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;","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:1398"}},{"Add":{"text":" \\path[dotted,->]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:1419"}},{"Add":{"text":" (l3) edge[gray] (l4')","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:1467"}},{"Add":{"text":" edge (l4) ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:1509"}},{"Add":{"text":" ;","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:1541"}},{"Add":{"text":"\\end{tikzpicture}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:1559"}}]}},{"FileAdd":{"meta":{"basename":"ch3_p4_program.tex","metadata":0},"context":[{"change":4,"pos":7550}],"contents":[{"Add":{"text":"% TeX root = 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[out=110,in=150,looseness=10,draw=rot-75]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:2801"}},{"Add":{"text":" node[swap]{$t_{1a}:\\begin{aligned} p&=0.5 \\\\ \\tau_1&= 1 \\leq x \\leq 3","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:2879"}},{"Add":{"text":" \\Land w = 0\\end{aligned}$} (l1)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:2919"}},{"Add":{"text":" (l1) edge [out=160,in=200,looseness=10,draw=rot-75]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:2979"}},{"Add":{"text":" node[swap]{$ t_{1b}: \\begin{aligned}p&=0.5 \\\\ \\tau_1&= 1 \\leq x \\leq 3","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3058"}},{"Add":{"text":" \\Land w = 0 \\\\ \\eta_{1b}(x)&= x + 1\\end{aligned}$} (l1)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3122"}},{"Add":{"text":" (l1) edge[bend left =30] node {$t_{2}:\\tau_2 = y > 0","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3183"}},{"Add":{"text":" \\Land w = 1$} (l2)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3210"}},{"Add":{"text":" (l2) edge[bend left=30] node {$t_{3}:\\begin{aligned} ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3272"}},{"Add":{"text":" p&=0.5 \\\\ \\eta_3(y) &= y - \\textsc{Bern}(0.5)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3330"}},{"Add":{"text":" \\end{aligned}$} (l1)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3359"}},{"Add":{"text":" % (l2) edge[bend left=30,draw=rwth-75] node[swap] {$t_{3a}: p=0.5$} (l1)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3440"}},{"Add":{"text":" % (l1) edge node {$t_4:x \\leq 0$} (l2)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3500"}},{"Add":{"text":" % edge[loop above] node {$t_2:\\begin{aligned} x &> 1 \\\\ x&:=","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3582"}},{"Add":{"text":" % x+1\\end{aligned}$} ()","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3627"}},{"Add":{"text":" % edge[loop below] node {$t_3:\\begin{aligned} x &> 0 \\\\ x&:=","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3710"}},{"Add":{"text":" % x-1\\end{aligned}$} ()","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3755"}},{"Add":{"text":" ;","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3777"}},{"Add":{"text":"\\end{tikzpicture}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3795"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3796"}}]}},{"FileAdd":{"meta":{"basename":"ch3_p3_lift.tex","metadata":0},"context":[{"change":4,"pos":7550}],"contents":[{"Add":{"text":"% TeX root = ../main.tex","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3863"}},{"Add":{"text":"\\begin{tikzpicture}[tree,initial above,font=\\scriptsize,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3920"}},{"Add":{"text":" every state/.append style={rectangle,rounded corners=10}]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:3983"}},{"Add":{"text":" \\node[state,initial] (l0) {$\\langle\\ell_0,\\texttt{true}\\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4053"}},{"Add":{"text":" \\node[state] (l1) [below=3cm of l0] {$\\langle\\ell_1,\\texttt{true}\\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4133"}},{"Add":{"text":" \\node[state] (l2) [below=3cm of l1] {$\\langle\\ell_2,\\texttt{true}\\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4213"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4214"}},{"Add":{"text":" \\path[->] ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4231"}},{"Add":{"text":" (l0) edge node { $t_0: \\eta_0(x) = u $ } (l1)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4285"}},{"Add":{"text":" (l1) edge [out=110,in=150,looseness=6,draw=rot-75]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4344"}},{"Add":{"text":" node[swap]{$t_{1a}:\\begin{aligned} p&=0.5 \\\\ \\tau_1&= 1 \\leq x \\leq 3","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4422"}},{"Add":{"text":" \\Land w = 0\\end{aligned}$} (l1)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4462"}},{"Add":{"text":" (l1) edge [out=160,in=200,looseness=6,draw=rot-75]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4521"}},{"Add":{"text":" node[swap]{$ t_{1b}: \\begin{aligned}p&=0.5 \\\\ \\tau_1&= 1 \\leq x \\leq 3","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4600"}},{"Add":{"text":" \\Land w = 0 \\\\ \\eta_{1b}(x)&= x + 1\\end{aligned}$} (l1)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4664"}},{"Add":{"text":" (l1) edge[bend left =30] node {$t_{2}:\\begin{aligned}\\tau_2 =& y > 0","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4741"}},{"Add":{"text":" \\Land w = 1\\end{aligned}$} (l2)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4781"}},{"Add":{"text":" (l2) edge[bend left=30] node {$t_{3}:\\begin{aligned} ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4843"}},{"Add":{"text":" p&=0.5 \\\\ \\eta_3(y) &= y - \\textsc{Bern}(0.5)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4901"}},{"Add":{"text":" \\end{aligned}$} (l1)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4930"}},{"Add":{"text":" ;","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4952"}},{"Add":{"text":"\\end{tikzpicture}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4970"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:4971"}}]}},{"FileAdd":{"meta":{"basename":"ch3_p3_full_pe_localized.tex","metadata":0},"context":[{"change":4,"pos":7550}],"contents":[{"Add":{"text":"% TeX root = ../main.tex","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5051"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5052"}},{"Add":{"text":"\\begin{tikzpicture}[program,initial above,font=\\scriptsize, ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5113"}},{"Add":{"text":" every state/.append style={rectangle,rounded corners=10},","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5175"}},{"Add":{"text":" ]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5181"}},{"Add":{"text":" \\node[state,initial] (l0t) {$\\langle \\ell_0, \\texttt{true} \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5259"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5260"}},{"Add":{"text":" \\node[state] (l1t) [below=2cm of l0t] {$\\langle \\ell_1, \\texttt{true} \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5345"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5346"}},{"Add":{"text":" \\node[state] (l1p134) [below=2cm of l1t] {$\\langle \\ell_1, \\lbrace \\psi_1, \\psi_3, \\psi_4 \\rbrace \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5459"}},{"Add":{"text":" \\node[state] (l1p124) [right=4cm of l1p134] {$\\langle \\ell_1, \\lbrace \\psi_1, \\psi_2, \\psi_4 \\rbrace \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5575"}},{"Add":{"text":" \\node[state] (l2py) [left=4cm of l1p134] {$\\langle \\ell_2, y > 0 \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5655"}},{"Add":{"text":" ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5660"}},{"Add":{"text":" \\node[state] (l2p134y) [below=2cm of l1p134] {$\\langle \\ell_2, \\psi_1 \\land \\psi_3 \\land \\psi_4 \\land y>0 \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5781"}},{"Add":{"text":" \\node[state] (l2p124y) [right=4cm of l2p134y] {$\\langle \\ell_2, \\psi_1 \\land \\psi_2 \\land \\psi_4 \\land y>0 \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:5904"}},{"Add":{"text":" \\node[state] (l1p56) [left=4cm of l2p1345] {$\\langle \\ell_1, \\lbrace \\psi_5, \\psi_6 \\rbrace \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6011"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6012"}},{"Add":{"text":" \\node[state,] (l1p13456) [below=2cm of l2p1345] {$\\langle \\ell_1, \\lbrace \\psi_1, \\psi_3, \\psi_4, \\psi_5, \\psi_6 \\rbrace \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6148"}},{"Add":{"text":" \\node[state,] (l1p12456) [right=4cm of l1p13456] {$\\langle \\ell_1, \\lbrace \\psi_1, \\psi_2, \\psi_4, \\psi_5, \\psi_6 \\rbrace \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6285"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6286"}},{"Add":{"text":" % Neu in Bild 2","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6306"}},{"Add":{"text":" \\node[state] (l2p13456y) [below=2.5cm of l1p13456] {$\\left\\langle\\ell_2,\\begin{aligned} \\psi_1 &\\land \\psi_3 \\land \\psi_4 \\land\\\\ \\psi_5 &\\land \\psi_6 \\land y > 0 \\end{aligned}\\right\\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6503"}},{"Add":{"text":" \\node[state] (l2p56y) [below=3cm of l1p56] {$\\langle \\ell_2, \\psi_5 \\land \\psi_6 \\land y > 0 \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6611"}},{"Add":{"text":" \\node[state] (l2p12456y) [right=4cm of l2p13456y] {$\\left\\langle\\ell_2, \\begin{aligned}\\psi_1 &\\land \\psi_2 \\land \\psi_4 \\land \\\\ \\psi_5 &\\land \\psi_6 \\land y > 0 \\end{aligned}\\right\\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6808"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6809"}},{"Add":{"text":" % Transitions","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6827"}},{"Add":{"text":" \\path[->] ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6844"}},{"Add":{"text":" (l0t) edge[] node (t01) { $t_{0,1}$ } (l1t)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6896"}},{"Add":{"text":" (l1t) edge[] node[swap] (t21) { $t_{2,1}$ } (l2p5)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:6955"}},{"Add":{"text":" (l1t) edge[bend right=20] node[swap] (t1a1) { $t_{1a,1}$ } (l1p134)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:7031"}},{"Add":{"text":" (l1t) edge[] node (t1b1) { $t_{1b,1}$ } (l1p124) %FEHLT IN DER SKIZZE","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:7110"}},{"Add":{"text":" (l2py) edge[] node[swap] (t31) { $t_{3,1}$ } (l1p56)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:7171"}},{"Add":{"text":" (l1p134) edge[bend right=10] node[swap] (t1b2) { $t_{1b,2}$ } (l1p124)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:7250"}},{"Add":{"text":" (l1p134) edge[loop above,transform canvas={xshift=5mm}] node[swap] (t1a2) { $t_{1a,2}$ } ()","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:7350"}},{"Add":{"text":" (l1p134) edge[] node (t22) { $t_{2,2}$ } (l2p134y)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:7409"}},{"Add":{"text":" (l1p124) edge[bend right=10] node[swap] (t1a3) { $t_{1a,3}$ } (l1p134)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:7488"}},{"Add":{"text":" (l1p124) edge[loop above] node (t1b3) { $t_{1b,3}$ } ()","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:7552"}},{"Add":{"text":" (l1p124) edge[] node (t23) { $t_{2,3}$ } (l2p124y)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:7611"}},{"Add":{"text":" (l1p56) edge[bend right=30] node (t1a6) { $t_{1a,6}$ } (l1p13456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:7685"}},{"Add":{"text":" (l1p56) edge[out=290,in=205] node[swap,pos=.40] (t1b6) { $t_{1b,6}$ } (l1p12456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:7774"}},{"Add":{"text":" (l2p134y) edge[] node[swap] (t32) { $t_{3,2}$ } (l1p13456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:7841"}},{"Add":{"text":" (l2p124y) edge[] node[swap] (t33) { $t_{3,3}$ } (l1p12456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:7908"}},{"Add":{"text":" (l1p13456) edge[loop above,transform canvas={xshift=12mm}] node[swap] (t1a4) { $t_{1a,4}$ } ()","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:8011"}},{"Add":{"text":" (l1p12456) edge[loop above,transform canvas={xshift=12mm}] node[swap] (t1b5) { $t_{1b,5}$ } ()","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:8114"}},{"Add":{"text":" (l1p13456) edge[bend left=10] node (t1b4) { $t_{1b,4}$ } (l1p12456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:8190"}},{"Add":{"text":" (l1p12456) edge[bend left=10] node (t1a5) { $t_{1a,5}$ } (l1p13456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:8266"}},{"Add":{"text":" % (l1p56) edge[rot, bend right=20] node[swap] (t25) { $t_{2,5}$ } (l2p5)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:8347"}},{"Add":{"text":" (l1p56) edge[bend left=15,transform canvas={xshift=-4mm}] node[] (t26) { $t_{2,6}$ } (l2p56y)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:8449"}},{"Add":{"text":" (l2p56y) edge[bend left=15,transform canvas={xshift=-4mm}] node[] (t36) { $t_{3,6}$ } (l1p56)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:8551"}},{"Add":{"text":" % (l1p13456) edge[bend right=20] node[swap] (t24) { $t_{2,4}$ } (l2p1345)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:8633"}},{"Add":{"text":" (l1p13456) edge[bend left=20] node[pos=.55] (t26) { $t_{2,4}$ } (l2p13456y)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:8717"}},{"Add":{"text":" (l2p13456y) edge[bend left=20] node[pos=.45] (t34) { $t_{3,4}$ } (l1p13456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:8801"}},{"Add":{"text":" % (l1p12456) edge[bend right=20] node[swap] (t26) { $t_{2,6}$ } (l2p1245)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:8883"}},{"Add":{"text":" (l1p12456) edge[bend right=20] node[swap] (t25) { $t_{2,5}$ } (l2p12456y)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:8965"}},{"Add":{"text":" (l2p12456y) edge[bend right=20] node[swap] (t35) { $t_{3,5}$ } (l1p12456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9047"}},{"Add":{"text":" ;","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9057"}},{"Add":{"text":"\\end{tikzpicture}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9075"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9076"}}]}},{"FileAdd":{"meta":{"basename":"ch3_p3_full_pe.tex","metadata":0},"context":[{"change":4,"pos":7550}],"contents":[{"Add":{"text":"% TeX root = ../main.tex","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9146"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9147"}},{"Add":{"text":"\\begin{tikzpicture}[program,initial above,font=\\scriptsize, ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9208"}},{"Add":{"text":" every state/.append style={rectangle,rounded corners=10},","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9270"}},{"Add":{"text":" every initial by arrow/.style={grun}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9311"}},{"Add":{"text":" ]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9317"}},{"Add":{"text":" \\node[state,initial,grun] (l0t) {$\\langle \\ell_0, \\texttt{true} \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9421"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9422"}},{"Add":{"text":" \\node[state,grun] (l1t) [below=2cm of l0t] {$\\langle \\ell_1, \\texttt{true} \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9512"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9513"}},{"Add":{"text":" \\node[state,rwth] (l1p134) [below=2cm of l1t] {$\\langle \\ell_1, \\lbrace \\psi_1, \\psi_3, \\psi_4 \\rbrace \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9631"}},{"Add":{"text":" \\node[state,rwth] (l1p124) [right=4cm of l1p134] {$\\langle \\ell_1, \\lbrace \\psi_1, \\psi_2, \\psi_4 \\rbrace \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9752"}},{"Add":{"text":" \\node[state,rwth] (l2p5) [left=4cm of l1p134] {$\\langle \\ell_2, \\lbrace \\psi_5 \\rbrace \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9854"}},{"Add":{"text":" ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9859"}},{"Add":{"text":" \\node[state,rot] (l2p1345) [below=2cm of l1p134] {$\\langle \\ell_2, \\lbrace \\psi_1, \\psi_3, \\psi_4, \\psi_5 \\rbrace \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:9988"}},{"Add":{"text":" \\node[state,rot] (l2p1245) [right=4cm of l2p1345] {$\\langle \\ell_2, \\lbrace \\psi_1, \\psi_2, \\psi_4, \\psi_5 \\rbrace \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10118"}},{"Add":{"text":" \\node[state,rot] (l1p56) [left=4cm of l2p1345] {$\\langle \\ell_1, \\lbrace \\psi_5, \\psi_6 \\rbrace \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10229"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10230"}},{"Add":{"text":" \\node[state,] (l1p13456) [below=2cm of l2p1345] {$\\langle \\ell_1, \\lbrace \\psi_1, \\psi_3, \\psi_4, \\psi_5, \\psi_6 \\rbrace \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10366"}},{"Add":{"text":" \\node[state,] (l1p12456) [right=4cm of l1p13456] {$\\langle \\ell_1, \\lbrace \\psi_1, \\psi_2, \\psi_4, \\psi_5, \\psi_6 \\rbrace \\rangle$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10503"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10504"}},{"Add":{"text":" % Transitions","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10522"}},{"Add":{"text":" \\path[->] ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10539"}},{"Add":{"text":" (l0t) edge[grun] node (t01) { $t_{0,1}$ } (l1t)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10595"}},{"Add":{"text":" (l1t) edge[rwth] node[swap] (t21) { $t_{2,1}$ } (l2p5)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10658"}},{"Add":{"text":" (l1t) edge[rwth,bend right=20] node[swap] (t1a1) { $t_{1a,1}$ } (l1p134)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10739"}},{"Add":{"text":" (l1t) edge[rwth] node (t1b1) { $t_{1b,1}$ } (l1p124)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10800"}},{"Add":{"text":" (l2p5) edge[rot, bend right=20] node[swap] (t31) { $t_{3,1}$ } (l1p56)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10879"}},{"Add":{"text":" (l1p134) edge[rot] node (t22) { $t_{2,2}$ } (l2p1345)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:10941"}},{"Add":{"text":" (l1p134) edge[rot, bend right=10] node[swap] (t1b2) { $t_{1b,2}$ } (l1p124)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:11025"}},{"Add":{"text":" (l1p134) edge[rot,loop above,transform canvas={xshift=5mm}] node[swap] (t1a2) { $t_{1a,2}$ } ()","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:11129"}},{"Add":{"text":" (l1p134) edge[rot] node (t22) { $t_{2,2}$ } (l2p1345)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:11191"}},{"Add":{"text":" (l1p124) edge[rot, bend right=10] node[swap] (t1a3) { $t_{1a,3}$ } (l1p134)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:11275"}},{"Add":{"text":" (l1p124) edge[rot,loop above] node (t1b3) { $t_{1b,3}$ } ()","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:11343"}},{"Add":{"text":" (l1p124) edge[rot] node (t23) { $t_{2,3}$ } (l2p1245)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:11405"}},{"Add":{"text":" (l1p56) edge[rot, bend right=20] node[swap] (t25) { $t_{2,5}$ } (l2p5)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:11484"}},{"Add":{"text":" (l1p56) edge[bend right=30] node (t1a5) { $t_{1a,5}$ } (l1p13456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:11558"}},{"Add":{"text":" (l1p56) edge[out=270,in=210] node[swap] (t1b5) { $t_{1b,5}$ } (l1p12456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:11639"}},{"Add":{"text":" (l2p1345) edge[bend right=20] node[swap] (t32) { $t_{3,2}$ } (l1p13456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:11719"}},{"Add":{"text":" (l1p13456) edge[bend right=20] node[swap] (t24) { $t_{2,4}$ } (l2p1345)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:11799"}},{"Add":{"text":" (l2p1245) edge[bend right=20] node[swap] (t33) { $t_{3,3}$ } (l1p12456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:11879"}},{"Add":{"text":" (l1p12456) edge[bend right=20] node[swap] (t26) { $t_{2,6}$ } (l2p1245)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:11959"}},{"Add":{"text":" (l1p13456) edge[loop above,transform canvas={xshift=12mm}] node[swap] (t1a4) { $t_{1a,4}$ } ()","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:12062"}},{"Add":{"text":" (l1p12456) edge[loop above,transform canvas={xshift=12mm}] node[swap] (t1b6) { $t_{1b,6}$ } ()","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:12165"}},{"Add":{"text":" (l1p13456) edge[bend left=10] node (t1b4) { $t_{1b,4}$ } (l1p12456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:12241"}},{"Add":{"text":" (l1p12456) edge[bend left=10] node (t1a6) { $t_{1a,6}$ } (l1p13456)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:12317"}},{"Add":{"text":" 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{$G_2$};","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14339"}},{"Add":{"text":"\\end{tikzpicture}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14357"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14358"}}]}},{"Edit":{"inode":{"change":4,"pos":7593},"path":"ch6_appendix.tex","line":2,"atom":{"NewVertex":{"up_context":[{"change":4,"pos":7616}],"down_context":[{"change":4,"pos":16849}],"flag":"BLOCK","start":14360,"end":14377,"inode":{"change":4,"pos":7593}}},"contents":[{"Add":{"text":"\\label{sec:apdx}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14376"}}]}},{"Replacement":{"inode":{"change":4,"pos":7792},"path":"ch4_implementation.tex","line":32,"ins":{"up_context":[{"change":2,"pos":2233}],"down_context":[{"change":2,"pos":2315}],"flag":"BLOCK","start":14378,"end":14462,"inode":{"change":4,"pos":7792}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | 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This motivates control-flow","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14700"}},{"Add":{"text":"refinement in order to find a program that is equivalent to the original one,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14778"}},{"Add":{"text":"which opens the possibility for the analysis to find tighter runtime-complexity","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14858"}}],"del_contents":[{"Del":{"text":"\\gls{koat}s analysis goal is to find tight upper bounds for \\glspl{pip}. The","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:3970"}},{"Del":{"text":"analysis techniques used such as \\acrfull{mprf}, have their limits and cannot","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:25974"}},{"Del":{"text":"always find tight bounds for every program, which motivates control-flow","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:26047"}},{"Del":{"text":"refinement in order to find an equivalent program to the original one, that","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:26123"}},{"Del":{"text":"opens the possibility for the analysis to find tighter runtime-complexity","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:26197"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":9,"ins":{"up_context":[{"change":4,"pos":72}],"down_context":[{"change":4,"pos":4049}],"flag":"BLOCK","start":14860,"end":14938,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":72},"to":{"change":4,"start":3972,"end":4049},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"Recall Definition \\ref{def:runtime_complexity}; the newly constructed refined","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:14937"}}],"del_contents":[{"Del":{"text":"Recall Definition\\ref{def:runtime_complexity}; the newly constructed refined","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:4048"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":11,"ins":{"up_context":[{"change":4,"pos":4123}],"down_context":[{"change":4,"pos":4199}],"flag":"BLOCK","start":14939,"end":15020,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":4123},"to":{"change":4,"start":4123,"end":4199},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"original program, which implies that the runtime-complexity bounds found for the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15019"}}],"del_contents":[{"Del":{"text":"original program which implies that runtime-complexity bounds found for the","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:4198"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":13,"ins":{"up_context":[{"change":4,"pos":4272}],"down_context":[{"change":4,"pos":557}],"flag":"BLOCK","start":15021,"end":15357,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":4272},"to":{"change":4,"start":4272,"end":4708},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"Naturally, the control-flow refinement will add and replace transitions and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15096"}},{"Add":{"text":"locations. We will show that each transition in the refined program is similar","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15175"}},{"Add":{"text":"to a transition in the original program and consequently every run in the new","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15253"}},{"Add":{"text":"program has a run of equal run~time complexity and probability in the original","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15332"}},{"Add":{"text":"program and vice-versa.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15356"}}],"del_contents":[{"Del":{"text":"Naturally the control-flow refinement will add and replace transitions and","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:4346"}},{"Del":{"text":"locations, hence the number runtime-complexity bounds and size-bounds of each","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:4424"}},{"Del":{"text":"transition won't be preserved. However we will show, that each transition in the","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:4505"}},{"Del":{"text":"refined program is a similar to a transition in the original program and hence","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:4584"}},{"Del":{"text":"every run in the new program has a run of equal runtime-complexity and","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:4655"}},{"Del":{"text":"probability in the original program and vice-versa.","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:4707"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":20,"ins":{"up_context":[{"change":4,"pos":1266}],"down_context":[{"change":4,"pos":1497}],"flag":"BLOCK","start":15358,"end":15584,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":1266},"to":{"change":4,"start":26945,"end":26979},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":26979},"to":{"change":4,"start":4709,"end":4773},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":4773},"to":{"change":4,"start":27057,"end":27186},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":27057},"to":{"change":4,"start":27057,"end":27186},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"evaluation technique presented by \\citeauthor{Domenech19} \\cite{Domenech19} for","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15437"}},{"Add":{"text":"\\glspl{pip}. The partial evaluation will be constructed in a similar way but","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15514"}},{"Add":{"text":"explicitly taking probability and general transitions into account. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15583"}}],"del_contents":[{"Del":{"text":"evaluation technique presented by","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:26978"}},{"Del":{"text":"\\citeauthor{Domenech19}\\cite{Domenech19} for \\acrlong{pip}. The","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:4772"}},{"Del":{"text":"partial evaluation will be constructed in a similar way but explicitly taking","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:27134"}},{"Del":{"text":"probability and general transitions into account. ","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:27185"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":24,"ins":{"up_context":[{"change":4,"pos":1498}],"down_context":[{"change":4,"pos":27336}],"flag":"BLOCK","start":15585,"end":15974,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":1498},"to":{"change":4,"start":1498,"end":1650},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":1650},"to":{"change":4,"start":27187,"end":27336},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"The construction is done in three steps. First, an unfolding operation is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15658"}},{"Add":{"text":"presented. Second, an evaluation step is presented, which uses the unfolding.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15736"}},{"Add":{"text":"Third, the evaluation step is repeated until a fix-point is found. The unfolding","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15817"}},{"Add":{"text":"and evaluation will be adapted multiple times in order to 1. find finite","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15890"}},{"Add":{"text":"partial evaluation graphs, and 2. evaluate only the relevant parts of the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15964"}},{"Add":{"text":"program.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:15973"}}],"del_contents":[{"Del":{"text":"The construction is done in three steps. First the single evaluation step","pos":"KVSF5AFJEIMIYAC264OUAA56SGP5MFAUG3AECKOTUPIXPGULVFQQC:1571"}},{"Del":{"text":"(unfolding) of transitions in a probabilistic program is presented. Second, a","pos":"KVSF5AFJEIMIYAC264OUAA56SGP5MFAUG3AECKOTUPIXPGULVFQQC:1649"}},{"Del":{"text":"possibly infinite evaluation graph is constructed that will finally be collapsed","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:27267"}},{"Del":{"text":"into a finite evaluation graph by the use of an abstraction layer. ","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:27335"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":31,"ins":{"up_context":[{"change":4,"pos":27337}],"down_context":[{"change":4,"pos":27490}],"flag":"BLOCK","start":15975,"end":16131,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":27337},"to":{"change":4,"start":27337,"end":27490},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"Afterwards, we will show that the partial evaluation is sound for \\glspl{pip}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16052"}},{"Add":{"text":"and prove that the expected runtime complexity is preserved and thus also the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16130"}}],"del_contents":[{"Del":{"text":"Afterwards we will show that the partial evaluation is sound for \\gls{pip} and","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:27415"}},{"Del":{"text":"prove that the expected runtime-complexity is preserved and thus also the","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:27489"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":38,"ins":{"up_context":[{"change":4,"pos":27606}],"down_context":[{"change":4,"pos":27716}],"flag":"BLOCK","start":16132,"end":16247,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":27606},"to":{"change":4,"start":27606,"end":27716},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\input{figures/ch3_p3_program}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16170"}},{"Add":{"text":" \\caption{Representation of $\\Prog_3$, a variation of $\\Prog_2$ from","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16246"}}],"del_contents":[{"Del":{"text":" \\input{figures/ch1_prob}","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:27638"}},{"Del":{"text":" \\caption{Representation of $\\Prog_3$, a variataion of $\\Prog_2$ from","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:27715"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":54,"ins":{"up_context":[{"change":4,"pos":28626}],"down_context":[{"change":4,"pos":29165}],"flag":"BLOCK","start":16248,"end":16787,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":28626},"to":{"change":4,"start":28626,"end":29165},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\{t_2\\}$, and assigns the value for $w$ accordingly. The program can","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16320"}},{"Add":{"text":" alternate between the loops non-deterministically after every iteration.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16397"}},{"Add":{"text":" Since both loops are independent of one another, the non-determinism does","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16475"}},{"Add":{"text":" not change anything about the expected runtime complexity of the program. On","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16556"}},{"Add":{"text":" the general transition $g_1 = \\{t_{1a}, t_{1b}\\}$ at location $\\ell_1$ the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16635"}},{"Add":{"text":" program branches probabilistically, with a probability of $p=0.5$. On the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16713"}},{"Add":{"text":" transition $t_3$, the value with which $y$ is decremented is sampled","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16786"}}],"del_contents":[{"Del":{"text":" \\{t_2\\}$, and assigns the value for $w$ accordingly. ","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:28683"}},{"Del":{"text":" The program can alter between the loops non-deterministically after every","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:28761"}},{"Del":{"text":" iteration. Since both loops are independent of one another, the","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:28829"}},{"Del":{"text":" non-determinism doesn't change anything about the expected","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:28892"}},{"Del":{"text":" runtime-complexity of the program.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:28931"}},{"Del":{"text":" On the general transition $g_1 = \\{t_{1a}, t_{1b}\\}$ at location $\\ell_1$","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:29009"}},{"Del":{"text":" the program branches probabilistically, with a probability of $p=0.5$. On","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:29087"}},{"Del":{"text":" the transition $t_3$, the value with which $y$ is decremented is sampled","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:29164"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":65,"ins":{"up_context":[{"change":4,"pos":10500}],"down_context":[{"change":4,"pos":2054}],"flag":"BLOCK","start":16788,"end":16890,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":10500},"to":{"change":4,"start":2690,"end":2735},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":2735},"to":{"change":4,"start":5010,"end":5067},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"\\section{Partial Evaluation of Probabilistic Programs}\\label{sec:theory}\\label{sec:partialevaluation}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16889"}}],"del_contents":[{"Del":{"text":"\\section{Partial evaluation on probabilistic","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:2734"}},{"Del":{"text":"programs}\\label{sec:theory}\\label{sec:partialevaluation}","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:5066"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":67,"ins":{"up_context":[{"change":4,"pos":2055}],"down_context":[{"change":4,"pos":29376}],"flag":"BLOCK","start":16891,"end":16967,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":2055},"to":{"change":4,"start":29301,"end":29376},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"During a normal execution, a program visits locations one after another and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:16966"}}],"del_contents":[{"Del":{"text":"During a normal execution a program visits locations one after another and","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:29375"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":71,"ins":{"up_context":[{"change":4,"pos":29607}],"down_context":[{"change":4,"pos":5146}],"flag":"BLOCK","start":16968,"end":17309,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":29607},"to":{"change":4,"start":29607,"end":29867},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":29867},"to":{"change":4,"start":5068,"end":5146},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"impossible: First, the number of starting states is infinite, and second the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17044"}},{"Add":{"text":"program is not guaranteed to terminate, hence the simulation of a specific run","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17123"}},{"Add":{"text":"might not terminate either.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17151"}},{"Add":{"text":"For non-deterministic and probabilistic programs, the problem becomes even more","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17231"}},{"Add":{"text":"complex, since the transitions taken do not only depend on the current state","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17308"}}],"del_contents":[{"Del":{"text":"impossible: First, the number of starting states is infinite; second the program","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:29687"}},{"Del":{"text":"is not guaranteed to terminate and hence the simulation of a specific run might","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:29767"}},{"Del":{"text":"not terminate either.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:29789"}},{"Del":{"text":"For non-deterministic and probabilistic programs, the problem gets even more","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:29866"}},{"Del":{"text":"complex, since the transition taken does not only depend on the current state","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:5145"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":77,"ins":{"up_context":[{"change":4,"pos":5220}],"down_context":[{"change":4,"pos":1532}],"flag":"BLOCK","start":17310,"end":17811,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":5220},"to":{"change":4,"start":5220,"end":5709},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"simulating every single possible run, it would be more efficient to treat","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17383"}},{"Add":{"text":"\\emph{similar} states as one. Instead of simulating a single assignment, the set","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17464"}},{"Add":{"text":"of all possible assignments after a step is described by a set of constraints.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17543"}},{"Add":{"text":"The resulting graph is called a \\emph{partial evaluation graph}. It was","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17615"}},{"Add":{"text":"formalised for \\glspl{chc} by","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17645"}},{"Add":{"text":"\\citeauthor{gallagher2019eptcs} \\cite{gallagher2019eptcs}. In this thesis, the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17724"}},{"Add":{"text":"definitions are adapted to better fit the algorithm and the formalism of","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17797"}},{"Add":{"text":"\\Glspl{pip}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17810"}}],"del_contents":[{"Del":{"text":"simulating every single possible run, it would be nice to treat \\emph{similar}","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:5298"}},{"Del":{"text":"states as one. Instead of simulating a single assignment, the set of all","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:5371"}},{"Del":{"text":"possible assignments after a step is described by a set of constraints. The","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:5447"}},{"Del":{"text":"resulting graph is called a \\emph{partial evaluation graph}. It was formalized","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:5526"}},{"Del":{"text":"for \\glspl{chc} by \\citeauthor{gallagher2019eptcs}\\cite{gallagher2019eptcs}. In","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:5606"}},{"Del":{"text":"this thesis the definitions are adapted to better fit the algorithm and the","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:5682"}},{"Del":{"text":"formalism of \\Glspl{pip}.","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:5708"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":98,"ins":{"up_context":[{"change":4,"pos":5623}],"down_context":[{"change":4,"pos":5701}],"flag":"BLOCK","start":17812,"end":17892,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":5623},"to":{"change":4,"start":5623,"end":5701},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"all partial evaluation graphs for a program $\\Prog$ is defined as $\\PEG_\\Prog =","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17891"}}],"del_contents":[{"Del":{"text":"all partial evaluation graphs for a program $\\Prog$ is defined as $\\G_\\Prog =","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:5700"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":103,"ins":{"up_context":[{"change":4,"pos":32312}],"down_context":[{"change":4,"pos":5126}],"flag":"BLOCK","start":17893,"end":18467,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":32312},"to":{"change":4,"start":5812,"end":6154},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"\\ell, \\texttt{true}\\rangle \\in \\vers_\\Prog$. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17938"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17939"}},{"Add":{"text":"\\begin{definition}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:17958"}},{"Add":{"text":" We define the lift $G\\lift\\in \\PEG$ of a graph $G \\in \\G$ denoted as","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18031"}},{"Add":{"text":" \\begin{equation*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18053"}},{"Add":{"text":" G\\lift = (\\set{\\langle\\ell,\\texttt{true}\\rangle}{\\ell \\in V(G)},","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18126"}},{"Add":{"text":" \\set{(\\langle\\ell,\\texttt{true}\\rangle,p,\\tau,\\eta,\\langle\\ell',","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18199"}},{"Add":{"text":" \\texttt{true})}{t \\in E(G)})","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18236"}},{"Add":{"text":" \\end{equation*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18256"}},{"Add":{"text":"\\end{definition}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18273"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18274"}},{"Add":{"text":"\\begin{example}\\label{ex:lift}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18305"}},{"Add":{"text":" The lift of the graph $G_{\\Prog_3}$ of the program $\\Prog_3$ is displayed in","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18386"}},{"Add":{"text":" \\fref{fig:ex_lift}. It has the very same structure as the original program.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18466"}}],"del_contents":[{"Del":{"text":"\\ell, \\texttt{true}\\rangle \\in \\vers_\\Prog$. Obviously, the lift of a program is","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:5892"}},{"Del":{"text":"not useful for the analysis, as it doesn't capture any interesting properties of","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:5973"}},{"Del":{"text":"the program, besides what is already known in the original program. In the","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:6048"}},{"Del":{"text":"following we will present a construction for a partial evaluation graph using","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:6126"}},{"Del":{"text":"unfolding and abstraction.","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:6153"}}]}},{"Edit":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":118,"atom":{"NewVertex":{"up_context":[{"change":4,"pos":5127}],"down_context":[{"change":4,"pos":3723}],"flag":"BLOCK","start":18468,"end":18981,"inode":{"change":4,"pos":7949}}},"contents":[{"Add":{"text":" \\begin{figure}[ht]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18490"}},{"Add":{"text":" \\centering","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18509"}},{"Add":{"text":" \\input{figures/ch3_p3_lift}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18545"}},{"Add":{"text":" \\caption{Lift of the graph $G_{\\Prog_3}$ of the program","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18609"}},{"Add":{"text":" $\\Prog_3$.\\label{fig:ex_lift}}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18648"}},{"Add":{"text":" \\end{figure}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18665"}},{"Add":{"text":"\\end{example}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18679"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18680"}},{"Add":{"text":"Obviously, the lift of a program is not useful for the analysis, as it does not","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18760"}},{"Add":{"text":"capture any interesting properties of the program, besides what is already","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18835"}},{"Add":{"text":"known in the original program. In the following we will present a construction","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18914"}},{"Add":{"text":"for a partial evaluation graph using unfolding and abstraction. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18979"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:18980"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":142,"ins":{"up_context":[{"change":4,"pos":5751}],"down_context":[{"change":4,"pos":5894}],"flag":"BLOCK","start":18982,"end":19125,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":5751},"to":{"change":4,"start":5751,"end":5894},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"probabilistic sampling is over-approximated with a non-probabilistic update","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19057"}},{"Add":{"text":"$\\tilde{\\eta}: \\PV \\rightarrow \\Z[\\V]$ and a constraint denoted as","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19124"}}],"del_contents":[{"Del":{"text":"probabilistic sampling is over-approximated with an non-probabilistic update","pos":"KVSF5AFJEIMIYAC264OUAA56SGP5MFAUG3AECKOTUPIXPGULVFQQC:5827"}},{"Del":{"text":"$\\tilde{\\eta}: \\PV \\rightarrow Z[\\V]$ and a constraint denoted as","pos":"KVSF5AFJEIMIYAC264OUAA56SGP5MFAUG3AECKOTUPIXPGULVFQQC:5893"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":145,"ins":{"up_context":[{"change":4,"pos":5975}],"down_context":[{"change":4,"pos":6052}],"flag":"BLOCK","start":19126,"end":19204,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":5975},"to":{"change":4,"start":5975,"end":6052},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"$\\eta$ are replaced by fresh temporary variables that are then constrained by","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19203"}}],"del_contents":[{"Del":{"text":"$\\eta$ are replaced by fresh temporary variables that are then constraint by","pos":"KVSF5AFJEIMIYAC264OUAA56SGP5MFAUG3AECKOTUPIXPGULVFQQC:6051"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":156,"ins":{"up_context":[{"change":4,"pos":5748}],"down_context":[{"change":4,"pos":6605}],"flag":"BLOCK","start":19205,"end":19252,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":4,"pos":5748},"to":{"change":4,"start":6559,"end":6605},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":6559},"to":{"change":4,"start":6559,"end":6605},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\texttt{false} & \\text{otherwise.}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19251"}}],"del_contents":[{"Del":{"text":" \\texttt{false} & \\text{otherwise}","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:6604"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":185,"ins":{"up_context":[{"change":4,"pos":6863}],"down_context":[{"change":4,"pos":6936}],"flag":"BLOCK","start":19253,"end":19327,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":6863},"to":{"change":4,"start":6863,"end":6936},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" the inverse does not hold. The satisfiability check marks unreachable","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19326"}}],"del_contents":[{"Del":{"text":" the inverse doesn't hold. The satisfiability check marks unreachable","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:6935"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":191,"ins":{"up_context":[{"change":4,"pos":34062}],"down_context":[{"change":4,"pos":34133}],"flag":"BLOCK","start":19328,"end":19398,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":34062},"to":{"change":4,"start":34062,"end":34133},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" has a probabilistic update $\\eta(y) = y - \\textsc{Bern}(0.5)$ and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19397"}}],"del_contents":[{"Del":{"text":" has an probabilistic update $\\eta(y) = y - \\textsc{Bern}(0.5)$ and","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:34132"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":236,"ins":{"up_context":[{"change":4,"pos":8159}],"down_context":[{"change":4,"pos":8320}],"flag":"BLOCK","start":19399,"end":19621,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":8159},"to":{"change":4,"start":7911,"end":8067},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":8067},"to":{"change":2,"start":9924,"end":9968},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"We define a function $\\evaluate_\\Prog : \\G_\\Prog \\rightarrow \\G_\\Prog $ that","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19475"}},{"Add":{"text":"unfolds all versions $\\langle \\ell, \\varphi \\rangle \\in V(G)$ in the graph $G","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19553"}},{"Add":{"text":"\\in \\G_\\Prog$ with each outgoing transition $t \\in \\TPout(\\ell)$. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19620"}}],"del_contents":[{"Del":{"text":"We define a function $\\evaluate_\\Prog(T) : \\G_\\Prog \\rightarrow \\G_\\Prog $ that","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:7990"}},{"Del":{"text":"unfolds all versions $\\langle \\ell, \\varphi \\rangle$ in the graph with each","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:8066"}},{"Del":{"text":"outgoing transitions $t \\in \\TPout(\\ell)$. ","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:9967"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":242,"ins":{"up_context":[{"change":4,"pos":8190}],"down_context":[{"change":2,"pos":9969}],"flag":"BLOCK","start":19622,"end":19688,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":8190},"to":{"change":4,"start":8190,"end":8256},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\langle \\ell, \\varphi\\rangle \\in V(G) \\text{, and } t \\in","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19687"}}],"del_contents":[{"Del":{"text":" \\langle \\ell, \\varphi\\rangle \\in E(G) \\text{, and } t \\in","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:8255"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":246,"ins":{"up_context":[{"change":4,"pos":21314}],"down_context":[{"change":4,"pos":8351}],"flag":"BLOCK","start":19689,"end":19763,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":21314},"to":{"change":4,"start":8278,"end":8351},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"For a program $\\Prog$ with an initial location $\\ell_0$, the unfolding is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19762"}}],"del_contents":[{"Del":{"text":"For a program $\\Prog$ with a initial location $\\ell_0$, the unfolding is","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:8350"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":257,"ins":{"up_context":[{"change":4,"pos":8811}],"down_context":[{"change":4,"pos":8835}],"flag":"BLOCK","start":19764,"end":19788,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":8811},"to":{"change":4,"start":8811,"end":8835},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\text{ times}}(G_0)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19787"}}],"del_contents":[{"Del":{"text":" \\text{ times}}(T_0)","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:8834"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":261,"ins":{"up_context":[{"change":4,"pos":8932}],"down_context":[{"change":4,"pos":9012}],"flag":"BLOCK","start":19789,"end":19870,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":8932},"to":{"change":4,"start":8932,"end":9012},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"$n \\in \\N$. Let $G^* = \\evaluate_\\Prog^*$ be the limit the evaluation approaches","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19869"}}],"del_contents":[{"Del":{"text":"$n \\in \\N$. Let $G^* = \\evaluate_\\Prog^*$ be the limit the unfolding approaches","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:9011"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":280,"ins":{"up_context":[{"change":4,"pos":9558}],"down_context":[{"change":4,"pos":12761}],"flag":"BLOCK","start":19871,"end":19984,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":4,"pos":9558},"to":{"change":4,"start":4805,"end":4858},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":12696},"to":{"change":4,"start":4805,"end":4858},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":17342},"to":{"change":4,"start":4805,"end":4858},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":170},"to":{"change":4,"start":4805,"end":4858},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":4858},"to":{"change":4,"start":12697,"end":12761},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\input{figures/ch3_p4_program.tex}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19917"}},{"Add":{"text":" \\caption{A Program $\\Prog_4$\\label{fig:ch3_inf_prog}}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:19983"}}],"del_contents":[{"Del":{"text":" \\input{figures/ch3_example1_program.tex}","pos":"HF25ZQRDZXNE2IM3YGE3NAPWGDE3U252H764IBIJSIZS3LRLYIMAC:4857"}},{"Del":{"text":" \\caption{A Program $\\Prog$\\label{fig:ch3_inf_prog}}","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:12760"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":284,"ins":{"up_context":[{"change":4,"pos":9609}],"down_context":[{"change":4,"pos":12952}],"flag":"BLOCK","start":19985,"end":20131,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":4,"pos":9609},"to":{"change":4,"start":4920,"end":4970},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":12841},"to":{"change":4,"start":4920,"end":4970},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":17408},"to":{"change":4,"start":4920,"end":4970},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":4920},"to":{"change":4,"start":4920,"end":4970},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":4970},"to":{"change":4,"start":12842,"end":12952},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\input{figures/ch3_p4_tree.tex}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20028"}},{"Add":{"text":" \\caption{Extract of the infinite evaluation \\\\graph of $\\Prog_4$\\label{fig:ch3_inf_tree}}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20130"}}],"del_contents":[{"Del":{"text":" \\input{figures/ch3_example1_tree.tex}","pos":"HF25ZQRDZXNE2IM3YGE3NAPWGDE3U252H764IBIJSIZS3LRLYIMAC:4969"}},{"Del":{"text":" \\caption{Extract of the infinite evaluation graph of","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:12906"}},{"Del":{"text":" $\\Prog$\\label{fig:ch3_inf_tree}}","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:12951"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":287,"ins":{"up_context":[{"change":4,"pos":12982}],"down_context":[{"change":4,"pos":13080}],"flag":"BLOCK","start":20132,"end":20239,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":12982},"to":{"change":4,"start":6081,"end":6178},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\caption{A simple program $\\Prog_4$ with an infinite evaluation","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20203"}},{"Add":{"text":" graph.\\label{fig:ch3_inf}}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20238"}}],"del_contents":[{"Del":{"text":" \\caption{A simple example program with an infinite evaluation graph.\\label{fig:ch3_inf}}","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:6177"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":291,"ins":{"up_context":[{"change":4,"pos":2267}],"down_context":[{"change":4,"pos":6179}],"flag":"BLOCK","start":20240,"end":20378,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":2267},"to":{"change":4,"start":9610,"end":9738},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" Consider the program $\\Prog_4$ shown in \\fref{fig:ch3_inf_prog}. On the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20315"}},{"Add":{"text":" initial state $s_0 \\in \\Sigma$ with $s_0(x) = 0$, the run","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20377"}}],"del_contents":[{"Del":{"text":" Consider the program shown in \\fref{fig:ch3_inf_prog}. On the initial state","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:9689"}},{"Del":{"text":" $s_0 \\in \\Sigma$ with $s_0(x) = 0$, The run","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:9737"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":296,"ins":{"up_context":[{"change":4,"pos":9974}],"down_context":[{"change":4,"pos":10034}],"flag":"BLOCK","start":20379,"end":20449,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":9974},"to":{"change":4,"start":9974,"end":10034},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" behaviour, and creates the infinite evaluation graph displayed in","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20448"}}],"del_contents":[{"Del":{"text":" behavior, and creates the evaluation graph displayed in","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:10033"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":300,"ins":{"up_context":[{"change":4,"pos":6419}],"down_context":[{"change":4,"pos":6498}],"flag":"BLOCK","start":20450,"end":20529,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":6419},"to":{"change":4,"start":6419,"end":6498},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" evaluation step, unfolds $\\langle \\ell_1, \\textit{true}\\rangle$ over $t_3$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20528"}}],"del_contents":[{"Del":{"text":" unfolding steps, unfolds $\\langle \\ell_1, \\textit{true}\\rangle$ over $t_3$","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:6497"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":302,"ins":{"up_context":[{"change":4,"pos":6572}],"down_context":[{"change":4,"pos":6652}],"flag":"BLOCK","start":20530,"end":20610,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":6572},"to":{"change":4,"start":6572,"end":6652},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" x>1\\rangle$. $\\ell_2$ has no outgoing transitions. Thus, no new transitions","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20609"}}],"del_contents":[{"Del":{"text":" x>1\\rangle$. $\\ell_2$ has no outgoing transitions, hence no new transitions","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:6651"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":305,"ins":{"up_context":[{"change":4,"pos":6809}],"down_context":[{"change":4,"pos":6889}],"flag":"BLOCK","start":20611,"end":20690,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":6809},"to":{"change":4,"start":6809,"end":6889},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" within $x > 2$. The new version is unfolded over and over again, until the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20689"}}],"del_contents":[{"Del":{"text":" within $x > 2$. The new version is unfolded over and after again, until the","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:6888"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":325,"ins":{"up_context":[{"change":2,"pos":10226}],"down_context":[{"change":2,"pos":10301}],"flag":"BLOCK","start":20691,"end":20762,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":10226},"to":{"change":2,"start":10226,"end":10301},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"from a finite set of properties, that are all entailed by the original","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20761"}}],"del_contents":[{"Del":{"text":"from a set finite set of properties, that are all entailed by the original","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:10300"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":332,"ins":{"up_context":[{"change":2,"pos":10589}],"down_context":[{"change":2,"pos":10667}],"flag":"BLOCK","start":20763,"end":20885,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":10589},"to":{"change":2,"start":10589,"end":10667},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" 2^{\\Psi \\union \\neg \\Psi} \\union \\{\\texttt{true},\\texttt{false}\\}$ which is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20842"}},{"Add":{"text":" also finite. The abstraction function","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20884"}}],"del_contents":[{"Del":{"text":" 2^{\\Psi \\union \\neg \\Psi}$ which is also finite. The abstraction function","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:10666"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":373,"ins":{"up_context":[{"change":4,"pos":4778}],"down_context":[{"change":4,"pos":5170}],"flag":"BLOCK","start":20886,"end":21092,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":4778},"to":{"change":4,"start":12468,"end":12666},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"% \\begin{lemma}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20901"}},{"Add":{"text":"% Let $\\alpha: \\Phi \\rightarrow \\A$ be an abstraction function over a finite","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:20982"}},{"Add":{"text":"% abstraction space $\\A\\subset\\C$. The partial evaluation graph $G_\\alpha^*$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21063"}},{"Add":{"text":"% exists and is finite.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21091"}}],"del_contents":[{"Del":{"text":"\\begin{lemma}","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:12481"}},{"Del":{"text":" Let $\\alpha: \\Phi \\rightarrow \\A$ be an abstraction function over a finite","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:12560"}},{"Del":{"text":" abstraction space $\\A\\subset\\C$. The partial evaluation graph $G_\\alpha^*$","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:12639"}},{"Del":{"text":" exists and is finite.","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:12665"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":378,"ins":{"up_context":[{"change":4,"pos":5171}],"down_context":[{"change":4,"pos":5666}],"flag":"BLOCK","start":21093,"end":21912,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":5171},"to":{"change":4,"start":12667,"end":13074},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":13074},"to":{"change":2,"start":11462,"end":11618},"introduced_by":2},{"previous":"","flag":"DELETED","from":{"change":2,"pos":11618},"to":{"change":4,"start":13228,"end":13458},"introduced_by":2},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":13228},"to":{"change":4,"start":13228,"end":13458},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"% \\begin{proof}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21112"}},{"Add":{"text":"% The abstraction $\\alpha$ can map to only finitely many values in the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21191"}},{"Add":{"text":"% finite abstraction space $\\A$. Every evaluation step adds only versions","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21273"}},{"Add":{"text":"% from a reduced and finite set of version $\\Loc_\\Prog \\times \\A \\subset","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21354"}},{"Add":{"text":"% \\vers_\\Prog$. Hence the set of vertices of partial evaluation graph","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21432"}},{"Add":{"text":"% equally finite. A transition is added after a version $\\langle \\ell,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21511"}},{"Add":{"text":"% \\_\\rangle$ for every outgoing transitions $t \\in \\TPout(\\ell)$ only. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21591"}},{"Add":{"text":"% Since $\\TPout(\\ell) \\subseteq \\T_\\Prog$ is finite, only finitely many","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21671"}},{"Add":{"text":"% transitions can exist between any two versions.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21729"}},{"Add":{"text":"% The graph grows with every evaluation step, hence the limit exists and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21810"}},{"Add":{"text":"% it must be reached in a finite number of evaluation steps.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21879"}},{"Add":{"text":"% \\end{proof}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21897"}},{"Add":{"text":"% \\end{lemma}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21911"}}],"del_contents":[{"Del":{"text":" \\begin{proof}","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:12684"}},{"Del":{"text":" The abstraction $\\alpha$ can map to only finitely many values in the","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:12761"}},{"Del":{"text":" finite abstraction space $\\A$. Every evaluation step adds only versions","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:12841"}},{"Del":{"text":" from a reduced and finite set of version $\\Loc_\\Prog \\times \\A \\subset","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:12920"}},{"Del":{"text":" \\vers_\\Prog$. Hence the set of vertices of partial evaluation graph","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:12996"}},{"Del":{"text":" equally finite. A transition is added after a version $\\langle \\ell,","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:13073"}},{"Del":{"text":" \\_\\rangle$ for every outgoing transitions $t \\in \\TPout(\\ell)$ only. ","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:11539"}},{"Del":{"text":" Since $\\TPout(\\ell) \\subseteq \\T_\\Prog$ is finite, only finitely many","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:11617"}},{"Del":{"text":" transitions can exist between any two versions.","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:13283"}},{"Del":{"text":" The graph grows with every evaluation step, hence the limit exists and","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:13362"}},{"Del":{"text":" it must be reached in a finite number of evaluation steps.","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:13429"}},{"Del":{"text":" \\end{proof}","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:13445"}},{"Del":{"text":"\\end{lemma}","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:13457"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":397,"ins":{"up_context":[{"change":4,"pos":13684}],"down_context":[{"change":4,"pos":13755}],"flag":"BLOCK","start":21913,"end":21989,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":13684},"to":{"change":4,"start":13684,"end":13755},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" $\\psi_1,\\psi_2,\\psi_3 \\in \\C$ and an abstraction space $\\A_\\Psi$ where ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:21988"}}],"del_contents":[{"Del":{"text":" $\\psi_1,\\psi_2,\\psi_3 \\in \\C$ and an abstraction space $\\A$ where ","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:13754"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":400,"ins":{"up_context":[{"change":4,"pos":13802}],"down_context":[{"change":4,"pos":14225}],"flag":"BLOCK","start":21990,"end":22554,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":13802},"to":{"change":4,"start":13802,"end":14225},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\begin{aligned}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22017"}},{"Add":{"text":" \\psi_1 &= 1 \\leq x, \\hspace{1cm} \\neg\\psi_1 = 1 > x,\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22088"}},{"Add":{"text":" \\psi_2 &= 2 \\leq x, \\hspace{1cm} \\neg\\psi_2 = 2 > x,\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22151"}},{"Add":{"text":" \\psi_3 &= x \\leq 3, \\hspace{1cm} \\neg\\psi_3 = x > 3,\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22214"}},{"Add":{"text":" \\psi_4 &= x \\leq 4, \\hspace{1cm} \\neg\\psi_4 = x > 4,\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22277"}},{"Add":{"text":" \\psi_5 &= y > 0,\\hspace{1cm} \\neg\\psi_5 = y \\leq 0,\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22339"}},{"Add":{"text":" \\psi_6 &= y \\geq 0,\\hspace{1cm} \\neg\\psi_6 = y < 0, \\text{ and}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22419"}},{"Add":{"text":" \\end{aligned}\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22447"}},{"Add":{"text":" \\A_\\Psi =","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22465"}},{"Add":{"text":" \\{\\psi_1,\\neg\\psi_1,\\dots,\\psi_6,\\neg\\psi_6,\\texttt{true},\\texttt{false}\\}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22553"}}],"del_contents":[{"Del":{"text":" \\psi_1 = 1 \\leq x, \\hspace{1cm} \\neg\\psi_1 = 1 > x\\\\","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:13862"}},{"Del":{"text":" \\psi_2 = 2 \\leq x, \\hspace{1cm} \\neg\\psi_2 = 2 > x\\\\","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:13923"}},{"Del":{"text":" \\psi_3 = x \\leq 3, \\hspace{1cm} \\neg\\psi_3 = x > 3\\\\","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:13984"}},{"Del":{"text":" \\psi_4 = x \\leq 4, \\hspace{1cm} \\neg\\psi_4 = x > 4\\\\","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:14045"}},{"Del":{"text":" \\psi_5 = y > 0,\\hspace{1cm} \\neg\\psi_5 = y \\leq 0\\\\","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:14105"}},{"Del":{"text":" \\psi_6 = y \\geq 0,\\hspace{1cm} \\neg\\psi_6 = y < 0\\\\","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:14165"}},{"Del":{"text":" \\A = \\{\\psi_1,\\neg\\psi_1,\\dots,\\psi_6,\\neg\\psi_6\\}","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:14224"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":412,"ins":{"up_context":[{"change":4,"pos":14268}],"down_context":[{"change":4,"pos":14342}],"flag":"BLOCK","start":22555,"end":22650,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":14268},"to":{"change":2,"start":11659,"end":11734},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" The property based abstraction $\\alpha_\\Psi$ uses the abstraction space","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22630"}},{"Add":{"text":" $\\A_\\Psi$. The","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22649"}}],"del_contents":[{"Del":{"text":" The abstraction function $\\alpha$ uses the abstraction space $\\A$. The","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:11733"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":419,"ins":{"up_context":[{"change":4,"pos":14573}],"down_context":[{"change":4,"pos":14652}],"flag":"BLOCK","start":22651,"end":22729,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":14573},"to":{"change":4,"start":14573,"end":14652},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" On the first evaluation step, the transition $t_0$ is unfolded yielding a","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22728"}}],"del_contents":[{"Del":{"text":" On the first evaluation step, the transition $t_0$ is unfolded, yielding a","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:14651"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":434,"ins":{"up_context":[{"change":4,"pos":15363}],"down_context":[{"change":4,"pos":15426}],"flag":"BLOCK","start":22730,"end":22800,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":15363},"to":{"change":2,"start":11807,"end":11876},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\eta_{1b},\\langle \\ell_1, \\{\\psi_1,\\psi_2,\\psi_4\\}\\rangle) \\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22799"}}],"del_contents":[{"Del":{"text":" \\eta_{1b},\\langle \\ell_1, \\{\\psi_1\\psi_2,\\psi_4\\}\\rangle) \\\\","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:11875"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":458,"ins":{"up_context":[{"change":2,"pos":12680}],"down_context":[{"change":2,"pos":12755}],"flag":"BLOCK","start":22801,"end":22876,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":12680},"to":{"change":2,"start":12680,"end":12755},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" t_{1b,3} &= (\\langle \\ell_1, \\{\\psi_1,\\psi_3,\\psi_4\\}\\rangle, 0.5,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22875"}}],"del_contents":[{"Del":{"text":" t_{1b,3} &= (\\langle \\ell_1, \\{\\psi_1,\\psi_2,\\psi_4\\}\\rangle, 0.5,","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:12754"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":460,"ins":{"up_context":[{"change":2,"pos":12829}],"down_context":[{"change":2,"pos":12874}],"flag":"BLOCK","start":22877,"end":22929,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":12829},"to":{"change":2,"start":12829,"end":12874},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\ell_1, \\{\\psi_1,\\psi_2,\\psi_4\\}\\rangle) \\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:22928"}}],"del_contents":[{"Del":{"text":" \\ell_1, \\{\\psi_1,\\psi_2\\}\\rangle) \\\\","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:12873"}}]}},{"Edit":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":469,"atom":{"EdgeMap":{"edges":[{"previous":"BLOCK","flag":"BLOCK | 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$t_{1a,2},t_{1b,2},t_{1a,3}$, and $t_{1b,3}$ transition to already","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23010"}},{"Add":{"text":" existing versions $\\langle \\ell_1, \\{\\psi_1,\\psi_3,\\psi_4\\}\\rangle$ and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23086"}},{"Add":{"text":" $\\langle \\ell_1, \\{\\psi_1,\\psi_2,\\psi_4\\}\\rangle$.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23141"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":474,"ins":{"up_context":[{"change":4,"pos":17489}],"down_context":[{"change":4,"pos":17598}],"flag":"BLOCK","start":23143,"end":23231,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | 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\\fref{fig:ex_prob_fpe}.\\todo{update","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:17565"}},{"Del":{"text":" figure}","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:17577"}},{"Del":{"text":"","pos":"M3WK5HF3XU5MN2JKJMRJVDMC74F4MZAQ23FTGJRW4CGU5ZL72MHAC:7548"}},{"Del":{"text":" \\begin{figure}","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:17597"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":477,"ins":{"up_context":[{"change":4,"pos":17617}],"down_context":[{"change":4,"pos":17732}],"flag":"BLOCK","start":23232,"end":23354,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":17617},"to":{"change":4,"start":17617,"end":17732},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\input{figures/ch3_p3_full_pe}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23270"}},{"Add":{"text":" \\caption{Partial evaluation graph $G_{\\Prog_3,\\alpha_\\Psi}^*$ of $\\Prog_3$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23353"}}],"del_contents":[{"Del":{"text":" \\input{figures/ch1_prob_fpe}","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:17653"}},{"Del":{"text":" \\caption{Partial evaluation graph $G_{\\Prog_3,\\alpha}^*$ of $\\Prog_3$","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:17731"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":483,"ins":{"up_context":[{"change":2,"pos":13287}],"down_context":[{"change":2,"pos":13394}],"flag":"BLOCK","start":23355,"end":23463,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":13287},"to":{"change":2,"start":13287,"end":13394},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"\\subsection{Localized abstraction}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23389"}},{"Add":{"text":"Example \\ref{ex:global_abstr} illustrates that a program gets large very","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23462"}}],"del_contents":[{"Del":{"text":"\\subsection{Dynamic abstraction}","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:13319"}},{"Del":{"text":"Example \\ref{ex:global_abstr} illustrates that a program get's large very","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:13393"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":510,"ins":{"up_context":[{"change":2,"pos":14649}],"down_context":[{"change":2,"pos":14721}],"flag":"BLOCK","start":23464,"end":23537,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | 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in","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23643"}},{"Add":{"text":" \\fref{fig:ex_localized_pba} It looks very similar to before, but we saved","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23721"}},{"Add":{"text":" some entailment checks on location $\\ell_2$, in return the versions at","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23796"}},{"Add":{"text":" $\\ell_2$ are not collapsed anymore. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23837"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23838"}},{"Add":{"text":" \\begin{figure}[ht]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23861"}},{"Add":{"text":" \\centering","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23880"}},{"Add":{"text":" \\input{figures/ch3_p3_full_pe_localized}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:23929"}},{"Add":{"text":" \\caption{Partial evaluation of $\\Prog_3$ with a localized property based","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24010"}},{"Add":{"text":" abstraction $S_{\\bot,\\Psi_{\\ell_1},\\bot}$.\\label{fig:ex_localized_pba}}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24090"}},{"Add":{"text":" \\end{figure}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24107"}}],"del_contents":[{"Del":{"text":" evaluation step, and just show the final evaluation graph. It looks very","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:16853"}},{"Del":{"text":" similar to before, but we saved some entailment checks on location $\\ell_2$,","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:16934"}},{"Del":{"text":" in return the versions at $\\ell_2$ are not collapsed anymore.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17000"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":576,"ins":{"up_context":[{"change":4,"pos":7589}],"down_context":[{"change":2,"pos":17381}],"flag":"BLOCK","start":24109,"end":24465,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":7589},"to":{"change":2,"start":17302,"end":17381},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"Clearly, a global abstraction function is a special case of a localized ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24181"}},{"Add":{"text":"abstraction function, where every location is abstracted with the same","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24252"}},{"Add":{"text":"abstraction function. How to select properties for the localized property-based","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24332"}},{"Add":{"text":"abstraction is discussed in \\Cref{ch:implementation}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24386"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24387"}},{"Add":{"text":"The property based abstraction crafted in Example \\ref{ex:localized_pba} was","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24464"}}],"del_contents":[{"Del":{"text":"The property based abstraction crafted in Example \\ref{ex:localized_abstr} was","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17380"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":584,"ins":{"up_context":[{"change":2,"pos":17539}],"down_context":[{"change":2,"pos":17616}],"flag":"BLOCK","start":24466,"end":24867,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":17539},"to":{"change":2,"start":17539,"end":17616},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"per loop in the original program. This is formalized by the lemma","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24531"}},{"Add":{"text":"\\ref{lem:localized_pba_finite}. But before, we will adapt the evaluation one","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24608"}},{"Add":{"text":"last time. Instead of evaluating the entire program, the evaluation will unfold","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24688"}},{"Add":{"text":"only transitions in a given component $T$. The component is not required to be","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24767"}},{"Add":{"text":"connected. How to select the component for best results ins discussed in","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24840"}},{"Add":{"text":"\\Cref{ch:implementation}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24866"}}],"del_contents":[{"Del":{"text":"per loop in the original program. This is formalized by the following lemma.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17615"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":591,"ins":{"up_context":[{"change":2,"pos":17617}],"down_context":[{"change":2,"pos":17899}],"flag":"BLOCK","start":24868,"end":25163,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":17617},"to":{"change":2,"start":17617,"end":17899},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"\\begin{definition}[Component-level Evaluation]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24914"}},{"Add":{"text":" Let $\\Prog$ be a \\gls{pip} and $T \\subseteq \\T_\\Prog$ be a component of","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:24990"}},{"Add":{"text":" $\\Prog$. For a localized abstraction $S : \\vers_\\Prog \\rightarrow \\C$ an","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25067"}},{"Add":{"text":" evaluation step $\\evaluate_{\\Prog,S,T} : \\G_\\Prog \\rightarrow \\G_\\Prog$ is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25146"}},{"Add":{"text":" defined as ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25162"}}],"del_contents":[{"Del":{"text":"\\begin{lemma}","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17630"}},{"Del":{"text":" Let $S_\\alpha$ be a dynamic abstraction function. The partial evaluation","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17707"}},{"Del":{"text":" graph $G_{S_\\alpha}^*$ is finite if every loop $L$ in $\\Prog$ contains a","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17784"}},{"Del":{"text":" location $\\ell \\in L$ such that","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17820"}},{"Del":{"text":" $\\set{S_\\alpha(\\langle\\ell,\\varphi\\rangle)}{\\varphi \\in \\Phi}$ is finite.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17898"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":597,"ins":{"up_context":[{"change":2,"pos":17900}],"down_context":[{"change":2,"pos":17967}],"flag":"BLOCK","start":25164,"end":25622,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":17900},"to":{"change":2,"start":17900,"end":17967},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\begin{align*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25182"}},{"Add":{"text":" \\evaluate_{\\Prog,S}(G) = G ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25218"}},{"Add":{"text":" &+ \\set{ \\unfold_{S}(\\langle \\ell, \\varphi \\rangle, t)}{ \\langle \\ell,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25297"}},{"Add":{"text":" \\varphi\\rangle \\in E(G) \\text{, and } t \\in \\TPout(\\ell) \\cap T}\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25372"}},{"Add":{"text":" &+ \\set{","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25389"}},{"Add":{"text":" (\\langle\\ell,\\varphi\\rangle,p,\\tau,\\eta,\\langle\\ell',\\texttt{true}\\rangle)}{","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25478"}},{"Add":{"text":" \\langle \\ell, \\varphi\\rangle \\in E(G) \\text{, and } t \\in","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25548"}},{"Add":{"text":" \\TPout(\\ell) \\backslash T}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25587"}},{"Add":{"text":" \\end{align*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25604"}},{"Add":{"text":"\\end{definition}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25621"}}],"del_contents":[{"Del":{"text":" \\begin{proof}","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17917"}},{"Del":{"text":" \\todo{Proof}","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17938"}},{"Del":{"text":" \\end{proof}","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17954"}},{"Del":{"text":"\\end{lemma}","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:17966"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":608,"ins":{"up_context":[{"change":2,"pos":17968}],"down_context":[{"change":2,"pos":18243}],"flag":"BLOCK","start":25623,"end":26005,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":17968},"to":{"change":2,"start":17968,"end":18243},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"The partial evaluation is bootstrapped with the lift of the program with the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25699"}},{"Add":{"text":"component removed.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25718"}},{"Add":{"text":"\\begin{equation*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25736"}},{"Add":{"text":" G_0 = (G_\\Prog - T)\\lift","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25765"}},{"Add":{"text":"\\end{equation*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25781"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25782"}},{"Add":{"text":"The component-level evaluation is then repeated on $G_0$ until a fix-point is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25860"}},{"Add":{"text":"found. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25868"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25869"}},{"Add":{"text":"\\begin{equation*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25887"}},{"Add":{"text":" G_{S,T}^* = \\evaluate_{\\Prog,S}^*(G_0) =","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25932"}},{"Add":{"text":" \\lim_{n\\rightarrow\\infty}\\evaluate_{\\Prog,S}^n(G_0)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:25988"}},{"Add":{"text":"\\end{equation*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26004"}}],"del_contents":[{"Del":{"text":"Clearly, a global abstraction function is a special case of a dynamic","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:18037"}},{"Del":{"text":"abstraction function, where every location is abstracted with the same","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:18108"}},{"Del":{"text":"abstraction function. How to select properties for the localized property-based","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:18188"}},{"Del":{"text":"abstraction is discussed in \\Cref{ch:implementation}.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:18242"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":622,"ins":{"up_context":[{"change":2,"pos":18244}],"down_context":[{"change":2,"pos":18540}],"flag":"BLOCK","start":26006,"end":26738,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":18244},"to":{"change":2,"start":18244,"end":18540},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"\\begin{example}\\label{ex:component_pe}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26044"}},{"Add":{"text":" In this example the partial evaluation of $\\Prog_3$ is restricted to the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26121"}},{"Add":{"text":" component $T = \\{t_{1a}, t_{1b}\\}$, because the two loops are the only","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26196"}},{"Add":{"text":" problematic transitions during analysis with \\gls{mprf}s. Also we reduce the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26277"}},{"Add":{"text":" number of properties. The evaluation uses the localized property-based","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26352"}},{"Add":{"text":" abstraction $S_{\\bot,\\Psi_{\\ell_1},\\bot}$ with $\\Psi_1 = {\\psi_1, \\psi_2,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26430"}},{"Add":{"text":" \\psi_3, \\psi_4}$ where","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26457"}},{"Add":{"text":" \\begin{align*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26476"}},{"Add":{"text":" \\psi_1 = 1 \\leq x, \\hspace{1cm} \\neg\\psi_1 = 1 > x\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26537"}},{"Add":{"text":" \\psi_2 = 2 \\leq x, \\hspace{1cm} \\neg\\psi_2 = 2 > x\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26598"}},{"Add":{"text":" \\psi_3 = x \\leq 3, \\hspace{1cm} \\neg\\psi_3 = x > 3\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26659"}},{"Add":{"text":" \\psi_4 = x \\leq 4, \\hspace{1cm} \\neg\\psi_4 = x > 4\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26720"}},{"Add":{"text":" \\end{align*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26737"}}],"del_contents":[{"Del":{"text":"Next, we will adapt the evaluation a final time. Instead of evaluating the","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:18318"}},{"Del":{"text":"entire program, the evaluation will unfold only transitions in a given component","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:18399"}},{"Del":{"text":"$T$. The component is not required to be connected. How to select the component","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:18479"}},{"Del":{"text":"for best results ins discussed in \\Cref{ch:implementation}.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:18539"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":636,"ins":{"up_context":[{"change":2,"pos":18541}],"down_context":[{"change":2,"pos":18562}],"flag":"BLOCK","start":26739,"end":27152,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":18541},"to":{"change":2,"start":18541,"end":18562},"introduced_by":2}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" The program is lifted as shown in Example \\ref{ex:lift} and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26802"}},{"Add":{"text":" \\fref{fig:ex_lift}, but with the transitions $t_{1a}$ and $t_{1b}$ removed.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26882"}},{"Add":{"text":" The partial evaluation starts on this graph $G_0$ with ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26942"}},{"Add":{"text":" \\begin{align*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:26961"}},{"Add":{"text":" V(G_0) &= \\{\\langle\\ell_0,\\texttt{true}\\rangle,\\langle\\ell_1,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27031"}},{"Add":{"text":" \\texttt{true}\\rangle,\\langle\\ell_2,\\texttt{true}\\rangle\\}\\\\ ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27100"}},{"Add":{"text":" E(G_0) &= \\{t_0,t_2,t_3\\}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27134"}},{"Add":{"text":" \\end{align*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27151"}}],"del_contents":[{"Del":{"text":"\\todo{continue here}","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:18561"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":645,"ins":{"up_context":[{"change":2,"pos":18563}],"down_context":[{"change":4,"pos":7986}],"flag":"BLOCK","start":27153,"end":29692,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":2,"pos":18563},"to":{"change":4,"start":17821,"end":17959},"introduced_by":2},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":7589},"to":{"change":4,"start":17821,"end":17959},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" The transitions $t_{1a}$ and $t_{1b}$ are unfolded already in the first","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27228"}},{"Add":{"text":" evaluation step. We find to new transitions by unfolding $t_{1a}, t_{1b} \\in","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27309"}},{"Add":{"text":" \\TPout(\\ell_1) \\cap T$ from $\\langle\\ell_1,\\texttt{true}\\rangle$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27378"}},{"Add":{"text":" \\begin{align*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27397"}},{"Add":{"text":" t_{1a,1} &= (\\langle \\ell_1, \\texttt{true}\\rangle, 0.5, \\tau_1,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27469"}},{"Add":{"text":" \\eta_{1a}, \\langle \\ell_1, \\{\\psi_1,\\psi_3,\\psi_4\\}\\rangle) \\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27540"}},{"Add":{"text":" t_{1b,1} &= (\\langle \\ell_1, \\texttt{true}\\rangle, 0.5, \\tau_1,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27612"}},{"Add":{"text":" \\eta_{1b},\\langle \\ell_1, \\{\\psi_1,\\psi_2,\\psi_4\\}\\rangle) \\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27682"}},{"Add":{"text":" G_1 &= G_0 + \\{t_{1a,1},t_{1b,1}\\}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27725"}},{"Add":{"text":" \\end{align*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27742"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27743"}},{"Add":{"text":" Note, the transition $t_2$ is not in $T$. Instead of unfolding it, the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27818"}},{"Add":{"text":" transition $t_2' = (\\langle\\ell_1,\\texttt{true}\\rangle,1,\\tau_2,\\eta_2,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27894"}},{"Add":{"text":" \\langle\\ell_2,\\texttt{true}\\rangle)$ is added to the graph, which is already","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:27975"}},{"Add":{"text":" contained as the lift of $t_2$. The same holds for $t_0$ and $t_3$.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28047"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28048"}},{"Add":{"text":" In the second step the new versions $\\langle \\ell_1,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28105"}},{"Add":{"text":" \\{\\psi_1,\\psi_3,\\psi_4\\}\\rangle$ and $\\langle \\ell_1,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28163"}},{"Add":{"text":" \\{\\psi_1\\psi_2,\\psi_4\\}\\rangle$ are unfolded. We find new transitions ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28238"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28239"}},{"Add":{"text":" \\begin{align*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28258"}},{"Add":{"text":" t_{1a,2} &= (\\langle \\ell_1, \\{\\psi_1,\\psi_3,\\psi_4\\}\\rangle, 0.5,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28333"}},{"Add":{"text":" \\tau_1 \\Land \\psi_1 \\Land \\psi_3 \\Land \\psi_4, \\eta_{1a}, \\langle","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28407"}},{"Add":{"text":" \\ell_1, \\{\\psi_1,\\psi_3,\\psi_4\\}\\rangle) \\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28459"}},{"Add":{"text":" t_{1b,2} &= (\\langle \\ell_1, \\{\\psi_1,\\psi_3,\\psi_4\\}\\rangle, 0.5,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28534"}},{"Add":{"text":" \\tau_1 \\Land \\psi_1 \\Land \\psi_3 \\Land \\psi_4, \\eta_{1b}, \\langle","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28608"}},{"Add":{"text":" \\ell_1, \\{\\psi_1,\\psi_2,\\psi_4\\}\\rangle) \\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28660"}},{"Add":{"text":" t_{2,1} &= (\\langle \\ell_1, \\{\\psi_1,\\psi_3,\\psi_4\\}\\rangle, 1, \\tau_2","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28739"}},{"Add":{"text":" \\Land \\psi_1 \\Land \\psi_3 \\Land \\psi_4, \\eta_2,\\langle \\ell_2,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28810"}},{"Add":{"text":" \\texttt{true}\\rangle)\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28842"}},{"Add":{"text":" t_{1a,3} &= (\\langle \\ell_1, \\{\\psi_1,\\psi_2,\\psi_4\\}\\rangle, 0.5,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28917"}},{"Add":{"text":" \\tau_1 \\Land \\psi_1 \\Land \\psi_2 \\Land \\psi_4, \\eta_{1a}, \\langle","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:28991"}},{"Add":{"text":" \\ell_1, \\{\\psi_1,\\psi_3,\\psi_4\\}\\rangle) \\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29043"}},{"Add":{"text":" t_{1b,3} &= (\\langle \\ell_1, \\{\\psi_1,\\psi_2,\\psi_4\\}\\rangle, 0.5,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29118"}},{"Add":{"text":" \\tau_1 \\Land \\psi_1 \\Land \\psi_2 \\Land \\psi_4, \\eta_{1b}, \\langle","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29192"}},{"Add":{"text":" \\ell_1, \\{\\psi_1,\\psi_2,\\psi_4\\}\\rangle) \\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29244"}},{"Add":{"text":" t_{2,2} &= (\\langle \\ell_1, \\{\\psi_1,\\psi_2,\\psi_4\\}\\rangle, 1, \\tau_2","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29323"}},{"Add":{"text":" \\Land \\psi_1 \\Land \\psi_2 \\Land \\psi_4, \\eta_2, \\langle \\ell_2,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29395"}},{"Add":{"text":" \\texttt{true}\\rangle) \\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29428"}},{"Add":{"text":" G_2 &= G_1+ \\{t_{1a,2},t_{1b,2},t_{2,1},t_{1a,3},t_{1b,3},t_{2,2}\\}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29504"}},{"Add":{"text":" \\end{align*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29521"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29522"}},{"Add":{"text":" Note, the transition $t_{2,1}$ and $t_{2,2}$ are new transitions but they","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29600"}},{"Add":{"text":" transition to the trivial version instead beeing unfolded due to $t_2","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29674"}},{"Add":{"text":" \\not\\in\\T$. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29691"}}],"del_contents":[{"Del":{"text":"We will proceed with proving, that the graph can be used for constructing an new","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:17901"}},{"Del":{"text":"program $\\Prog'$ with equal expected runtime-complexity.","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:17958"}}]}},{"Edit":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":691,"atom":{"NewVertex":{"up_context":[{"change":4,"pos":7987}],"down_context":[{"change":4,"pos":17960}],"flag":"BLOCK","start":29693,"end":30849,"inode":{"change":4,"pos":7949}}},"contents":[{"Add":{"text":" The evaluation ends here, since no new versions were added in the previous","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29771"}},{"Add":{"text":" step. We get $G_3 = G_2 = G^*_{\\Prog_3,S_{\\bot,\\Psi_{\\ell_1},\\bot}}$.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29845"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29846"}},{"Add":{"text":" The full evaluation graph is displayed in \\fref{fig:ex_component_pe}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29920"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29921"}},{"Add":{"text":" \\begin{figure}[ht]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29944"}},{"Add":{"text":" \\centering","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:29963"}},{"Add":{"text":" \\input{figures/ch3_p3_component_pe}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30007"}},{"Add":{"text":" \\caption{Component-level partial evaluation graph of $\\Prog_3$ from","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30083"}},{"Add":{"text":" \\fref{fig:ex_prob} on a component $T = \\{t_{1a},t_{1b}\\}$ and with a","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30160"}},{"Add":{"text":" localized property-based abstraction","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30205"}},{"Add":{"text":" $S_{\\bot,\\Psi_{\\ell_1},\\bot}$.\\label{fig:ex_component_pe}}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30272"}},{"Add":{"text":" \\end{figure}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30289"}},{"Add":{"text":"\\end{example}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30303"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30304"}},{"Add":{"text":"\\rem{","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30310"}},{"Add":{"text":" In practice versions that have no incoming transitions in $G_0$ are removed,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30391"}},{"Add":{"text":" before the evaluation, to avoid unnecessary unfoldings. If the version is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30469"}},{"Add":{"text":" reachable in the partial evaluation nonetheless, it is readded by the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30543"}},{"Add":{"text":" evaluation and evaluated on in later step. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30591"}},{"Add":{"text":"}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30593"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30594"}},{"Add":{"text":"The partial evaluation over the entire program is a special case of the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30666"}},{"Add":{"text":"component level evaluation with a component $T = \\T_\\Prog$. All following lemmas","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30747"}},{"Add":{"text":"and theorems hold equally for a (global) abstraction and partial evaluations of","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30827"}},{"Add":{"text":"the entire program.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30847"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30848"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":719,"ins":{"up_context":[{"change":4,"pos":18027}],"down_context":[{"change":4,"pos":18335}],"flag":"BLOCK","start":30850,"end":31158,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":18027},"to":{"change":4,"start":18027,"end":18335},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" Let $G^*_{\\Prog,S}$ be the evaluation graph of a \\gls{pip} $\\Prog$ for any","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:30928"}},{"Add":{"text":" localized abstraction function $S$. For every transition $t = (\\langle","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31003"}},{"Add":{"text":" \\ell_1, \\varphi_1\\rangle, p, \\tau, \\eta, \\langle \\ell_2, \\varphi_2\\rangle)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31082"}},{"Add":{"text":" \\in E(G^*_{\\Prog,\\alpha})$ one can find a unique transition denoted as","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31157"}}],"del_contents":[{"Del":{"text":" Let $G^*_{\\Prog,\\alpha}$ be the evaluation graph of a \\gls{pip} $\\Prog$ for","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:18106"}},{"Del":{"text":" any abstraction function $\\alpha$. For every transition $t = (\\langle \\ell_1,","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:18188"}},{"Del":{"text":" \\varphi_1\\rangle, p, \\tau, \\eta, \\langle \\ell_2, \\varphi_2\\rangle) \\in","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:18263"}},{"Del":{"text":" E(G^*_{\\Prog,\\alpha})$ one can find a unique transition denoted as","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:18334"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":729,"ins":{"up_context":[{"change":4,"pos":25176}],"down_context":[{"change":4,"pos":18615}],"flag":"BLOCK","start":31159,"end":31246,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":25176},"to":{"change":4,"start":18550,"end":18615},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\proof{Follows directly from construction of the unfolding and the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31229"}},{"Add":{"text":" evaluation}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31245"}}],"del_contents":[{"Del":{"text":" \\proof{Follows directly from construction of the unfolding.}","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:18614"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":734,"ins":{"up_context":[{"change":4,"pos":18706}],"down_context":[{"change":4,"pos":25407}],"flag":"BLOCK","start":31247,"end":33406,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":18706},"to":{"change":4,"start":18706,"end":18748},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"with Lemma \\ref{lem:original_transition}. In Example \\ref{ex:global_abstr} and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31325"}},{"Add":{"text":"\\ref{ex:component_pe} the localized property-based abstraction was carefully","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31402"}},{"Add":{"text":"crafted so that the evaluation graph is finite. As was already foreshadowed the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31482"}},{"Add":{"text":"partial evaluation graph is for a localized abstraction function is finite when at least one location","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31584"}},{"Add":{"text":"per loop in the original program is abstracted to a finite abstraction space. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31663"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31664"}},{"Add":{"text":"\\begin{lemma}\\label{lem:localized_pba_finite}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31710"}},{"Add":{"text":" Let $S$ be a localized abstraction function. The partial evaluation graph","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31788"}},{"Add":{"text":" $G_{\\Prog,S}^*$ is finite if every loop $L$ in $\\Prog$ contains a location","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31867"}},{"Add":{"text":" $\\ell \\in L$ such that $\\set{S(\\langle\\ell,\\varphi\\rangle)}{\\varphi \\in","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31943"}},{"Add":{"text":" \\Phi}$ is finite.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31965"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31966"}},{"Add":{"text":" \\begin{proof}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:31984"}},{"Add":{"text":" Let $G_{\\Prog,S}^*$ be the partial evaluation graph of a \\gls{pip}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32059"}},{"Add":{"text":" $\\Prog$ with a localized abstraction $S$, and let","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32117"}},{"Add":{"text":" $\\set{S(\\langle\\ell,\\varphi\\rangle)}{\\varphi \\in \\Phi}$ for some $\\ell","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32196"}},{"Add":{"text":" \\in L$ on any loop $L$ of $Prog$.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32238"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32239"}},{"Add":{"text":" The proof will be done by contradiction. Assume the partial evaluation","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32318"}},{"Add":{"text":" graph $G_{\\Prog,S}^*$ is infinite. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32362"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32363"}},{"Add":{"text":" We can construct an infinite path $p = v_0v_1\\dots = \\langle\\ell_0","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32438"}},{"Add":{"text":" \\_\\rangle\\langle\\ell_1,\\_\\rangle\\dots \\in \\vers_\\Prog^\\omega$, where all","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32519"}},{"Add":{"text":" versions on this path are pairwise different. Let $n \\in \\N$. Since","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32595"}},{"Add":{"text":" $\\evaluate_\\Prog^*$ doesn't converge there must exist a fresh version","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32673"}},{"Add":{"text":" $v_n \\in E(G_n) \\backslash E(G_{n-1})$ that was added by unfolding or","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32751"}},{"Add":{"text":" copying a transition $t_n =(\\ell_{n-1},\\_, \\_, \\_,\\ell_n)$ from a","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32825"}},{"Add":{"text":" version $v_{n-1} = \\langle \\ell_{n-1},\\_\\rangle \\in E(G_{n-1})$.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32898"}},{"Add":{"text":" $v_{n-1}$ must have been freshly added in $G_{n-1}$ or else $v_n$ would","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:32978"}},{"Add":{"text":" have been part of $G_{n-1}$ already. By backwards induction we can","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33053"}},{"Add":{"text":" create a path $v_0v_1\\dots{}v_n$ of $G_n$ for arbitrarily large $n \\in N$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33135"}},{"Add":{"text":" with pairwise different versions which implies that there exists an","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33211"}},{"Add":{"text":" infinite path $p = v_0v_1\\dots = \\langle\\ell_0","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33266"}},{"Add":{"text":" \\_\\rangle\\langle\\ell_1,\\_\\rangle\\dots \\in \\vers_\\Prog^\\omega$ with","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33341"}},{"Add":{"text":" pairwise different versions in $G_{\\Prog,S}^*$ as well.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33405"}}],"del_contents":[{"Del":{"text":"with Lemma \\ref{lem:original_transition}.","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:18747"}}]}},{"Edit":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":770,"atom":{"NewVertex":{"up_context":[{"change":4,"pos":25408}],"down_context":[{"change":4,"pos":18749}],"flag":"BLOCK","start":33407,"end":34755,"inode":{"change":4,"pos":7949}}},"contents":[{"Add":{"text":" Every version was added by unfolding or copying a transition $t_i","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33480"}},{"Add":{"text":" =(\\ell_{i-1},\\_, \\_, \\_,\\ell_i) \\in \\T_\\Prog$. Consequently, there must","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33560"}},{"Add":{"text":" exist an infinite path $p' = \\ell_0\\ell_1\\dots \\in \\Loc_\\Prog^\\omega$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33638"}},{"Add":{"text":" and since $\\Prog$ is finite the path must contain a loop $L \\in \\Loc^*$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33718"}},{"Add":{"text":" that is appears an infinite number of times on $p'$. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33780"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33781"}},{"Add":{"text":" By assumption there must exists a location $\\ell^* \\in L$ for which","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33857"}},{"Add":{"text":" $\\set{S(\\langle\\ell,\\varphi\\rangle)}{\\varphi \\in \\Phi}$ is finite. At","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:33935"}},{"Add":{"text":" the same time versions $\\langle\\ell^*,\\_\\rangle$ must appear an infinite","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34016"}},{"Add":{"text":" number of times on $p$, and there can be only abstracted to finitely","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34093"}},{"Add":{"text":" many possible versions, which is a contradiction to the versions on $p$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34173"}},{"Add":{"text":" being pairwise different.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34207"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34208"}},{"Add":{"text":" $G_{\\Prog,S}^*$ cannot be infinite.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34252"}},{"Add":{"text":" \\end{proof}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34268"}},{"Add":{"text":"\\end{lemma}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34280"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34281"}},{"Add":{"text":"We will proceed with proving, that the partial evaluation graph of a component","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34360"}},{"Add":{"text":"level partial evaluation can be used for constructing an new program $\\Prog'$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34438"}},{"Add":{"text":"with equal expected runtime-complexity. All results equally hold for partial","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34515"}},{"Add":{"text":"evaluation of entire program and independently of the choice of the (global or","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34594"}},{"Add":{"text":"localized) abstraction function as long as the localized evaluation function","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34671"}},{"Add":{"text":"asserts a finite partial evaluation graph by Lemma","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34722"}},{"Add":{"text":"\\ref{lem:localized_pba_finite}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34754"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":795,"ins":{"up_context":[{"change":4,"pos":18800}],"down_context":[{"change":4,"pos":19032}],"flag":"BLOCK","start":34756,"end":34987,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":18800},"to":{"change":4,"start":18800,"end":19032},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" Let $G^*_{\\Prog,S,T}$ be the evaluation graph of a \\gls{pip} $\\Prog$ with a","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34835"}},{"Add":{"text":" localized abstraction function $S$ on a component $T \\subseteq \\TP$. The","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34912"}},{"Add":{"text":" transitions of $E(G^*_{\\Prog,S,T})$ can be split into finite pairwise","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:34986"}}],"del_contents":[{"Del":{"text":" Let $G^*_{\\Prog,\\alpha}$ be the evaluation graph of a \\gls{pip} $\\Prog$ with","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:18880"}},{"Del":{"text":" general transitions $\\GTP$ and the abstraction function $\\alpha$. The","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:18954"}},{"Del":{"text":" transitions of $E(G^*_{\\Prog,\\alpha})$ can be split into finite pairwise","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:19031"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":801,"ins":{"up_context":[{"change":4,"pos":25430}],"down_context":[{"change":4,"pos":19418}],"flag":"BLOCK","start":34988,"end":35501,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":25430},"to":{"change":4,"start":19144,"end":19299},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":19299},"to":{"change":4,"start":26080,"end":26110},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":26080},"to":{"change":4,"start":26080,"end":26110},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":26110},"to":{"change":4,"start":19300,"end":19417},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":19417},"to":{"change":4,"start":26164,"end":26192},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":26164},"to":{"change":4,"start":26164,"end":26192},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" \\item there exist a unique general transition $\\bar{g}_i \\in \\GTP$ and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35066"}},{"Add":{"text":" some $\\varphi \\in \\Phi$ for which the following property holds:","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35142"}},{"Add":{"text":" \\begin{align*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35169"}},{"Add":{"text":" g_i &= \\set{t}{\\bar{t} \\in \\bar{g}_i, t \\in E(G^*_{\\Prog,S,T})} \\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35252"}},{"Add":{"text":" &= \\set{(\\langle\\ell_{\\bar{g}_i},\\varphi\\rangle,p,\\varphi","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35330"}},{"Add":{"text":" \\land \\tau_{g_i}, \\eta, \\langle\\ell',\\varphi'\\rangle}{","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35405"}},{"Add":{"text":" (\\ell_g, p, \\tau_g, \\eta, \\varphi') \\in g'_i}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35475"}},{"Add":{"text":" \\end{align*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35500"}}],"del_contents":[{"Del":{"text":" \\item there exist a unique general transition $\\bar{g_i} \\in \\GTP$ and","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:19222"}},{"Del":{"text":" some $\\varphi \\in \\Psi$ for which the following property holds:","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:19298"}},{"Del":{"text":" \\begin{equation*}","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:26109"}},{"Del":{"text":" g_i = \\set{\\unfold(\\langle \\ell_{g'_i}, \\varphi \\rangle,","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:19372"}},{"Del":{"text":" \\bar{t})}{\\bar{t} \\in g'_i}","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:19416"}},{"Del":{"text":" \\end{equation*}","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:26191"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":815,"ins":{"up_context":[{"change":4,"pos":26231}],"down_context":[{"change":4,"pos":27263}],"flag":"BLOCK","start":35502,"end":36517,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":26231},"to":{"change":4,"start":19591,"end":19612},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" Let $t =(\\langle \\ell, \\varphi\\rangle, p, \\tau, \\eta, \\_) \\in","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35571"}},{"Add":{"text":" E(G^*_{\\Prog,S,T})$ be a transitions in the partial evaluation graph","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35648"}},{"Add":{"text":" $G^*_{\\Prog,S,T}$ of a \\gls{pip} $\\Prog$ with a component $T \\subset","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35725"}},{"Add":{"text":" \\TP$. There exists a unique general","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35769"}},{"Add":{"text":" transition $g \\in \\GTP$ with $\\bar{t} \\in g$. Let $ g = \\{ t'_1, \\dots,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35849"}},{"Add":{"text":" t'_n \\}$. We set $\\bar{g} = g$. All $t'_i$ share a common guard $\\tau_g$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35930"}},{"Add":{"text":" and source location $\\ell_g = \\ell$. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35976"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:35977"}},{"Add":{"text":" Let $1 \\leq i \\leq n$ an $t'_i = (\\ell, p, \\tau_g, \\_, \\_)$. By","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36049"}},{"Add":{"text":" construction of $\\evaluate_\\Prog,S,T$ there was added either a","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36120"}},{"Add":{"text":" transition $t_i = \\unfold(\\langle\\ell,\\varphi\\rangle, t_i) =","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36189"}},{"Add":{"text":" (\\langle\\ell,\\varphi\\rangle, p, \\varphi \\land \\tau_g, \\_, \\_)$ if $t'_i","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36269"}},{"Add":{"text":" \\in T$ or $t_i = (\\langle\\ell,\\varphi\\rangle, p, \\varphi \\land \\tau_g,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36348"}},{"Add":{"text":" \\_, \\_)$ if $t'_i \\not\\in T$. In any case, all $t'_i$ share a common","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36425"}},{"Add":{"text":" guard $\\tau_g' = \\varphi \\land \\tau_g$ and the probabilities add up to","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36504"}},{"Add":{"text":" 1. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36516"}}],"del_contents":[{"Del":{"text":" \\todo{proof}","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:19611"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":838,"ins":{"up_context":[{"change":4,"pos":19837}],"down_context":[{"change":4,"pos":20277}],"flag":"BLOCK","start":36518,"end":36988,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":19837},"to":{"change":4,"start":19837,"end":20277},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":" Let $\\Prog$ as defined by Definition \\ref{def:pip} and $G^*_{\\Prog,S,T}$ be","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36597"}},{"Add":{"text":" the partial evaluation graph of $\\Prog$ for some localized abstraction","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36672"}},{"Add":{"text":" function $S$ and a component $T \\subseteq \\TP$. A new \\gls{pip} is defined","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36751"}},{"Add":{"text":" as $\\Prog' = (\\PV, \\Loc_{\\Prog'}, \\GTPn, \\langle l_0, \\textit{true}\\rangle)$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36832"}},{"Add":{"text":" where $\\Loc_{\\Prog'} = E(G^*_{\\Prog,S,T}) \\subset \\vers_\\Prog$, and $\\GTPn =","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36913"}},{"Add":{"text":" \\mathcal{G}\\mathcal{T}_{G^*_{\\Prog,S,T}} \\subset 2^\\VTP$. $\\Prog'$ is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:36987"}}],"del_contents":[{"Del":{"text":" Let $\\Prog$ as defined by Definition \\ref{def:pip} and $G^*_{\\Prog,\\alpha}$","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:19916"}},{"Del":{"text":" be the partial evaluation graph of $\\Prog$ for some abstraction function","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:19993"}},{"Del":{"text":" $\\alpha$. A new \\gls{pip} is defined as $\\Prog' = (\\PV, \\Loc_{\\Prog'},","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:20068"}},{"Del":{"text":" \\GTPn, \\langle l_0, \\textit{true}\\rangle)$ where $\\Loc_{\\Prog'} =","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:20138"}},{"Del":{"text":" E(G^*_{\\Prog,\\alpha}) \\subset \\vers_\\Prog$, and $\\GTPn =","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:20199"}},{"Del":{"text":" \\mathcal{G}\\mathcal{T}_{G^*_{\\Prog,\\alpha}} \\subset 2^\\VTP$. $\\Prog'$ is","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:20276"}}]}},{"Replacement":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":851,"ins":{"up_context":[{"change":4,"pos":20597}],"down_context":[{"change":4,"pos":20971}],"flag":"BLOCK","start":36989,"end":37366,"inode":{"change":4,"pos":7949}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":20597},"to":{"change":4,"start":20597,"end":20971},"introduced_by":4}],"inode":{"change":4,"pos":7949}},"ins_contents":[{"Add":{"text":"Theorem \\ref{thm:correctness} which will shows that any runtime-complexity","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:37063"}},{"Add":{"text":"bounds for the partial evaluation are runtime-complexity bounds of the original","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:37143"}},{"Add":{"text":"program, which demonstrates the correctness of the construction. Lemma","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:37214"}},{"Add":{"text":"\\ref{lem:constrfgeq} and Theorem \\ref{thm:thightness} will show that worst-case","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:37294"}},{"Add":{"text":"expected runtime-complexity doesn't get worse with partial evaluation.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:37365"}}],"del_contents":[{"Del":{"text":"Theorem \\ref{thm:correctness} which will shows that runtime-complexity bounds","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:20674"}},{"Del":{"text":"for the partial evaluation are runtime-complexity bounds of the original","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:20747"}},{"Del":{"text":"program, which demonstrates the correctness of the construction. ","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:20813"}},{"Del":{"text":"Lemma \\ref{lem:constrfgeq} and Theorem \\ref{thm:thightness} will show that","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:20888"}},{"Del":{"text":"worst-case expected runtime-complexity doesn't get worse with partial","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:20958"}},{"Del":{"text":"evaluation.","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:20970"}}]}},{"Edit":{"inode":{"change":4,"pos":7949},"path":"ch3_theory.tex","line":1245,"atom":{"EdgeMap":{"edges":[{"previous":"","flag":"DELETED","from":{"change":4,"pos":3892},"to":{"change":4,"start":33904,"end":33905},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":9585},"to":{"change":4,"start":33904,"end":33905},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":33904},"to":{"change":4,"start":33904,"end":33905},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":33905},"to":{"change":4,"start":9586,"end":10113},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":10113},"to":{"change":4,"start":4949,"end":4950},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":4949},"to":{"change":4,"start":4949,"end":4950},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":4950},"to":{"change":4,"start":10114,"end":10240},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":10240},"to":{"change":4,"start":5944,"end":5961},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":5944},"to":{"change":4,"start":5944,"end":5961},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":431},"to":{"change":4,"start":3838,"end":3839},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":5961},"to":{"change":4,"start":3838,"end":3839},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":3838},"to":{"change":4,"start":3838,"end":3839},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":3839},"to":{"change":4,"start":10241,"end":10449},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":10449},"to":{"change":4,"start":6174,"end":6175},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":6174},"to":{"change":4,"start":6174,"end":6175},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":6175},"to":{"change":4,"start":10450,"end":10471},"introduced_by":4}],"inode":{"change":4,"pos":7949}}},"contents":[{"Del":{"text":"","pos":"TKJMPTB73W5TLLTNUOC4FGRQ7DHE5GRBTL7C5QBXRCNQPMJKGEQAC:33904"}},{"Del":{"text":"\\section{History Dependent Abstractions}","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:9626"}},{"Del":{"text":"In \\Sref{sec:partialevaluation} the abstraction the details of the abstraction","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:9705"}},{"Del":{"text":"were omitted. Recall Definition \\ref{def:abstraction}, an abstraction space must","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:9786"}},{"Del":{"text":"be finite and the abstraction function must map any probability to elements of","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:9865"}},{"Del":{"text":"this abstraction space. The abstraction function was kept simple, in order to","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:9943"}},{"Del":{"text":"keep the proofs consise. In this section we will expand to take the evaluation","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:10022"}},{"Del":{"text":"history into account, which opens the possibility for more refined abstraction","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:10101"}},{"Del":{"text":"functions.","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:10112"}},{"Del":{"text":"","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:4949"}},{"Del":{"text":"\\begin{definition}[History dependent abstraction]","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:10163"}},{"Del":{"text":" A history dependent abstraction function is a function $\\G \\rightarrow$","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:10239"}},{"Del":{"text":"\\end{definition}","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:5960"}},{"Del":{"text":"","pos":"QCVDPRXH57UFLU4YV3P2I5RQZGCBUKROYUZHDWOCHL43KDYN7MPAC:3838"}},{"Del":{"text":"In this section we will present the property-based abstraction","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:10303"}},{"Del":{"text":"proposed by \\citeauthor{Domenech19}\\cite{Domenech19} for non-probabilistic","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:10378"}},{"Del":{"text":"programs but which can be applied equally to probabilistic programs. ","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:10448"}},{"Del":{"text":"","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:6174"}},{"Del":{"text":"\\todo{continue here}","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:10470"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":46,"ins":{"up_context":[{"change":4,"pos":7138}],"down_context":[{"change":4,"pos":9693}],"flag":"BLOCK","start":37367,"end":38020,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":7138},"to":{"change":4,"start":1239,"end":1808},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":1808},"to":{"change":4,"start":42869,"end":42941},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"one symbol is replaced by another is denoted with $[\\cdot/\\cdot]$. For example","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:37445"}},{"Add":{"text":"$(x 0}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:46874"}},{"Add":{"text":" \\end{align*}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:46891"}}],"del_contents":[{"Del":{"text":" \\begin{align}","pos":"RVA3DOCNBWFBWA2MLC2TVQOEY25AUCYGEJSWMD72QRQ26AFLVZUAC:23303"}},{"Del":{"text":" \\text{supp}(\\mu) = \\set{x \\in \\N}{P(X = x) > 0}","pos":"RVA3DOCNBWFBWA2MLC2TVQOEY25AUCYGEJSWMD72QRQ26AFLVZUAC:23359"}},{"Del":{"text":" \\end{align}","pos":"RVA3DOCNBWFBWA2MLC2TVQOEY25AUCYGEJSWMD72QRQ26AFLVZUAC:23375"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":532,"ins":{"up_context":[{"change":4,"pos":3758}],"down_context":[{"change":4,"pos":35049}],"flag":"BLOCK","start":46893,"end":47115,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":3758},"to":{"change":4,"start":50034,"end":50251},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"When considering a random variable, one is usually interested in the value to","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:46970"}},{"Add":{"text":"expect in general. For example, most possible outcomes of a binomial","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47039"}},{"Add":{"text":"distribution are very improbable. This is described by the expected value.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47114"}}],"del_contents":[{"Del":{"text":"When considering a random variable one is usually interested the value to expect","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:50114"}},{"Del":{"text":"in general. For example most possible outcomes of a binomial distribution are","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:50192"}},{"Del":{"text":"very improbable. This is described by the expected value.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:50250"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":542,"ins":{"up_context":[{"change":4,"pos":35170}],"down_context":[{"change":4,"pos":35277}],"flag":"BLOCK","start":47116,"end":47224,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":35170},"to":{"change":4,"start":35170,"end":35277},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" \\E(X) &= \\sum_{r \\in \\overline{N}} r \\cdot P(X = r)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47175"}},{"Add":{"text":" = \\sum_{r \\in \\overline{N}} P(X \\leq r)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47223"}}],"del_contents":[{"Del":{"text":" \\E(X) &= \\sum_{r \\in \\bar{\\N}} r \\cdot P(X = r)","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:35225"}},{"Del":{"text":" &= \\sum_{r \\in \\bar{\\N}} \\cdot P(X \\leq r)","pos":"Q3TJZVCBWD7FDMS4MWTT7VUNOI3R3MQCOU6JJCJ5UVIDJPXKMLWAC:35276"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":551,"ins":{"up_context":[{"change":4,"pos":23551}],"down_context":[{"change":4,"pos":4150}],"flag":"BLOCK","start":47225,"end":47426,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":23551},"to":{"change":4,"start":23551,"end":23753},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"They will be briefly introduced in this section, following the definitions from","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47304"}},{"Add":{"text":"\\cite{kraaikamp2005modern}. Examples of the various probability distributions","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47382"}},{"Add":{"text":"are displayed in \\fref{fig:distributions}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47425"}}],"del_contents":[{"Del":{"text":"They will be briefly introduced in this section, following the definitions","pos":"RVA3DOCNBWFBWA2MLC2TVQOEY25AUCYGEJSWMD72QRQ26AFLVZUAC:23625"}},{"Del":{"text":"from \\cite{kraaikamp2005modern}. Examples for the various probability","pos":"RVA3DOCNBWFBWA2MLC2TVQOEY25AUCYGEJSWMD72QRQ26AFLVZUAC:23695"}},{"Del":{"text":"distributions are displayed in \\fref{fig:distributions}.","pos":"RVA3DOCNBWFBWA2MLC2TVQOEY25AUCYGEJSWMD72QRQ26AFLVZUAC:23752"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":557,"ins":{"up_context":[{"change":4,"pos":18004}],"down_context":[{"change":4,"pos":23977}],"flag":"BLOCK","start":47427,"end":47534,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":18004},"to":{"change":4,"start":18004,"end":18046},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":18046},"to":{"change":4,"start":22561,"end":22625},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":23934},"to":{"change":4,"start":22561,"end":22625},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"$0$, for example decided by a coin toss. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47468"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47469"}},{"Add":{"text":"\\begin{definition}[Bernoulli distribution]\\label{def:bernoulli}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47533"}}],"del_contents":[{"Del":{"text":"$0$, for example decided by a coin-flip. ","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:18045"}},{"Del":{"text":"\\begin{definition}[Bernoulli Distribution]\\label{def:bernoulli}","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:22624"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":594,"ins":{"up_context":[{"change":4,"pos":25083}],"down_context":[{"change":4,"pos":25119}],"flag":"BLOCK","start":47535,"end":47588,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":25083},"to":{"change":4,"start":22720,"end":22773},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"\\subsection{Geometric Distribution}\\label{ssec:geom}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47587"}}],"del_contents":[{"Del":{"text":"\\subsection{Geometric distribution}\\label{ssec:geom}","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:22772"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":597,"ins":{"up_context":[{"change":4,"pos":25278}],"down_context":[{"change":4,"pos":25349}],"flag":"BLOCK","start":47589,"end":47650,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":25278},"to":{"change":4,"start":25278,"end":25349},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"The number of repetitions follows a geometric distribution. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47649"}}],"del_contents":[{"Del":{"text":"The number of repetition is the result to the geometric distribution. ","pos":"RVA3DOCNBWFBWA2MLC2TVQOEY25AUCYGEJSWMD72QRQ26AFLVZUAC:25348"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":601,"ins":{"up_context":[{"change":4,"pos":25446}],"down_context":[{"change":4,"pos":25526}],"flag":"BLOCK","start":47651,"end":47731,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":25446},"to":{"change":4,"start":25446,"end":25526},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" parameterised by $p$ (the probability of the repeated Bernoulli event) with","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47730"}}],"del_contents":[{"Del":{"text":" parameterized by $p$ (the probability of the repeated Bernoulli event) with","pos":"RVA3DOCNBWFBWA2MLC2TVQOEY25AUCYGEJSWMD72QRQ26AFLVZUAC:25525"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":615,"ins":{"up_context":[{"change":4,"pos":25933}],"down_context":[{"change":4,"pos":26009}],"flag":"BLOCK","start":47732,"end":47810,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":25933},"to":{"change":4,"start":25933,"end":26009},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"sampling from a population with a certain number of successes without putting","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47809"}}],"del_contents":[{"Del":{"text":"sampling from population with a certain number of successes without putting","pos":"RVA3DOCNBWFBWA2MLC2TVQOEY25AUCYGEJSWMD72QRQ26AFLVZUAC:26008"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":680,"ins":{"up_context":[{"change":4,"pos":30977}],"down_context":[{"change":4,"pos":50775}],"flag":"BLOCK","start":47811,"end":47883,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":30977},"to":{"change":4,"start":50695,"end":50775},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"Let $\\D$ be the set of probability distributions. $\\Z[\\V\\union \\D]$, is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47882"}}],"del_contents":[{"Del":{"text":"Let $\\D$ be the set of (valid) probability distributions. $\\Z[\\V\\union \\D]$, is","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:50774"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":684,"ins":{"up_context":[{"change":4,"pos":51008}],"down_context":[{"change":4,"pos":51080}],"flag":"BLOCK","start":47884,"end":47959,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":51008},"to":{"change":4,"start":51008,"end":51080},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"as an argument. As such we can define a function $[\\cdot]: \\Z[\\V\\union \\D]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:47958"}}],"del_contents":[{"Del":{"text":"as argument. As such we can define a function $[\\cdot]: \\Z[\\V\\union \\D]","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:51079"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":689,"ins":{"up_context":[{"change":4,"pos":13712}],"down_context":[{"change":4,"pos":13793}],"flag":"BLOCK","start":47960,"end":48041,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":13712},"to":{"change":4,"start":13712,"end":13793},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"The values of the variables are fixed by a syntactical substitution; where every","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48040"}}],"del_contents":[{"Del":{"text":"The values of the variables are fixed by a syntactical substitution, where every","pos":"KVSF5AFJEIMIYAC264OUAA56SGP5MFAUG3AECKOTUPIXPGULVFQQC:13792"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":694,"ins":{"up_context":[{"change":4,"pos":51521}],"down_context":[{"change":4,"pos":14151}],"flag":"BLOCK","start":48042,"end":48119,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":51521},"to":{"change":4,"start":51521,"end":51594},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"within $f$. The random variable has the following probability distribution: ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48118"}}],"del_contents":[{"Del":{"text":"within. The random variable has the following probability distribution: ","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:51593"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":712,"ins":{"up_context":[{"change":4,"pos":14984}],"down_context":[{"change":4,"pos":15068}],"flag":"BLOCK","start":48120,"end":48218,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":14984},"to":{"change":4,"start":51813,"end":51903},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" $s(x) = 2$. $[\\varf{}](s)$ is the random variable described by the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48190"}},{"Add":{"text":" polynomial $\\varf[x/2]","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48217"}}],"del_contents":[{"Del":{"text":" $s(x) = 2$. $[\\varf{}](s)$ is the random variable described by the polynomial $f[x/2]","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:51902"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":737,"ins":{"up_context":[{"change":4,"pos":31913}],"down_context":[{"change":4,"pos":52217}],"flag":"BLOCK","start":48219,"end":48431,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":31913},"to":{"change":4,"start":52008,"end":52217},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"The value of a random variable can be approximated by a constraint for the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48293"}},{"Add":{"text":"bounds of the support of the probability distribution. For any probability","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48368"}},{"Add":{"text":"distribution $D \\in \\D$ and a variable $d \\in \\V$, we define ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48430"}}],"del_contents":[{"Del":{"text":"The value of a random variable can approximated by a constraint for the bounds","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:52086"}},{"Del":{"text":"of the support of the probability distribution. For any probability distribution","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:52167"}},{"Del":{"text":"$D \\in \\D$ and a variable $d \\in \\V$, we define ","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:52216"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":743,"ins":{"up_context":[{"change":4,"pos":52344}],"down_context":[{"change":4,"pos":52418}],"flag":"BLOCK","start":48432,"end":48509,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":52344},"to":{"change":4,"start":52344,"end":52418},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" \\min(\\support(D)) \\text{and} \\max (\\support(D)) \\text{ is known}, \\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48508"}}],"del_contents":[{"Del":{"text":" \\min(\\support(D)) \\text{and} \\max (\\support(D)) \\text{ exist}, \\\\","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:52417"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":745,"ins":{"up_context":[{"change":4,"pos":52482}],"down_context":[{"change":4,"pos":52591}],"flag":"BLOCK","start":48510,"end":48623,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":52482},"to":{"change":4,"start":52482,"end":52591},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" \\text{ is known}, \\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48538"}},{"Add":{"text":" d \\leq \\max(\\support(D)) & \\text{if} \\max(\\support(D)) \\text{ is known},\\\\","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48622"}}],"del_contents":[{"Del":{"text":" \\text{ exists}, \\\\","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:52508"}},{"Del":{"text":" d \\leq \\max(\\support(D)) & \\text{if} \\max(\\support(D)) \\text{ exists},\\\\","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:52590"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":758,"ins":{"up_context":[{"change":4,"pos":52744}],"down_context":[{"change":4,"pos":52817}],"flag":"BLOCK","start":48624,"end":48701,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":52744},"to":{"change":4,"start":52744,"end":52817},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" with probability distributions $D_1,\\dots,D_n$ used within $f$. Then the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48700"}}],"del_contents":[{"Del":{"text":" with probability distributions $D_1,\\dots,D_n$ used within. Then the","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:52816"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":779,"ins":{"up_context":[{"change":4,"pos":53557}],"down_context":[{"change":4,"pos":53638}],"flag":"BLOCK","start":48702,"end":48784,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":53557},"to":{"change":4,"start":53557,"end":53638},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" Hence, a variable $d$ assigned from the distribution satisfies the constraint","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48783"}}],"del_contents":[{"Del":{"text":" Hence a variable $d$ assigned from the distribution satisfies the constraint","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:53637"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":782,"ins":{"up_context":[{"change":4,"pos":53792}],"down_context":[{"change":4,"pos":53872}],"flag":"BLOCK","start":48785,"end":48868,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":53792},"to":{"change":4,"start":53792,"end":53872},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" variable $d$. We obtain the new approximated polynomial $\\tilde{\\varf} = x^2 +","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48867"}}],"del_contents":[{"Del":{"text":" variable $d$. We get the new approximated polynomial $\\tilde{\\varf} = x^2 +","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:53871"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":787,"ins":{"up_context":[{"change":4,"pos":32784}],"down_context":[{"change":4,"pos":32857}],"flag":"BLOCK","start":48869,"end":48945,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":32784},"to":{"change":4,"start":32784,"end":32857},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"over-approximation of all the probabilistic polynomials used in the update.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:48944"}}],"del_contents":[{"Del":{"text":"over-approximation of all used probabilistic polynomials in the update. ","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:32856"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":888,"ins":{"up_context":[{"change":4,"pos":80}],"down_context":[{"change":4,"pos":23015}],"flag":"BLOCK","start":48946,"end":49027,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":80},"to":{"change":4,"start":54817,"end":54898},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"over the years to formalise integer programs. The one used in this thesis is the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49026"}}],"del_contents":[{"Del":{"text":"over the years to formalize integer programs. The one used in this thesis is the","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:54897"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":890,"ins":{"up_context":[{"change":4,"pos":23093}],"down_context":[{"change":4,"pos":23166}],"flag":"BLOCK","start":49028,"end":49102,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":23093},"to":{"change":4,"start":23093,"end":23166},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"integer programs, the various concepts of integer programs are introduced","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49101"}}],"del_contents":[{"Del":{"text":"integer programs the various concepts of integer programs are introduced","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:23165"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":894,"ins":{"up_context":[{"change":4,"pos":54975}],"down_context":[{"change":4,"pos":55515}],"flag":"BLOCK","start":49103,"end":49808,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":4,"pos":54975},"to":{"change":4,"start":23279,"end":23432},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":23279},"to":{"change":4,"start":23279,"end":23432},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":23432},"to":{"change":4,"start":54976,"end":55054},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":55054},"to":{"change":4,"start":11823,"end":11897},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":11897},"to":{"change":4,"start":55132,"end":55515},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":55132},"to":{"change":4,"start":55132,"end":55515},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"where every configuration contains the location which represents the program","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49179"}},{"Add":{"text":"counter, and a state that holds the current values of the programs ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49247"}},{"Add":{"text":"variables. A transition contains a condition $\\tau$, called \\textit{guard}, that","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49328"}},{"Add":{"text":"must be true for the current state to be taken. Whenever a transition is taken","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49407"}},{"Add":{"text":"it modifies the state with some \\textit{update} $\\eta$. An update assigns a new","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49487"}},{"Add":{"text":"value to each program variable depending on the previous state. Every integer","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49565"}},{"Add":{"text":"program starts at a unique location usually called $\\ell_0$. Whenever no","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49638"}},{"Add":{"text":"transition guard is satisfied the program terminates. The arguments to the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49713"}},{"Add":{"text":"program are given by some starting state $s_0$, assigning a value for every","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49789"}},{"Add":{"text":"program variable.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49807"}}],"del_contents":[{"Del":{"text":"where every configuration contains the location at which the program is and a","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:23356"}},{"Del":{"text":"state that holds the current values of the program variables. A transition","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:23431"}},{"Del":{"text":"contains a condition $\\tau$, called \\textit{guard}, that must be true for the","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:55053"}},{"Del":{"text":"current state to be taken. Whenever a transition is taken it modifies the","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:11896"}},{"Del":{"text":"state with some \\textit{update} $\\eta$. An update assigns a new value to each","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:55209"}},{"Del":{"text":"program variable depending on the previous state. Every integer program starts","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:55288"}},{"Del":{"text":"at a unique location usually called $\\ell_0$. Whenever no transition guard is","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:55366"}},{"Del":{"text":"satisfied the program terminates. The arguments to the program is given by some","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:55446"}},{"Del":{"text":"starting state $s_0$, assigning a value for every program variable.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:55514"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":906,"ins":{"up_context":[{"change":4,"pos":11979}],"down_context":[{"change":4,"pos":12048}],"flag":"BLOCK","start":49809,"end":49878,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":11979},"to":{"change":4,"start":11979,"end":12048},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"Updates assign a value to every program variable; however for better","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49877"}}],"del_contents":[{"Del":{"text":"Updates assign a value to every program variable, however for better","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:12047"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":922,"ins":{"up_context":[{"change":4,"pos":24320}],"down_context":[{"change":4,"pos":24398}],"flag":"BLOCK","start":49879,"end":49956,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":24320},"to":{"change":4,"start":55755,"end":55831},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" $\\ell_1$ with the state $s(x) = 3$, both transitions $t_2$ and $t_3$ are","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49955"}}],"del_contents":[{"Del":{"text":" $\\ell_1$ with the state $s(x) = 3$ both transitions $t_2$ and $t_3$ are","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:55830"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":925,"ins":{"up_context":[{"change":4,"pos":24541}],"down_context":[{"change":4,"pos":24565}],"flag":"BLOCK","start":49957,"end":49983,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":24541},"to":{"change":4,"start":24541,"end":24565},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" (non-deterministic). ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:49982"}}],"del_contents":[{"Del":{"text":" (non-determistic). ","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:24564"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":930,"ins":{"up_context":[{"change":4,"pos":24737}],"down_context":[{"change":4,"pos":20308}],"flag":"BLOCK","start":49984,"end":50063,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":24737},"to":{"change":4,"start":24737,"end":24816},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"fully arbitrarily, the transitions are picked by their respective probability.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:50062"}}],"del_contents":[{"Del":{"text":"fully arbitrarily, the transitions are picked with the respective probability.","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:24815"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":933,"ins":{"up_context":[{"change":4,"pos":24966}],"down_context":[{"change":4,"pos":25039}],"flag":"BLOCK","start":50064,"end":50137,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":24966},"to":{"change":4,"start":24966,"end":25039},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"concept of \\textit{general transitions}; they are sets of (conventional)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:50136"}}],"del_contents":[{"Del":{"text":"concept of \\textit{general transitions}: they are sets of (conventional)","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:25038"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":946,"ins":{"up_context":[{"change":4,"pos":25632}],"down_context":[{"change":4,"pos":25706}],"flag":"BLOCK","start":50138,"end":50214,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":25632},"to":{"change":4,"start":25632,"end":25706},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" Take the program displayed in \\fref{fig:ex_pip_prob} as an example. The","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:50213"}}],"del_contents":[{"Del":{"text":" Take the program displayed in \\fref{fig:ex_pip_prob} for example. The","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:25705"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":948,"ins":{"up_context":[{"change":4,"pos":25782}],"down_context":[{"change":4,"pos":26098}],"flag":"BLOCK","start":50215,"end":50667,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":25782},"to":{"change":4,"start":25782,"end":25931},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":25931},"to":{"change":4,"start":12133,"end":12202},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":12202},"to":{"change":4,"start":26002,"end":26078},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":55906},"to":{"change":4,"start":26002,"end":26078},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":26002},"to":{"change":4,"start":26002,"end":26078},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":26078},"to":{"change":4,"start":20385,"end":20466},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":20466},"to":{"change":4,"start":55907,"end":55980},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" red). For simplicity, all the other transitions are unique members of their","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:50294"}},{"Add":{"text":" own general transition and the trivial probability of 1 is omitted. At","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:50369"}},{"Add":{"text":" location $\\ell_1$ with a variable assignment $s(x) = 3$, three transitions","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:50448"}},{"Add":{"text":" $t_2$, $t_3$ and $t_4$ are possible. The non-determinism is resolved first.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:50528"}},{"Add":{"text":" Then if the general transition containing $t_2$ and $t_3$ is selected, a","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:50605"}},{"Add":{"text":" coin is thrown to decide whether to take $t_2$ or $t_3$.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:50666"}}],"del_contents":[{"Del":{"text":" red). For simplicity, all other transitions are unique members of their own","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:25861"}},{"Del":{"text":" general transition and the trivial probability of 1 is omitted. ","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:25930"}},{"Del":{"text":" At location $\\ell_1$ with a variable assignment $s(x) = 3$ three","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:12201"}},{"Del":{"text":" transitions $t_2$, $t_3$ and $t_4$ are possible. The non-determinism is","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:26077"}},{"Del":{"text":" resolved first. Then if the general transition containing $t_2$ and $t_3$ is","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:20465"}},{"Del":{"text":" selected, a coin is thrown to decide whether to take $t_2$ or $t_3$.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:55979"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":961,"ins":{"up_context":[{"change":4,"pos":56295}],"down_context":[{"change":4,"pos":26833}],"flag":"BLOCK","start":50668,"end":50975,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":56295},"to":{"change":4,"start":56295,"end":56524},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":56524},"to":{"change":4,"start":26742,"end":26833},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":26742},"to":{"change":4,"start":26742,"end":26833},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"from all the integers, but can be constrained by the guard, in which case the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:50745"}},{"Add":{"text":"transition is only taken if the non-deterministic sampling fulfils the guard. If","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:50826"}},{"Add":{"text":"the sampled value fulfils no transition's guard, the program terminates.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:50899"}},{"Add":{"text":"Non-deterministic sampling is resolved before non-deterministic branching.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:50974"}}],"del_contents":[{"Del":{"text":"from all the integers, but can be constraint by the guard, in which case the","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:56371"}},{"Del":{"text":"transition is only taken if the non-deterministic sampling fulfills the guard.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:56450"}},{"Del":{"text":"If the sampled value fulfills no transition's guard it means the program","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:56523"}},{"Del":{"text":"terminates. Non-deterministic sampling is resolved before the non-deterministic","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:26821"}},{"Del":{"text":"branching.","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:26832"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":975,"ins":{"up_context":[{"change":4,"pos":27151}],"down_context":[{"change":4,"pos":12277}],"flag":"BLOCK","start":50976,"end":51051,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":27151},"to":{"change":4,"start":27151,"end":27224},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" Take the program displayed in \\fref{fig:ex_pip_nds} as an example. The","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:51050"}}],"del_contents":[{"Del":{"text":" Take the program displayed in \\fref{fig:ex_pip_nds} for example. The","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:27223"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":977,"ins":{"up_context":[{"change":4,"pos":12348}],"down_context":[{"change":4,"pos":12428}],"flag":"BLOCK","start":51052,"end":51133,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":12348},"to":{"change":4,"start":12348,"end":12428},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" Furthermore, the temporary variable is constrained to $1 \\leq u \\leq 3$. The","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:51132"}}],"del_contents":[{"Del":{"text":" Furthermore, the temporary variable is constraint to $1 \\leq u \\leq 3$. The","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:12427"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":986,"ins":{"up_context":[{"change":4,"pos":20758}],"down_context":[{"change":4,"pos":56751}],"flag":"BLOCK","start":51134,"end":51208,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":20758},"to":{"change":4,"start":12778,"end":12851},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"\\Sref{sec:probability}). Updates are expanded to integer polynomials with","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:51207"}}],"del_contents":[{"Del":{"text":"\\Sref{sec:probability}. Updates are expanded to integer polynomials with","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:12850"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":988,"ins":{"up_context":[{"change":4,"pos":56832}],"down_context":[{"change":4,"pos":28132}],"flag":"BLOCK","start":51209,"end":51290,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":56832},"to":{"change":4,"start":56832,"end":56945},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"equal to the (possibly temporary) variable or is sampled from the distribution. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:51289"}}],"del_contents":[{"Del":{"text":"equal to the (possibly temporary) variable or sampled from the distribution with","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:56912"}},{"Del":{"text":"the corresponding probability. ","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:56944"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":998,"ins":{"up_context":[{"change":4,"pos":28448}],"down_context":[{"change":4,"pos":28521}],"flag":"BLOCK","start":51291,"end":51366,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":28448},"to":{"change":4,"start":28448,"end":28521},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" Take the program depicted in \\fref{fig:ex_pip_probs} as an example. In","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:51365"}}],"del_contents":[{"Del":{"text":" Take the program depicted in \\fref{fig:ex_pip_probs} for example. In","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:28520"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1000,"ins":{"up_context":[{"change":4,"pos":28593}],"down_context":[{"change":4,"pos":28849}],"flag":"BLOCK","start":51367,"end":51610,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":28593},"to":{"change":4,"start":28593,"end":28849},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" \\textsc{Bern}(0.5)$. The Bernoulli distribution can take the value of 1 or 2","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:51447"}},{"Add":{"text":" with a probability of 0.5 in both cases. In the case that the distribution is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:51529"}},{"Add":{"text":" sampled to 1, the variable x is decremented, and stays the same otherwise. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:51609"}}],"del_contents":[{"Del":{"text":" \\textsc{Bern}(0.5)$. The value of the Bernoulli distribution can take the","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:28670"}},{"Del":{"text":" value of 1 or 2 with a probability of 0.5 in both cases. In case that the","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:28748"}},{"Del":{"text":" distribution is sampled to 1, the variable x is decremented, and stays the","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:28827"}},{"Del":{"text":" same otherwise. ","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:28848"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1006,"ins":{"up_context":[{"change":4,"pos":28941}],"down_context":[{"change":4,"pos":29018}],"flag":"BLOCK","start":51611,"end":51689,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":28941},"to":{"change":4,"start":28941,"end":29018},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"programs, where every general transition consists of a single transition, all","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:51688"}}],"del_contents":[{"Del":{"text":"programs, where every general transition consist of a single transition, all","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:29017"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1008,"ins":{"up_context":[{"change":4,"pos":29090}],"down_context":[{"change":4,"pos":29169}],"flag":"BLOCK","start":51690,"end":51770,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":29090},"to":{"change":4,"start":29090,"end":29169},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"probabilistic sampling is done in the transitions. 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Similarly non-deterministic","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:29168"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1015,"ins":{"up_context":[{"change":4,"pos":21159}],"down_context":[{"change":4,"pos":21240}],"flag":"BLOCK","start":51771,"end":51849,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":21159},"to":{"change":4,"start":6457,"end":6538},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"then after a small excursion to Markov decision processes, the semantics are ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:51848"}}],"del_contents":[{"Del":{"text":"then after a small excursion to Markov decision processes, the semantics are are","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:6537"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1019,"ins":{"up_context":[{"change":4,"pos":19013}],"down_context":[{"change":4,"pos":19227}],"flag":"BLOCK","start":51850,"end":52069,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":19013},"to":{"change":4,"start":56946,"end":57024},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":57024},"to":{"change":4,"start":19088,"end":19227},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":19088},"to":{"change":4,"start":19088,"end":19227},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"of locations, and $\\T = \\Loc \\times [0,1] \\times (\\PV \\rightarrow","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:51915"}},{"Add":{"text":"\\Z[\\V\\union\\D]) \\times \\R \\times \\Loc$ be the set of transitions over locations","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:51995"}},{"Add":{"text":"$\\Loc$. Let $\\GT = 2^{\\T}$ be a set of general transitions over $\\Loc$. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:52068"}}],"del_contents":[{"Del":{"text":"of locations, $\\T = \\Loc \\times [0,1] \\times (\\PV \\rightarrow \\Z[\\V\\union\\D])","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:57023"}},{"Del":{"text":"\\times \\R \\times \\Loc$ be the set of transitions over locations $\\Loc$. Let $\\GT","pos":"M3WK5HF3XU5MN2JKJMRJVDMC74F4MZAQ23FTGJRW4CGU5ZL72MHAC:19168"}},{"Del":{"text":"= 2^{\\T}$ be set set of general transitions over $\\Loc$. ","pos":"M3WK5HF3XU5MN2JKJMRJVDMC74F4MZAQ23FTGJRW4CGU5ZL72MHAC:19226"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1023,"ins":{"up_context":[{"change":4,"pos":19228}],"down_context":[{"change":4,"pos":57025}],"flag":"BLOCK","start":52070,"end":52157,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":4,"pos":19228},"to":{"change":4,"start":15369,"end":15455},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":29542},"to":{"change":4,"start":15369,"end":15455},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"\\begin{definition}[Probabilistic Integer Program \\cite{meyer2021tacas}]\\label{def:pip}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:52156"}}],"del_contents":[{"Del":{"text":"\\begin{definition}[Probabilistic Integer Program\\cite{meyer2021tacas}]\\label{def:pip}","pos":"KVSF5AFJEIMIYAC264OUAA56SGP5MFAUG3AECKOTUPIXPGULVFQQC:15454"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1025,"ins":{"up_context":[{"change":4,"pos":57106}],"down_context":[{"change":4,"pos":11429}],"flag":"BLOCK","start":52158,"end":52176,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":57106},"to":{"change":4,"start":57106,"end":57124},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" system where:","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:52175"}}],"del_contents":[{"Del":{"text":" system where ","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:57123"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1028,"ins":{"up_context":[{"change":4,"pos":21308}],"down_context":[{"change":4,"pos":57206}],"flag":"BLOCK","start":52177,"end":52259,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":21308},"to":{"change":4,"start":57125,"end":57206},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" \\item $\\Loc_\\Prog \\subseteq \\Loc$ is a finite non-empty set of locations,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:52258"}}],"del_contents":[{"Del":{"text":" \\item $\\Loc_\\Prog \\subseteq \\Loc$ is a finite non-empty set of location,","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:57205"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1045,"ins":{"up_context":[{"change":4,"pos":21788}],"down_context":[{"change":4,"pos":57720}],"flag":"BLOCK","start":52260,"end":52339,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":21788},"to":{"change":4,"start":57642,"end":57720},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" transitions share a common start location, which we call $\\ell_g$.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:52338"}}],"del_contents":[{"Del":{"text":" transitions share a common start location which we call $\\ell_g$.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:57719"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1050,"ins":{"up_context":[{"change":4,"pos":15861}],"down_context":[{"change":4,"pos":57871}],"flag":"BLOCK","start":52340,"end":52421,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":15861},"to":{"change":4,"start":57791,"end":57871},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"Before defining the semantics of a \\gls{pip}, a new special location $\\ell_\\bot$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:52420"}}],"del_contents":[{"Del":{"text":"Before defining the semantics of a \\gls{pip} a new special location $\\ell_\\bot$","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:57870"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1053,"ins":{"up_context":[{"change":4,"pos":58030}],"down_context":[{"change":4,"pos":58111}],"flag":"BLOCK","start":52422,"end":52494,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":58030},"to":{"change":4,"start":58030,"end":58111},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"with the same purpose. Instead of just terminating, a program takes the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:52493"}}],"del_contents":[{"Del":{"text":"with the same purpose. 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Since no normal transition leaves the location","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:52574"}},{"Add":{"text":"$l_\\bot$ the run loops indefinitely on the location $\\ell_\\bot$. In addition, a","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:52654"}}],"del_contents":[{"Del":{"text":"transitions guard is satisfied. Since no normal transition leaves the location","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:22095"}},{"Del":{"text":"$l_\\bot$ the run loops indefinitely on the location $\\ell_\\bot$. 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This won't be relevant for the partial evaluation but is","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:58648"}},{"Del":{"text":"important to define the probabilistic properties of a \\gls{pip}. ","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:58714"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1074,"ins":{"up_context":[{"change":4,"pos":17422}],"down_context":[{"change":4,"pos":17501}],"flag":"BLOCK","start":52805,"end":52885,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":17422},"to":{"change":4,"start":17422,"end":17501},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" locations and transitions, that is, a configuration with a location that is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:52884"}}],"del_contents":[{"Del":{"text":" locations and transitions, that is a configuration with a location that is","pos":"KVSF5AFJEIMIYAC264OUAA56SGP5MFAUG3AECKOTUPIXPGULVFQQC:17500"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1077,"ins":{"up_context":[{"change":4,"pos":17659}],"down_context":[{"change":4,"pos":17740}],"flag":"BLOCK","start":52886,"end":52968,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":17659},"to":{"change":4,"start":17659,"end":17740},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" prefixes does not exclude invalid paths, e.g., two consecutive configurations","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:52967"}}],"del_contents":[{"Del":{"text":" prefixes does not exclude invalid paths, e.g. two consecutive configurations","pos":"KVSF5AFJEIMIYAC264OUAA56SGP5MFAUG3AECKOTUPIXPGULVFQQC:17739"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1083,"ins":{"up_context":[{"change":4,"pos":36478}],"down_context":[{"change":4,"pos":36530}],"flag":"BLOCK","start":52969,"end":53041,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":36478},"to":{"change":4,"start":6539,"end":6611},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"\\subsection{Probabilistic Integer Program as a Markov Decision Process}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53040"}}],"del_contents":[{"Del":{"text":"\\subsection{Probabilistic Integer Program as a Markov decision process}","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:6610"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1085,"ins":{"up_context":[{"change":4,"pos":36608}],"down_context":[{"change":4,"pos":36682}],"flag":"BLOCK","start":53042,"end":53117,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":36608},"to":{"change":4,"start":36608,"end":36682},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"the concept to a run of a \\gls{pip}. In general, a Markov decision process","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53116"}}],"del_contents":[{"Del":{"text":"the concept to a run of a \\gls{pip}. In general a Markov decision process","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:36681"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1088,"ins":{"up_context":[{"change":4,"pos":36721}],"down_context":[{"change":4,"pos":36800}],"flag":"BLOCK","start":53118,"end":53202,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":36721},"to":{"change":4,"start":22639,"end":22716},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" \\item \\textit{decision epochs}, which are points in time at which a decision is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53201"}}],"del_contents":[{"Del":{"text":" \\item \\textit{decision epochs} are points in time at which a decision is","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:22715"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1095,"ins":{"up_context":[{"change":4,"pos":23021}],"down_context":[{"change":4,"pos":37252}],"flag":"BLOCK","start":53203,"end":53256,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":4,"pos":23021},"to":{"change":4,"start":37200,"end":37252},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":37200},"to":{"change":4,"start":37200,"end":37252},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" with every action. \\cite{puterman1994markov}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53255"}}],"del_contents":[{"Del":{"text":" with every action.\\cite{puterman1994markov}","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:37251"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1105,"ins":{"up_context":[{"change":4,"pos":37673}],"down_context":[{"change":4,"pos":23255}],"flag":"BLOCK","start":53257,"end":53336,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":37673},"to":{"change":4,"start":23177,"end":23255},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"The highest reward is the optimisation goal, when taking decisions in a Markov","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53335"}}],"del_contents":[{"Del":{"text":"The highest reward is the optimization goal, when taking decision in a Markov","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:23254"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1107,"ins":{"up_context":[{"change":4,"pos":23330}],"down_context":[{"change":4,"pos":37968}],"flag":"BLOCK","start":53337,"end":53559,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":23330},"to":{"change":4,"start":23330,"end":23540},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"accumulated rewards over a whole process. In the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53385"}},{"Add":{"text":"literature \\cite{puterman1994markov, puterman1990markov}, the properties of a","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53463"}},{"Add":{"text":"given process with regard to the highest achievable rewards have been","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53533"}},{"Add":{"text":"extensively researched. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53558"}}],"del_contents":[{"Del":{"text":"accumulated rewards over a whole process. In literature\\cite{puterman1994markov,","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:23410"}},{"Del":{"text":"puterman1990markov}, the properties of a given process with regard to the","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:23484"}},{"Del":{"text":"highest achievable rewards is researched extensively. ","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:23539"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1113,"ins":{"up_context":[{"change":4,"pos":38042}],"down_context":[{"change":4,"pos":38227}],"flag":"BLOCK","start":53560,"end":53726,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":38042},"to":{"change":4,"start":23541,"end":23701},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"complexity. If one defines runtime as a reward, one can picture the policy for a","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53640"}},{"Add":{"text":"Markov decision process as an adversary. We call the policy the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53704"}},{"Add":{"text":"\\enquote{scheduler}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53725"}}],"del_contents":[{"Del":{"text":"complexity. If one defines runtime as reward, one can picture the policy for a","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:23619"}},{"Del":{"text":"Markov decision process as an adversary. We call the policy \\enquote{scheduler}.","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:23700"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1121,"ins":{"up_context":[{"change":4,"pos":13022}],"down_context":[{"change":4,"pos":38408}],"flag":"BLOCK","start":53727,"end":53802,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":4,"pos":13022},"to":{"change":4,"start":38334,"end":38408},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":38334},"to":{"change":4,"start":38334,"end":38408},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" infinitely many. Hence this is an infinite horizon Markov decision","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53801"}}],"del_contents":[{"Del":{"text":" infinitely many. Hence this is a infinite horizon Markov decision","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:38407"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1127,"ins":{"up_context":[{"change":4,"pos":23965}],"down_context":[{"change":4,"pos":23966}],"flag":"BLOCK","start":53803,"end":53865,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":4,"pos":23965},"to":{"change":4,"start":38685,"end":38747},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":38685},"to":{"change":4,"start":38685,"end":38747},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" \\item The transition probability will be defined later in","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53864"}}],"del_contents":[{"Del":{"text":" \\item the transition probability will be defined later in","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:38746"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1129,"ins":{"up_context":[{"change":4,"pos":24004}],"down_context":[{"change":4,"pos":38847}],"flag":"BLOCK","start":53866,"end":53930,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":24004},"to":{"change":4,"start":24004,"end":24068},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" \\item The reward is 1 for every non-terminating transition.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:53929"}}],"del_contents":[{"Del":{"text":" \\item the reward is 1 for every non terminating transition.","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:24067"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1139,"ins":{"up_context":[{"change":4,"pos":19111}],"down_context":[{"change":4,"pos":58931}],"flag":"BLOCK","start":53931,"end":54091,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":39278},"to":{"change":4,"start":39278,"end":39434},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"We differentiate between two types of schedulers: the first is a scheduler that","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:54010"}},{"Add":{"text":"at a given state always takes the same decision -- it has no memory of its past","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:54090"}}],"del_contents":[{"Del":{"text":"We differentiate between two types of schedulers: First the scheduler that at a","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:39357"}},{"Del":{"text":"given state always takes the same decision -- it has no memory of it's past","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:39433"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1166,"ins":{"up_context":[{"change":4,"pos":40278}],"down_context":[{"change":4,"pos":40405}],"flag":"BLOCK","start":54092,"end":54220,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":40278},"to":{"change":4,"start":40278,"end":40405},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" action, the new state is determined by the probability distribution","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:54163"}},{"Add":{"text":" described by the \\textit{transition probabilities}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:54219"}}],"del_contents":[{"Del":{"text":" action the new state is determined by the probability distribution described","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:40358"}},{"Del":{"text":" by the \\textit{transition probabilities}.","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:40404"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1175,"ins":{"up_context":[{"change":4,"pos":24700}],"down_context":[{"change":4,"pos":24990}],"flag":"BLOCK","start":54221,"end":54515,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":24700},"to":{"change":4,"start":24700,"end":24777},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":24777},"to":{"change":4,"start":59338,"end":59428},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | 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$c$.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:54514"}}],"del_contents":[{"Del":{"text":" \\{g_\\bot\\})$ is a history dependent scheduler if for every finite prefix","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:24776"}},{"Del":{"text":" $f\\in\\fpath_\\Prog$ and configuration $c=(\\ell,t,s) \\in \\confs_\\Prog$, $\\scheduler(fc)","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:59427"}},{"Del":{"text":" = (g, s')$ implies items \\ref{itm:md1} to \\ref{itm:md4} of Definition","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:13248"}},{"Del":{"text":" \\ref{def:mscheduler} for the last configuration $c$.","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:24989"}}]}},{"Edit":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1183,"atom":{"EdgeMap":{"edges":[{"previous":"BLOCK","flag":"BLOCK | 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","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:60104"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1184,"ins":{"up_context":[{"change":4,"pos":60106}],"down_context":[{"change":4,"pos":13337}],"flag":"BLOCK","start":54516,"end":54589,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":4,"pos":60106},"to":{"change":4,"start":40907,"end":40980},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":40907},"to":{"change":4,"start":40907,"end":40980},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"\\subsection{Probability Space of a Program\\label{ssec:probabilityspace}}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:54588"}}],"del_contents":[{"Del":{"text":"\\subsection{Probability space of a program\\label{ssec:probabilityspace}}","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:40979"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1193,"ins":{"up_context":[{"change":4,"pos":13945}],"down_context":[{"change":4,"pos":41618}],"flag":"BLOCK","start":54590,"end":54677,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":13945},"to":{"change":4,"start":13945,"end":14023},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"construction we refer the reader to \\citeauthor{meyer2021tacas} \\cite{meyer2021tacas}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:54676"}}],"del_contents":[{"Del":{"text":"construction we'll refer to \\citeauthor{meyer2021tacas}\\cite{meyer2021tacas}.","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:14022"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1195,"ins":{"up_context":[{"change":4,"pos":41619}],"down_context":[{"change":4,"pos":41697}],"flag":"BLOCK","start":54678,"end":54757,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":41619},"to":{"change":4,"start":41619,"end":41697},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"On any given configuration, a Markovian scheduler resolves the non-determinism","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:54756"}}],"del_contents":[{"Del":{"text":"On any given configuration a Markovian scheduler resolves the non-determinism","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:41696"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1197,"ins":{"up_context":[{"change":4,"pos":41776}],"down_context":[{"change":4,"pos":41850}],"flag":"BLOCK","start":54758,"end":54832,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":41776},"to":{"change":4,"start":41776,"end":41850},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"the previous configuration. The scheduler keeps the values of the program","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:54831"}}],"del_contents":[{"Del":{"text":"the previous configuration. The scheduler keeps the values of all program","pos":"R2A2OD7J5ATBRR7W4QPTZDVH6GGHQ7XBX2JE5R3GKVS5WIQ43O3AC:41849"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1201,"ins":{"up_context":[{"change":4,"pos":25370}],"down_context":[{"change":4,"pos":14095}],"flag":"BLOCK","start":54833,"end":54905,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":25370},"to":{"change":4,"start":14024,"end":14095},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"terminating state $\\ell_\\bot$ it takes the special transition $g_\\bot$,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:54904"}}],"del_contents":[{"Del":{"text":"terminating state $\\ell_\\bot$ it takes the special transition $g_\\bot$","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:14094"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1208,"ins":{"up_context":[{"change":4,"pos":60655}],"down_context":[{"change":4,"pos":20280}],"flag":"BLOCK","start":54906,"end":55071,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":60655},"to":{"change":4,"start":60655,"end":60815},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"other configuration are invalid starting configurations and they are assigned a","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:54985"}},{"Add":{"text":"zero probability by the probability measure $\\prSs : \\confs_\\Prog \\rightarrow","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55063"}},{"Add":{"text":"[0,1]$","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55070"}}],"del_contents":[{"Del":{"text":"other configuration are invalid starting configurations and get assigned a zero","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:60734"}},{"Del":{"text":"probability by the probability measure $\\prSs : \\confs_\\Prog \\rightarrow [0,1]$","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:60814"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1219,"ins":{"up_context":[{"change":4,"pos":20444}],"down_context":[{"change":4,"pos":21192}],"flag":"BLOCK","start":55072,"end":55842,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":20444},"to":{"change":4,"start":60875,"end":60952},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":60952},"to":{"change":4,"start":20518,"end":20676},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":20518},"to":{"change":4,"start":20518,"end":20676},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":20676},"to":{"change":4,"start":60953,"end":61185},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":61185},"to":{"change":4,"start":14195,"end":14427},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":14427},"to":{"change":4,"start":61347,"end":61411},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":42628},"to":{"change":4,"start":61347,"end":61411},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"Let us say that the program ran up to a configuration $c=(\\ell,t,s)$. The","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55145"}},{"Add":{"text":"scheduler resolves the non-deterministic sampling and branching to","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55212"}},{"Add":{"text":"$\\scheduler(c) = (g, \\tilde{s})$. The probability for the event $c \\rightarrow","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55291"}},{"Add":{"text":"c'$ that $c'$ follows the configuration $c$ with $c' = (\\ell', t', s')$ and $t'","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55371"}},{"Add":{"text":"= (\\ell, p, \\tau, \\eta, \\_, \\ell') \\in g$ depends on three things: First the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55448"}},{"Add":{"text":"transition $t \\in g$ is taken with the probability $p$; second the probability","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55527"}},{"Add":{"text":"that each program variable $x\\in\\PV$ is updated to the value $s'(x)$ by the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55603"}},{"Add":{"text":"probabilistic update -- recall Definition \\ref{def:prob_update}; third that the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55683"}},{"Add":{"text":"temporary variables are sampled by the scheduler equal to the target state. The","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55763"}},{"Add":{"text":"result is the probability measure $\\prSs: \\confs_\\Prog^2 \\rightarrow [0,1]$. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55841"}}],"del_contents":[{"Del":{"text":"Lets say the program ran up to a configuration $c=(\\ell,t,s)$. The scheduler","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:60951"}},{"Del":{"text":"resolves the non-deterministic sampling and branching to $\\scheduler(c) = (g,","pos":"KVSF5AFJEIMIYAC264OUAA56SGP5MFAUG3AECKOTUPIXPGULVFQQC:20595"}},{"Del":{"text":"\\tilde{s})$. The probability for the event $c \\rightarrow c'$ that $c'$ follows","pos":"KVSF5AFJEIMIYAC264OUAA56SGP5MFAUG3AECKOTUPIXPGULVFQQC:20675"}},{"Del":{"text":"the configuration $c$ with $c' = (\\ell', t', s')$ and $t' = (\\ell, p, \\tau,","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:61028"}},{"Del":{"text":"\\eta, \\_, \\ell') \\in g$ depends on three things: First the transition $t \\in g$","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:61108"}},{"Del":{"text":"is taken with the probability $p$, second the probability that each program","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:61184"}},{"Del":{"text":"variable $x\\in\\PV$ is updated to the value $s'(x)$ by the probabilistic update","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:14273"}},{"Del":{"text":"-- recall Definition \\ref{def:prob_update}, third that the temporary variables","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:14352"}},{"Del":{"text":"are sampled by the scheduler equal to the target state. The result is the","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:14426"}},{"Del":{"text":"probability measure $\\prSs: \\confs_\\Prog^2 \\rightarrow [0,1]$. ","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:61410"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1251,"ins":{"up_context":[{"change":4,"pos":21875}],"down_context":[{"change":4,"pos":18066}],"flag":"BLOCK","start":55843,"end":56034,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":21875},"to":{"change":4,"start":21875,"end":21952},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":21952},"to":{"change":4,"start":26112,"end":26220},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"By a cylindrical construction one arrives at the final probability measure for","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55921"}},{"Add":{"text":"the infinite runs $\\PrSs : \\runs_\\Prog \\rightarrow [0,1]$ and the probability","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:55999"}},{"Add":{"text":"space $(\\runs_\\Prog, \\F, \\PrSs)$.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56033"}}],"del_contents":[{"Del":{"text":"By a cylindrical construction one gets the final probability measure for the","pos":"KVSF5AFJEIMIYAC264OUAA56SGP5MFAUG3AECKOTUPIXPGULVFQQC:21951"}},{"Del":{"text":"infinite runs $\\PrSs : \\runs_\\Prog \\rightarrow [0,1]$ and the probability space","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:26191"}},{"Del":{"text":"$(\\runs_\\Prog, \\F, \\PrSs)$.","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:26219"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1268,"ins":{"up_context":[{"change":4,"pos":22576}],"down_context":[{"change":4,"pos":61581}],"flag":"BLOCK","start":56035,"end":56107,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":22576},"to":{"change":4,"start":22576,"end":22651},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" A run is terminating if it is admissible and it has a finite prefix","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56106"}}],"del_contents":[{"Del":{"text":" A run is is terminating if it is admissible and it has a finite prefix","pos":"KVSF5AFJEIMIYAC264OUAA56SGP5MFAUG3AECKOTUPIXPGULVFQQC:22650"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1278,"ins":{"up_context":[{"change":4,"pos":23071}],"down_context":[{"change":4,"pos":43595}],"flag":"BLOCK","start":56108,"end":56140,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":23071},"to":{"change":4,"start":18906,"end":18938},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"\\subsection{Runtime Complexity}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56139"}}],"del_contents":[{"Del":{"text":"\\subsection{Runtime-complexity}","pos":"45HZPHVHYJ2BXATQ4MKCMPMMPZFBNJECY7JV3WX63BSA4CSQN6HAC:18937"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1327,"ins":{"up_context":[{"change":4,"pos":27632}],"down_context":[{"change":4,"pos":27711}],"flag":"BLOCK","start":56141,"end":56221,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":27632},"to":{"change":4,"start":27632,"end":27711},"introduced_by":4}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" The expected runtime does not depend on the scheduler and is finite for all","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56220"}}],"del_contents":[{"Del":{"text":" The expected runtime doesn't depend on the scheduler and is finite for all","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:27710"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1339,"ins":{"up_context":[{"change":2,"pos":18917}],"down_context":[{"change":2,"pos":19005}],"flag":"BLOCK","start":56222,"end":56331,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":18917},"to":{"change":2,"start":18917,"end":19005},"introduced_by":2}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" For an arbitrary \\gls{pip} $\\Prog$ and an arbitrary initial state","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56291"}},{"Add":{"text":" $s_0\\in\\Sigma$ the following holds","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56330"}}],"del_contents":[{"Del":{"text":" For a \\gls{pip} $\\Prog$ and an initial state $s_0\\in\\Sigma$ the following","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:18994"}},{"Del":{"text":" holds","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:19004"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1346,"ins":{"up_context":[{"change":2,"pos":19140}],"down_context":[{"change":2,"pos":19295}],"flag":"BLOCK","start":56332,"end":56511,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":19140},"to":{"change":2,"start":19140,"end":19295},"introduced_by":2}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" That means, that it doesn't matter whether we use Markovian or history","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56406"}},{"Add":{"text":" dependent deterministic schedulers when considering the worst-case expected","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56486"}},{"Add":{"text":" runtime complexity.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56510"}}],"del_contents":[{"Del":{"text":" That means, that it doesn't matter if we use Markovian or history dependent","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:19219"}},{"Del":{"text":" scheduler when considering the worst-case expected runtime complexity.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:19294"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1351,"ins":{"up_context":[{"change":2,"pos":19314}],"down_context":[{"change":2,"pos":19520}],"flag":"BLOCK","start":56512,"end":56731,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":19314},"to":{"change":2,"start":19314,"end":19520},"introduced_by":2}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" We define $\\nu^\\scheduler$ to be the expected runtime for any","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56581"}},{"Add":{"text":" deterministic scheduler $\\scheduler$ -- Markovian or history dependent","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56660"}},{"Add":{"text":" -- on a program $\\Prog$ with an initial state $s_0\\in\\Sigma$.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56730"}}],"del_contents":[{"Del":{"text":" We define $\\nu^\\scheduler$ to be the expected runtime for any scheduler","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:19393"}},{"Del":{"text":" $\\scheduler$ -- Markovian or history dependent -- on a program $\\Prog$","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:19472"}},{"Del":{"text":" with an initial state $s_0\\in\\Sigma$.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:19518"}},{"Del":{"text":"","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:19519"}}]}},{"Edit":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1359,"atom":{"NewVertex":{"up_context":[{"change":2,"pos":19752}],"down_context":[{"change":2,"pos":19752}],"flag":"BLOCK","start":56732,"end":56947,"inode":{"change":4,"pos":8115}}},"contents":[{"Add":{"text":" Let $\\scheduler \\in \\HDS$ be a \\emph{history dependent} scheduler with","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56810"}},{"Add":{"text":" an expected reward $\\nu^\\scheduler$. We argue that $\\sup_{\\scheduler'","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56888"}},{"Add":{"text":" \\in \\MDS} \\nu^{\\scheduler'} \\geq \\nu^\\scheduler$.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:56946"}}]}},{"Edit":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1363,"atom":{"EdgeMap":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":19753},"to":{"change":2,"start":19753,"end":19978},"introduced_by":2}],"inode":{"change":4,"pos":8115}}},"contents":[{"Del":{"text":" Let $\\scheduler \\in HDS$ be a \\emph{history dependent} scheduler with an","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:19833"}},{"Del":{"text":" expected reward $\\nu^\\scheduler$. ","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:19876"}},{"Del":{"text":"","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:19877"}},{"Del":{"text":" We argue that $\\sup_{\\scheduler' \\in \\MDS} \\nu^{\\scheduler'} \\geq","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:19951"}},{"Del":{"text":" \\nu^\\scheduler$.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:19976"}},{"Del":{"text":"","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:19977"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1365,"ins":{"up_context":[{"change":2,"pos":20132}],"down_context":[{"change":2,"pos":20322}],"flag":"BLOCK","start":56948,"end":57218,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":20132},"to":{"change":2,"start":20132,"end":20322},"introduced_by":2}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" infinitely many prefixes, since there can be infinitely many assignments","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57028"}},{"Add":{"text":" caused by probabilistic sampling from finite distributions. We approach","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57108"}},{"Add":{"text":" the set $\\fpath_N$ with a series of finite sets $F_1, F_2, \\dots","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57181"}},{"Add":{"text":" \\subseteq\\fpath$ such that ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57217"}}],"del_contents":[{"Del":{"text":" infinitely many prefixes, since there can be infinitely many","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:20200"}},{"Del":{"text":" assignments. We approach the set $\\fpath_N$ with a series of sets $F_1,","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:20280"}},{"Del":{"text":" F_2, \\dots \\subseteq\\fpath$ and ","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:20321"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1382,"ins":{"up_context":[{"change":2,"pos":20927}],"down_context":[{"change":2,"pos":21020}],"flag":"BLOCK","start":57219,"end":57427,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":20927},"to":{"change":2,"start":20927,"end":21020},"introduced_by":2}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" $\\scheduler_\\text{opt} \\in \\MDS_{F_i}$ exists that maximises the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57291"}},{"Add":{"text":" expected reward when limited to the finite set of configurations. Let","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57369"}},{"Add":{"text":" $\\nu_i$ be the reward of this optimal scheduler.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57426"}}],"del_contents":[{"Del":{"text":" in $\\HDS_{F_i}$ exists. Let $\\nu_i$ be the reward of this optimal","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:21000"}},{"Del":{"text":" scheduler.","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:21019"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1386,"ins":{"up_context":[{"change":2,"pos":21021}],"down_context":[{"change":2,"pos":21209}],"flag":"BLOCK","start":57428,"end":57632,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":21021},"to":{"change":2,"start":21021,"end":21209},"introduced_by":2}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" In the limit when $i$ approaches infinity, the optimal scheduler must","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57505"}},{"Add":{"text":" yield an expected no less than the history dependent scheduler","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57576"}},{"Add":{"text":" $\\scheduler$. We get the following inequality.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57631"}}],"del_contents":[{"Del":{"text":" With $i$ approaching infinity the optimal scheduler must still always be","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:21101"}},{"Del":{"text":" as good or better than the history dependent scheduler $\\scheduler$. We","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:21181"}},{"Del":{"text":" get the following:","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:21208"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1404,"ins":{"up_context":[{"change":2,"pos":21850}],"down_context":[{"change":2,"pos":21958}],"flag":"BLOCK","start":57633,"end":57751,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":21850},"to":{"change":2,"start":21850,"end":21958},"introduced_by":2}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" Since the above holds for any history dependent scheduler, it also holds","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57713"}},{"Add":{"text":" for the supremum and we get ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57750"}}],"del_contents":[{"Del":{"text":" Since the above holds for any history scheduler, it also holds for the","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:21928"}},{"Del":{"text":" supremum and we get ","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:21957"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1407,"ins":{"up_context":[{"change":2,"pos":21984}],"down_context":[{"change":2,"pos":22067}],"flag":"BLOCK","start":57752,"end":57865,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":21984},"to":{"change":2,"start":21984,"end":22067},"introduced_by":2}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" \\sup_{\\scheduler \\in \\MDS_\\Prog} \\ESs(\\Rt_\\Prog)\\geq \\sup_{\\scheduler \\in \\HDS_\\Prog}\\ESs(\\Rt_\\Prog)","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57864"}}],"del_contents":[{"Del":{"text":" \\sup_{\\scheduler \\in \\MDS_\\Prog} \\geq \\sup_{\\scheduler \\in \\HDS_\\Prog}","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:22066"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1410,"ins":{"up_context":[{"change":2,"pos":22092}],"down_context":[{"change":2,"pos":22207}],"flag":"BLOCK","start":57866,"end":58002,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":22092},"to":{"change":2,"start":22092,"end":22207},"introduced_by":2}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" The inverse holds because the class of history dependent schedulers","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:57941"}},{"Add":{"text":" transcends its Markovian counterpart. One concludes","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:58001"}}],"del_contents":[{"Del":{"text":" The inverse holds because every Markovian scheduler can be used as a","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:22168"}},{"Del":{"text":" history dependent scheduler. ","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:22206"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1413,"ins":{"up_context":[{"change":2,"pos":22233}],"down_context":[{"change":2,"pos":22324}],"flag":"BLOCK","start":58003,"end":58122,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":2,"pos":22233},"to":{"change":2,"start":22233,"end":22324},"introduced_by":2}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":" \\sup_{\\scheduler \\in \\MDS_\\Prog}\\ESs(\\Rt_\\Prog) = \\sup_{\\scheduler \\in","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:58085"}},{"Add":{"text":" \\HDS_\\Prog}\\ESs(\\Rt_\\Prog);","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:58121"}}],"del_contents":[{"Del":{"text":" \\sup_{\\scheduler \\in \\MDS_\\Prog} \\leq \\sup_{\\scheduler \\in","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:22303"}},{"Del":{"text":" \\HDS_\\Prog}","pos":"KDY46C5VOI2QWEXPE6J6BW3U2KEMH3IYPUNYKH5FAISTRCWBWO3AC:22323"}}]}},{"Edit":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1416,"atom":{"NewVertex":{"up_context":[{"change":2,"pos":22348}],"down_context":[{"change":2,"pos":22348}],"flag":"BLOCK","start":58123,"end":58153,"inode":{"change":4,"pos":8115}}},"contents":[{"Add":{"text":" completing the proof.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:58152"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1434,"ins":{"up_context":[{"change":4,"pos":28209}],"down_context":[{"change":4,"pos":62906}],"flag":"BLOCK","start":58154,"end":58230,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | 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scheduler $\\scheduler$ and an initial state $s_0 \\in \\Sigma$ there is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:58311"}}],"del_contents":[{"Del":{"text":"case for a scheduler $\\scheduler$ and a initial state $s_0 \\in \\Sigma$ there is","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:63176"}}]}},{"Replacement":{"inode":{"change":4,"pos":8115},"path":"ch2_preliminaries.tex","line":1460,"ins":{"up_context":[{"change":4,"pos":30618}],"down_context":[{"change":2,"pos":22781}],"flag":"BLOCK","start":58313,"end":58393,"inode":{"change":4,"pos":8115}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":30618},"to":{"change":2,"start":22702,"end":22781},"introduced_by":2}],"inode":{"change":4,"pos":8115}},"ins_contents":[{"Add":{"text":"Adding a transition and possibly new vertices (locations) therein to a graph is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:58392"}}],"del_contents":[{"Del":{"text":"Adding 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famous problem at the heart of","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:58553"}},{"Add":{"text":"computer science and it is known to be undecidable. The so-called Halting","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:58627"}},{"Add":{"text":"Problem is strongly related to the search for upper and lower runtime complexity","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:58708"}},{"Add":{"text":"bounds, and it is equally undecidable. Nevertheless, the answers to these","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:58782"}},{"Add":{"text":"questions are very important in practice. For example, a compiler might want to","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:58862"}},{"Add":{"text":"warn the programmer about faulty code sections that were not marked explicitly","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:58941"}},{"Add":{"text":"to run indefinitely, or the developer could be forced to prove the efficiency of","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59022"}},{"Add":{"text":"his program in critical scenarios. In an ever more complex world, with ever more","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59103"}},{"Add":{"text":"complex algorithms, the need for automatic tools arise. Even though the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59175"}},{"Add":{"text":"question of runtime complexity cannot be answered in a general case, many","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59249"}},{"Add":{"text":"tools have been developed to automatically analyse the complexity of various","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59326"}},{"Add":{"text":"programming paradigms as tightly and quickly as possible. \\cite{giesl2017aprove,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59407"}}],"del_contents":[{"Del":{"text":"The question if a program terminates is a famous problem at the heart of","pos":"MQJMCLGFSBSFJ3JPCIHQSBZS4XZPGJI7NGGRASX5UVGJO6HES25AC:7560"}},{"Del":{"text":"computer science known to be undecidable. The so-called Halting Problem is","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:6830"}},{"Del":{"text":"strongly related to the search for upper and lower runtime complexity bounds,","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:15173"}},{"Del":{"text":"which is equally undecidable. Nevertheless, the answers to those questions are","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:15252"}},{"Del":{"text":"very important in practice. For example a compiler might want to warn the","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:15326"}},{"Del":{"text":"programmer about faulty code sections that were not marked explicitly to run","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:15403"}},{"Del":{"text":"indefinitely, or the developer could be forced to prove the efficiency of his","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:15481"}},{"Del":{"text":"program in critical scenarios. In an ever more complex world, with ever more","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:15558"}},{"Del":{"text":"complex algorithms, the need for automatic tools arised. Even though the","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:15631"}},{"Del":{"text":"question for runtime-complexity can not be answered in a general case, many","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:15707"}},{"Del":{"text":"tools were developed to automatically analyze the complexity of various","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:15779"}},{"Del":{"text":"programming paradigms as tight and fast as possible. \\cite{giesl2017aprove,","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:15855"}}]}},{"Replacement":{"inode":{"change":4,"pos":14309},"path":"ch1_introduction.tex","line":17,"ins":{"up_context":[{"change":4,"pos":8565}],"down_context":[{"change":4,"pos":14332}],"flag":"BLOCK","start":59409,"end":59649,"inode":{"change":4,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":8565},"to":{"change":4,"start":8565,"end":8794},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":8794},"to":{"change":4,"start":28665,"end":29149},"introduced_by":4}],"inode":{"change":4,"pos":14309}},"ins_contents":[{"Add":{"text":"Those tools are being constantly improved on and they are regularly compared to each","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59493"}},{"Add":{"text":"other in the \\gls{termcomp} \\cite{termcomp2015,termcomp2019}, the goal being to","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59573"}},{"Add":{"text":"find as many, and as tight bounds as possible for a set of test programs. ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59648"}}],"del_contents":[{"Del":{"text":"Those tools are improved on constantly and they are regularly compared to each","pos":"MQJMCLGFSBSFJ3JPCIHQSBZS4XZPGJI7NGGRASX5UVGJO6HES25AC:8643"}},{"Del":{"text":"other in the \\gls{termcomp}\\cite{termcomp2015,termcomp2019}, the goal being to","pos":"MQJMCLGFSBSFJ3JPCIHQSBZS4XZPGJI7NGGRASX5UVGJO6HES25AC:8722"}},{"Del":{"text":"find as many, as tight bounds for a set of test programs as possible. ","pos":"MQJMCLGFSBSFJ3JPCIHQSBZS4XZPGJI7NGGRASX5UVGJO6HES25AC:8793"}},{"Del":{"text":"% The tests are chosen to represent a large variety of commonly encountered","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:28740"}},{"Del":{"text":"% real-world programs, but also include known-to-be hard-to-solve problems in","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:28818"}},{"Del":{"text":"% order to challenge the competitors and improve their tools. Their collection of","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:28900"}},{"Del":{"text":"% test programs can be found in the \\gls{tpdb} which is publicly","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:28965"}},{"Del":{"text":"% available\\footnote{\\url{https://github.com/TermCOMP/TPDB}}. During the latests","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:29046"}},{"Del":{"text":"% competition in 2022, eleven candidates lined up in twenty-eight","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:29112"}},{"Del":{"text":"% categories\\cite{termcomp2022url}.","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:29148"}}]}},{"Edit":{"inode":{"change":4,"pos":14309},"path":"ch1_introduction.tex","line":22,"atom":{"NewVertex":{"up_context":[{"change":4,"pos":29171}],"down_context":[{"change":4,"pos":29171}],"flag":"BLOCK","start":59650,"end":60403,"inode":{"change":4,"pos":14309}}},"contents":[{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59650"}},{"Add":{"text":"Consider the program $\\Prog_1$ displayed in \\fref{fig:classic_c}. It contains","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59728"}},{"Add":{"text":"two program variables $x$, and $y$, and one temporary variable $u$, which is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59805"}},{"Add":{"text":"used to non-deterministically sample $x$ in the first transition $t_0$. The","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59881"}},{"Add":{"text":"program clearly terminates. The first loop increments $x$ up to 3 if $x$ is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:59957"}},{"Add":{"text":"positive to begin with. The second loop decrements $y$ while $y$ is positive.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60035"}},{"Add":{"text":"However, classical analysis using only \\gls{mprf} fails to find a finite","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60108"}},{"Add":{"text":"runtime-complexity bound for this program \\cite{giesl2022lncs}. With the help of","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60189"}},{"Add":{"text":"\\gls{cfr} on sub-\\gls{scc} level, the program (see \\fref{fig:classic_pe_c}) can","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60269"}},{"Add":{"text":"be transformed into an equivalent program where the analysis with \\gls{mprf}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60346"}},{"Add":{"text":"succeeds at finding a finite runtime-complexity bound.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60401"}},{"Add":{"text":"","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60402"}}]}},{"Edit":{"inode":{"change":4,"pos":14309},"path":"ch1_introduction.tex","line":52,"atom":{"EdgeMap":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":7994},"to":{"change":4,"start":64610,"end":64766},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":64766},"to":{"change":4,"start":8679,"end":8755},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":8679},"to":{"change":4,"start":8679,"end":8755},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":8755},"to":{"change":4,"start":64767,"end":64990},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":64990},"to":{"change":4,"start":15987,"end":16065},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":16065},"to":{"change":4,"start":64991,"end":65206},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":9058},"to":{"change":4,"start":64991,"end":65206},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":9271},"to":{"change":4,"start":30533,"end":30534},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":65206},"to":{"change":4,"start":30533,"end":30534},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":30533},"to":{"change":4,"start":30533,"end":30534},"introduced_by":4}],"inode":{"change":4,"pos":14309}}},"contents":[{"Del":{"text":"Consider the program $\\Prog_1$ displayed in \\fref{fig:classic_c}. It contains","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:64687"}},{"Del":{"text":"two program variables $x, y$ and one temporary variables $u$ which is used to","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:64765"}},{"Del":{"text":"non-deterministically sample $x$ in the first transition $t_0$. The program","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:8754"}},{"Del":{"text":"clearly terminates. The first loop increments $x$ up to 3 if $x$ is positive to","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:64846"}},{"Del":{"text":"begin with. The second loop decrements $y$ while $y$ is positive. However, the","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:64925"}},{"Del":{"text":"classical analysis using only \\gls{mprf} fails to find a finite","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:64989"}},{"Del":{"text":"runtime-complexity bounds for this program\\cite{giesl2022lncs}. With the help","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:16064"}},{"Del":{"text":"of \\gls{cfr} on sub-\\gls{scc} level, the program (see \\fref{fig:classic_pe_c})","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:65069"}},{"Del":{"text":"can be transformed into an equivalent program where the analysis with","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:65139"}},{"Del":{"text":"\\gls{mprf} succeeds at finding a finite runtime-complexity bound.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:65205"}},{"Del":{"text":"","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:30533"}}]}},{"Replacement":{"inode":{"change":4,"pos":14309},"path":"ch1_introduction.tex","line":53,"ins":{"up_context":[{"change":4,"pos":30601}],"down_context":[{"change":4,"pos":30889}],"flag":"BLOCK","start":60404,"end":60685,"inode":{"change":4,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":30601},"to":{"change":4,"start":16066,"end":16131},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":16131},"to":{"change":4,"start":30680,"end":30889},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":30680},"to":{"change":4,"start":30680,"end":30889},"introduced_by":4}],"inode":{"change":4,"pos":14309}},"ins_contents":[{"Add":{"text":"\\citeauthor{Domenech19} \\cite{Domenech19} and it is called \\gls{cfr}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60472"}},{"Add":{"text":"via partial evaluation. It was implemented in their analysis tool","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60538"}},{"Add":{"text":"iRankfinder \\cite{irankfinder2018wst}. It achieves very similar results to","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60613"}},{"Add":{"text":"\\gls{mprf} when used on entire programs or whole \\glspl{scc}. Its real","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60684"}}],"del_contents":[{"Del":{"text":"\\citeauthor{Domenech19}\\cite{Domenech19} and is called \\gls{cfr}","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:16130"}},{"Del":{"text":"via partial evaluation and implemented in their analysis tool","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:30741"}},{"Del":{"text":"iRankfinder\\cite{irankfinder2018wst}. It achieves very similar results to","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:30815"}},{"Del":{"text":"\\gls{mprf} when used on the entire programs or whole \\gls{scc}. Its real","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:30888"}}]}},{"Replacement":{"inode":{"change":4,"pos":14309},"path":"ch1_introduction.tex","line":58,"ins":{"up_context":[{"change":4,"pos":30959}],"down_context":[{"change":4,"pos":31200}],"flag":"BLOCK","start":60686,"end":60915,"inode":{"change":4,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":30959},"to":{"change":4,"start":30959,"end":31040},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":31040},"to":{"change":4,"start":16132,"end":16196},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":16196},"to":{"change":4,"start":31120,"end":31200},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":31120},"to":{"change":4,"start":31120,"end":31200},"introduced_by":4}],"inode":{"change":4,"pos":14309}},"ins_contents":[{"Add":{"text":"specifically the loops where \\glspl{mprf} fail to find size bounds. This technique","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60768"}},{"Add":{"text":"was presented by \\citeauthor{giesl2022lncs} \\cite{giesl2022lncs}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60833"}},{"Add":{"text":"and implemented in \\gls{koat} \\cite{giesl2014ijcar} the complexity analysis tool","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60914"}}],"del_contents":[{"Del":{"text":"specifically the loops where \\gls{mprf} fail to find size bounds. This technique","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:31039"}},{"Del":{"text":"was presented by \\citeauthor{giesl2022lncs}\\cite{giesl2022lncs}","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:16195"}},{"Del":{"text":"and implemented in \\gls{koat}\\cite{giesl2014ijcar} the complexity analysis tool","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:31199"}}]}},{"Replacement":{"inode":{"change":4,"pos":14309},"path":"ch1_introduction.tex","line":91,"ins":{"up_context":[{"change":4,"pos":65636}],"down_context":[{"change":4,"pos":32702}],"flag":"BLOCK","start":60916,"end":61350,"inode":{"change":4,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":65636},"to":{"change":4,"start":65636,"end":65785},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":65785},"to":{"change":4,"start":16265,"end":16340},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":16340},"to":{"change":4,"start":65858,"end":66063},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":65858},"to":{"change":4,"start":65858,"end":66063},"introduced_by":4}],"inode":{"change":4,"pos":14309}},"ins_contents":[{"Add":{"text":"terminate, since a longer runtime becomes increasingly unlikely. The current","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:60992"}},{"Add":{"text":"version of \\gls{koat} fails to find a finite runtime and expected complexity","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61069"}},{"Add":{"text":"bound for this program. It would be useful to transform $\\Prog_2$ into an","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61143"}},{"Add":{"text":"equivalent program like the one shown in \\fref{fig:prob_pe_c}, similar to the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61221"}},{"Add":{"text":"\\gls{cfr} technique presented by \\citeauthor{giesl2022lncs} \\cite{giesl2022lncs}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61302"}},{"Add":{"text":"and \\citeauthor{Domenech19} \\cite{Domenech19}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61349"}}],"del_contents":[{"Del":{"text":"terminate, since a longer runtime gets increasingly unlikely. The current","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:65709"}},{"Del":{"text":"version of \\gls{koat} fails to find finite runtime and expected complexity","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:65784"}},{"Del":{"text":"bounds for this program. It would be useful to transform $\\Prog_2$ into an","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:16339"}},{"Del":{"text":"equivalent program like the one shown in \\fref{fig:prob_pe_c} similarly to the","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:65936"}},{"Del":{"text":"\\gls{cfr} technique presented by \\citeauthor{giesl2022lncs}\\cite{giesl2022lncs}","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66016"}},{"Del":{"text":"and \\citeauthor{Domenech19}\\cite{Domenech19}.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66062"}}]}},{"Replacement":{"inode":{"change":4,"pos":14309},"path":"ch1_introduction.tex","line":99,"ins":{"up_context":[{"change":4,"pos":9941}],"down_context":[{"change":4,"pos":33366}],"flag":"BLOCK","start":61351,"end":62242,"inode":{"change":4,"pos":14309}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":4,"pos":9941},"to":{"change":4,"start":32727,"end":32884},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":32727},"to":{"change":4,"start":32727,"end":32884},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":32884},"to":{"change":4,"start":66064,"end":66796},"introduced_by":4}],"inode":{"change":4,"pos":14309}},"ins_contents":[{"Add":{"text":"\\subsection{\\acrshort{aprove}, \\acrshort{koat}, and \\acrshort{loat}} ","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61420"}},{"Add":{"text":"The tools \\gls{loat} \\cite{frohn2022ijcar} and \\gls{koat}","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61478"}},{"Add":{"text":"\\cite{brockschmidt2016acm} were developed at RWTH University and they aim to","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61555"}},{"Add":{"text":"find lower and upper (respectively) runtime-complexity bounds on transition","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61631"}},{"Add":{"text":"systems. At its core, \\gls{koat} computes bounds for subprograms using","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61702"}},{"Add":{"text":"\\gls{mprf} or a \\gls{twn}, and then puts the subprograms together in order to","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61780"}},{"Add":{"text":"generate upper runtime-complexity and size bounds for the whole program.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61853"}},{"Add":{"text":"Optionally to improve the analysis, \\gls{koat} can resort to partial evaluation","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61933"}},{"Add":{"text":"as a control-flow refinement technique on a sub-\\gls{scc} level","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:61997"}},{"Add":{"text":"\\cite{giesl2022lncs}. \\gls{aprove} is the overall tool, and it has taken part in","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62078"}},{"Add":{"text":"competitions. It works on various input formats and transforms them accordingly","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62158"}},{"Add":{"text":"so that they can be used with specialised tools such as \\gls{koat} and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62229"}},{"Add":{"text":"\\gls{loat}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62241"}}],"del_contents":[{"Del":{"text":"\\subsection{\\acrshort{aprove}, \\acrshort{koat}, and \\acrshort{loat}} The tools","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:32805"}},{"Del":{"text":"LoAT\\cite{frohn2022ijcar} and KoAT\\cite{brockschmidt2016acm} are developed at","pos":"GNUONJQNZG2MNEE55RW3XOQP3UICEBOKLVLKV42H3K64IBGLUGIQC:32883"}},{"Del":{"text":"RWTH University and aim at finding lower and respectively upper","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66127"}},{"Del":{"text":"runtime-complexity bounds on transition systems. At its core, KoAT computes","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66203"}},{"Del":{"text":"bounds for subprograms using \\gls{mprf} or \\gls{twn}, and then putting the","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66278"}},{"Del":{"text":"subprograms together in order to generate upper runtime-complexity and size","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66354"}},{"Del":{"text":"bounds for the whole program. Optionally to improve the analysis, \\gls{koat} can","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66435"}},{"Del":{"text":"resort to partial evaluation as a control-flow refinement techniques on a","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66509"}},{"Del":{"text":"sub-\\gls{scc} level\\cite{giesl2022lncs}.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66550"}},{"Del":{"text":"\\gls{apron} is the overall tool, taking part in competitions. It works on","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66624"}},{"Del":{"text":"various input formats and transforms them accordingly to be used with","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66694"}},{"Del":{"text":"specialized tools. \\gls{koat} and \\gls{loat} under the hood of \\gls{apron} for","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66773"}},{"Del":{"text":"the overall analysis.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66795"}}]}},{"Replacement":{"inode":{"change":4,"pos":14309},"path":"ch1_introduction.tex","line":121,"ins":{"up_context":[{"change":4,"pos":41123}],"down_context":[{"change":4,"pos":41196}],"flag":"BLOCK","start":62243,"end":62315,"inode":{"change":4,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":41123},"to":{"change":4,"start":41123,"end":41196},"introduced_by":4}],"inode":{"change":4,"pos":14309}},"ins_contents":[{"Add":{"text":"analysis techniques (e.g. linear, lexicographic, and multiphase ranking","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62314"}}],"del_contents":[{"Del":{"text":"analysis techniques (e.g. linear, lexicographic, and multiphase ranking ","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:41195"}}]}},{"Replacement":{"inode":{"change":4,"pos":14309},"path":"ch1_introduction.tex","line":123,"ins":{"up_context":[{"change":4,"pos":41266}],"down_context":[{"change":4,"pos":10824}],"flag":"BLOCK","start":62316,"end":62459,"inode":{"change":4,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":41266},"to":{"change":4,"start":41266,"end":41407},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":41407},"to":{"change":4,"start":66797,"end":66924},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":66924},"to":{"change":4,"start":16341,"end":16368},"introduced_by":4}],"inode":{"change":4,"pos":14309}},"ins_contents":[{"Add":{"text":"\\cite{gulwani2009}. iRankFinder implemented the control-flow refinement","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62387"}},{"Add":{"text":"technique by partial evaluation proposed by \\cite{gallagher2019eptcs}.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62458"}}],"del_contents":[{"Del":{"text":"\\cite{gulwani2009}. A typical example of such a program can be seen","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:41333"}},{"Del":{"text":"in example \\ref{ex:splitterloop}, where a single loop is replaced by two","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:41406"}},{"Del":{"text":"independent consecutive loops. iRankFinder implemented the control-flow","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66868"}},{"Del":{"text":"refinement technique by partial evaluation proposed by","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:66923"}},{"Del":{"text":"\\cite{gallagher2019eptcs}.","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:16367"}}]}},{"Replacement":{"inode":{"change":4,"pos":14309},"path":"ch1_introduction.tex","line":127,"ins":{"up_context":[{"change":4,"pos":35953}],"down_context":[{"change":4,"pos":41827}],"flag":"BLOCK","start":62460,"end":62697,"inode":{"change":4,"pos":14309}},"del":{"edges":[{"previous":"","flag":"DELETED","from":{"change":4,"pos":35953},"to":{"change":4,"start":41593,"end":41673},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":10840},"to":{"change":4,"start":41593,"end":41673},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":41673},"to":{"change":4,"start":16369,"end":16444},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":16444},"to":{"change":4,"start":41747,"end":41827},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":41747},"to":{"change":4,"start":41747,"end":41827},"introduced_by":4}],"inode":{"change":4,"pos":14309}},"ins_contents":[{"Add":{"text":"\\gls{koat} borrowed the implementation of iRankFinder as a black box, which came","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62540"}},{"Add":{"text":"with a number of limitations.\\cite{giesl2022lncs} Besides code-quality and","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62615"}},{"Add":{"text":"performance issues, the external library would not be able to profit from future","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62696"}}],"del_contents":[{"Del":{"text":"\\gls{koat} borrowed the implementation of iRankFinder as a black-box which came","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:41672"}},{"Del":{"text":"with a number of limitations\\cite{giesl2022lncs}. Besides code-quality and","pos":"NEWG2Q2EYR3BOPWVN44EXIPVH2GH23BYFHJXUSNYWWT7WWDEAIKQC:16443"}},{"Del":{"text":"performance issues, the external library wouldn't be able to profit from future","pos":"QIVGMR6INKBVT5IFCO2UJJQEO4PVW3ZM7IDTCTCOKOX7VKG7UPFAC:41826"}}]}},{"Replacement":{"inode":{"change":4,"pos":14309},"path":"ch1_introduction.tex","line":139,"ins":{"up_context":[{"change":4,"pos":67122}],"down_context":[{"change":4,"pos":67198}],"flag":"BLOCK","start":62698,"end":62769,"inode":{"change":4,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":67122},"to":{"change":4,"start":67122,"end":67198},"introduced_by":4}],"inode":{"change":4,"pos":14309}},"ins_contents":[{"Add":{"text":"\\Glspl{pip}. We prove the theoretical correctness of the construction.","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62768"}}],"del_contents":[{"Del":{"text":"\\Glspl{pip}. We will prove the theoretical correctness of the construction.","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:67197"}}]}},{"Replacement":{"inode":{"change":4,"pos":14309},"path":"ch1_introduction.tex","line":142,"ins":{"up_context":[{"change":4,"pos":67286}],"down_context":[{"change":4,"pos":67365}],"flag":"BLOCK","start":62770,"end":62846,"inode":{"change":4,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":67286},"to":{"change":4,"start":67286,"end":67365},"introduced_by":4}],"inode":{"change":4,"pos":14309}},"ins_contents":[{"Add":{"text":"\\Cref{ch:implementation} focuses on the changes made to \\gls{koat}, and the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62845"}}],"del_contents":[{"Del":{"text":"\\Cref{ch:implementation} will focus on the changes made to \\gls{koat}, and the","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:67364"}}]}},{"Replacement":{"inode":{"change":4,"pos":14309},"path":"ch1_introduction.tex","line":144,"ins":{"up_context":[{"change":4,"pos":67432}],"down_context":[{"change":4,"pos":67513}],"flag":"BLOCK","start":62847,"end":62923,"inode":{"change":4,"pos":14309}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":67432},"to":{"change":4,"start":67432,"end":67513},"introduced_by":4}],"inode":{"change":4,"pos":14309}},"ins_contents":[{"Add":{"text":"in \\Cref{ch:conclusion} we recapitulate the results of this thesis and give","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62922"}}],"del_contents":[{"Del":{"text":"in \\Cref{ch:conclusion} we will recapitulate the results of this thesis and give","pos":"7JUQQYQCE7G3ESFJETNGEPBCYGR5NGMAWHFPWI24RSQJPFM5YCTQC:67512"}}]}},{"Replacement":{"inode":{"change":4,"pos":14402},"path":"ch0_frontmatter.tex","line":48,"ins":{"up_context":[{"change":4,"pos":16028}],"down_context":[{"change":4,"pos":16239}],"flag":"BLOCK","start":62924,"end":63883,"inode":{"change":4,"pos":14402}},"del":{"edges":[{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":16028},"to":{"change":4,"start":67545,"end":67634},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":67634},"to":{"change":4,"start":10272,"end":11011},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":10272},"to":{"change":4,"start":10272,"end":11011},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":11011},"to":{"change":4,"start":67635,"end":67700},"introduced_by":4},{"previous":"","flag":"DELETED","from":{"change":4,"pos":67700},"to":{"change":4,"start":11075,"end":11146},"introduced_by":4},{"previous":"BLOCK","flag":"BLOCK | DELETED","from":{"change":4,"pos":11075},"to":{"change":4,"start":11075,"end":11146},"introduced_by":4}],"inode":{"change":4,"pos":14402}},"ins_contents":[{"Add":{"text":" \\acrshort{koat} is an automatic complexity analysis tool for integer","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:62996"}},{"Add":{"text":" programs, which was recently extended to the analysis of probabilistic","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:63071"}},{"Add":{"text":" integer programs. Currently, \\acrshort{koat} applies so-called control-flow","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:63151"}},{"Add":{"text":" refinement via partial evaluation to classic programs by calling the tool","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:63229"}},{"Add":{"text":" iRankFinder externally. In this thesis, we extend partial evaluation to","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:63305"}},{"Add":{"text":" probabilistic programs. A partially evaluated probabilistic integer program","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:63385"}},{"Add":{"text":" has the same expected runtime complexity as the original program. However,","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:63464"}},{"Add":{"text":" the transformed program might be easier to analyse. The new algorithm is","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:63541"}},{"Add":{"text":" implemented natively in \\acrshort{koat} and it is integrated into the","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:63615"}},{"Add":{"text":" existing analysis algorithm modular and on demand. Our evaluation shows that","pos":"YUEGUKXBV6IKGZPTBXJXW2ANYGYSRJILQFLYCYQUYD54LJVZS4CQC:63696"}},{"Add":{"text":" the native implementation is faster on classical programs. 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","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:10517"}},{"Del":{"text":" In this thesis, we extend partial evaluation to probabilistic programs. A","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:10595"}},{"Del":{"text":" partially evaluated probabilistic integer program has the same expected","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:10671"}},{"Del":{"text":" runtime-complexity as the original program. However, the transformed program","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:10752"}},{"Del":{"text":" might be easier to analyse.","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:10784"}},{"Del":{"text":" The new algorithm is implemented natively in \\acrshort{koat} and is","pos":"YS5O2UANXCWRI3W7KMRKKK7OT7R6V3QDL4JS4DRLXM2EZUIPFALAC:10856"}},{"Del":{"text":" integrated into the existing analysis algorithm modular and on-demand. 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