistic) integer programs. ","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48899"}},{"Add":{"text":"In \\Cref{ch:theory} the \\gls{cfr} techniques from \\citeauthor{Domenech19} are","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:48977"}},{"Add":{"text":"adapted to \\Glspl{pip}. We prove the theoretical correctness of the","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49045"}},{"Add":{"text":"construction. Furthermore, the technique is extended to refine only a part of","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49123"}},{"Add":{"text":"the original program.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49145"}},{"Add":{"text":"\\Cref{ch:implementation} focuses on the implementation and the integration into","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49225"}},{"Add":{"text":"the existing analysis. The performance of the new implementation is analysed.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49303"}},{"Add":{"text":"Finally in \\Cref{ch:conclusion}, we recapitulate the results of this thesis and","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49383"}},{"Add":{"text":"give an outlook on future research.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49419"}}],"del_contents":[{"Del":{"text":"\\Cref{ch:preliminaries}, focusing especially on constraints, probability","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66997"}},{"Del":{"text":"theory, and (probabilistic) integer programs. ","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:67044"}},{"Del":{"text":"In \\Cref{ch:theory} the \\gls{cfr} techniques from iRankFinder are adapted to","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:67121"}},{"Del":{"text":"\\Glspl{pip}. We prove the theoretical correctness of the construction.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62768"}},{"Del":{"text":"Furthermore, the technique of sub-\\gls{scc} level control-flow-refinement is","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:67274"}},{"Del":{"text":"presented.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:67285"}},{"Del":{"text":"\\Cref{ch:implementation} focuses on the changes made to \\gls{koat}, and the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62845"}},{"Del":{"text":"performance improvements made with the new implementation. Finally","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:67431"}},{"Del":{"text":"in \\Cref{ch:conclusion} we recapitulate the results of this thesis and give","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62922"}},{"Del":{"text":"an outlook on future research.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:67543"}}]}},{"Replacement":{"inode":{"change":3,"pos":14402},"path":"ch0_frontmatter.tex","line":65,"ins":{"up_context":[{"change":2,"pos":36634}],"down_context":[{"change":3,"pos":16254}],"flag":"BLOCK","start":49421,"end":49818,"inode":{"change":3,"pos":14402}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":36634},"to":{"change":2,"start":36634,"end":37050},"introduced_by":2}],"inode":{"change":3,"pos":14402}},"ins_contents":[{"Add":{"text":"\\begin{vplace}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49435"}},{"Add":{"text":" \\textit{","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49448"}},{"Add":{"text":" Hanna and Thibaud, I would like to express my deepest gratitude for your","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49525"}},{"Add":{"text":" continuous encouragement and assistance. Your unwavering belief in my","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49599"}},{"Add":{"text":" abilities and your willingness to lend a helping hand during challenging","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49676"}},{"Add":{"text":" times have been instrumental in the successful completion of this thesis.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49754"}},{"Add":{"text":" Thank you, from the bottom of my heart.","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49798"}},{"Add":{"text":" }","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49804"}},{"Add":{"text":"\\end{vplace}","pos":"5MHYFQWH3WWCAARBKU5OEMKNW2F5EMA2FJ5HS7PDDGEFH2HWKEPQC:49817"}}],"del_contents":[{"Del":{"text":"% \\begin{vplace}","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:36650"}},{"Del":{"text":"% \\textit{","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:36665"}},{"Del":{"text":"% Hanna and Thibaud, I would like to express my deepest gratitude","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:36735"}},{"Del":{"text":"% for your continuous encouragement and assistance. Your unwavering belief","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:36814"}},{"Del":{"text":"% in my abilities and your willingness to lend a helping hand during challenging","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:36899"}},{"Del":{"text":"% times have been instrumental in the successful completion of this thesis. Thank","pos":"USNRNHANGXUQHFFVHLAVWWA7WGPX2RZOYFQAV44HU5AYEW5YC27QC:36985"}},{"Del":{"text":"% you, from the bottom of my heart. 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