File:CritPair.pdf
Summary
| Description |
English: Shows a triangle diagram of a critical pair obtained from two rewrite rules s→t (upper row, left) and l→r (right). The substitution σ unifies the subterm s|p with l. The resulting overlay term sσ[lσ]p (lower row, middle) can be rewritten to the term tσ and sσ[rσ]p (lower row, left and right), respectively. The latter two terms form the critical pair. They can be eventually rewritten to a common term, if the rewrite rule set is confluent. |
| Date | |
| Source | Own work |
| Author | Jochen Burghardt |
| Other versions | File:CritPair svg.svg |
LaTeX source code |
|---|
\documentclass[12pt]{article}
\usepackage[pdftex]{color}
\usepackage[paperwidth=455mm,paperheight=305mm]{geometry}
\usepackage{amssymb}
\usepackage{amsmath}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{455mm}
\setlength{\textheight}{305mm}
\setlength{\oddsidemargin}{-2.5cm}
\setlength{\parindent}{0cm}
\setlength{\unitlength}{1mm}
% background colors
\definecolor{bL} {rgb}{0.99,0.80,0.80} % term l
\definecolor{bR} {rgb}{0.80,0.99,0.80} % term r
\definecolor{bS} {rgb}{0.80,0.80,0.99} % term s
\definecolor{bT} {rgb}{0.99,0.99,0.80} % term t
\definecolor{bSL} {rgb}{0.80,0.60,0.80} % s,l intersection
\definecolor{bSR} {rgb}{0.60,0.80,0.80} % s,r intersection
% foreground colors
\definecolor{fL} {rgb}{0.50,0.00,0.00} % term l
\definecolor{fR} {rgb}{0.00,0.50,0.00} % term r
\definecolor{fS} {rgb}{0.00,0.00,0.50} % term s
\definecolor{fT} {rgb}{0.50,0.50,0.00} % term t
\definecolor{fMeta} {rgb}{0.00,0.00,0.00} % diagram arrows etc.
\definecolor{fPath} {rgb}{0.30,0.30,0.30} % path p
\definecolor{fSbs} {rgb}{0.10,0.10,0.10} % substitution
\begin{document}
\begin{picture}(450,300)
%\put(0,0){\makebox(0,0){$+$}}%
%\put(450,300){\makebox(0,0){$+$}}%
\color{fMeta}
\thicklines
% rules
\put(135,230){\vector(-3,-1){60}}
\put(335,230){\vector(3,-1){60}}
%\put(105,225){\makebox(0,0)[r]{\bf Rule 1}}
%\put(365,225){\makebox(0,0)[l]{\bf Rule 2}}
% rewrites
\put(185,105){\vector(-3,-1){60}}
\put(285,105){\vector(3,-1){60}}
% local confluence?
\multiput(125,35)(30,-10){3}{\vector(3,-1){20}}
\multiput(345,35)(-30,-10){3}{\vector(-3,-1){20}}
\put(235,5){\makebox(0,0){\Huge\bf ?}}
%%%%%%%%%%%%%%%%% rule: t %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\color{bT}
\multiput(5,190)(0.050,0.100){300}{\line(1,0){30}}
\multiput(35,250)(0.050,-0.100){480}{\line(-1,-2){15}}
\multiput(35,170)(0,0.050){400}{\line(1,0){30}}
\multiput(75,170)(-0.050,0.100){320}{\line(-1,0){30}}
\color{fT}
%\put(35,250){\circle*{1}} % tip
\put(40,220){\makebox(0,0){\huge $t$}}
\put(5,190){\line(1,0){30}}
\put(35,170){\line(0,1){20}}
\put(35,170){\line(1,0){40}}
\put(5,190){\line(1,2){30}}
\put(35,250){\line(1,-2){40}}
%%%%%%%%%%%%%%%%% rule: s %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\color{bS}
\multiput(155,240)(0.050,0.100){300}{\line(1,0){30}}
\multiput(185,300)(0.050,-0.100){480}{\line(-1,-2){15}}
\multiput(185,220)(0,0.050){400}{\line(1,0){30}}
\multiput(225,220)(-0.050,0.100){320}{\line(-1,0){30}}
\color{fS}
%\put(185,300){\circle*{1}} % tip
\put(190,270){\makebox(0,0){\huge $s$}}
\put(155,240){\line(1,0){30}}
\put(185,220){\line(0,1){20}}
\put(185,220){\line(1,0){40}}
\put(155,240){\line(1,2){30}}
\put(185,300){\line(1,-2){40}}
%%%%%%%%%%%%%%%% rule: l %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\color{bL}
\multiput(265,240)(0.050,0.100){200}{\line(1,0){10}}
\multiput(280,270)(0.050,-0.100){200}{\line(-1,-2){5}}
\multiput(295,240)(-0.050,0.100){100}{\line(-1,0){20}}
\multiput(255,220)(0.050,0.100){200}{\line(1,0){20}}
\multiput(265,220)(0,0.100){200}{\line(1,0){20}}
\color{fL}
%\put(280,270){\circle*{1}} % tip
\put(280,250){\makebox(0,0){\huge $l$}}
\put(255,220){\line(1,0){29.500}}
\put(255,220){\line(1,2){25}}
\put(280,270){\line(1,-2){14.750}}
\put(284.500,220){\line(0,1){20.500}}
\put(284.500,240.500){\line(1,0){10.250}}
%%%%%%%%%%%%%%%% rule: r %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\color{bR}
\multiput(415,190)(0.050,0.100){200}{\line(1,0){10}}
\multiput(430,220)(0.050,-0.100){200}{\line(-1,-2){5}}
\multiput(445,190)(-0.050,0.100){100}{\line(-1,0){20}}
\multiput(405,170)(0.050,0.100){200}{\line(1,0){20}}
\multiput(415,170)(0,0.100){200}{\line(1,0){20}}
\color{fR}
%\put(430,220){\circle*{1}} % tip
\put(430,200){\makebox(0,0){\huge $r$}}
\put(405,170){\line(1,0){29.500}}
\put(405,170){\line(1,2){25}}
\put(430,220){\line(1,-2){14.750}}
\put(434.500,170){\line(0,1){20.500}}
\put(434.500,190.500){\line(1,0){10.250}}
%%%%%%%%%%%%%%%% rewrite result s->t %%%%%%%%%%%%%%%%%%%
\color{bT}
\multiput(55,65)(0.050,0.100){400}{\line(1,0){10}}
\multiput(85,125)(0.050,-0.100){480}{\line(-1,-2){10}}
\multiput(85,45)(0,0.050){400}{\line(1,0){30}}
\multiput(125,45)(-0.050,0.100){320}{\line(-1,0){20}}
\color{bT}
\multiput(65,65)(0.050,0.100){200}{\line(1,0){10}}
\multiput(80,95)(0.050,-0.100){200}{\line(-1,-2){5}}
\multiput(95,65)(-0.050,0.100){100}{\line(-1,0){20}}
\color{bL}
\multiput(55,45)(0.050,0.100){200}{\line(1,0){20}}
\multiput(65,45)(0,0.100){200}{\line(1,0){20}}
\color{fT}
%\put(85,125){\circle*{1}} % tip
\put(90,95){\makebox(0,0){\huge $t$}}
\put(55,65){\line(1,0){30}}
\put(85,45){\line(0,1){20}}
\put(85,45){\line(1,0){40}}
\put(55,65){\line(1,2){30}}
\put(85,125){\line(1,-2){40}}
% matching subst
\color{fSbs}
\put(79.750,54.500){\makebox(0,0){\huge $\sigma$}}
\put(65.250,64.500){\line(-1,-2){9.500}}
\put(55.750,45.500){\line(1,0){29}}
%%%%%%%%%%%%%%%% redex overlap %%%%%%%%%%%%%%%%%%%%%%%%%
\color{bS}
\multiput(205,115)(0.050,0.100){400}{\line(1,0){10}}
\multiput(235,175)(0.050,-0.100){480}{\line(-1,-2){10}}
\multiput(235,95)(0,0.050){400}{\line(1,0){30}}
\multiput(275,95)(-0.050,0.100){320}{\line(-1,0){20}}
\color{bSL}
\multiput(215,115)(0.050,0.100){200}{\line(1,0){10}}
\multiput(230,145)(0.050,-0.100){200}{\line(-1,-2){5}}
\multiput(245,115)(-0.050,0.100){100}{\line(-1,0){20}}
\color{bL}
\multiput(205,95)(0.050,0.100){200}{\line(1,0){20}}
\multiput(215,95)(0,0.100){200}{\line(1,0){20}}
% s
\color{fS}
%\put(235,175){\circle*{1}} % tip
\put(240,145){\makebox(0,0){\huge $s$}}
\put(205,115){\line(1,0){30}}
\put(235,95){\line(0,1){20}}
\put(235,95){\line(1,0){40}}
\put(205,115){\line(1,2){30}}
\put(235,175){\line(1,-2){40}}
% l
\color{fL}
%\put(230,145){\circle*{1}} % tip
\put(230,125){\makebox(0,0){\huge $l$}}
\put(205,95){\line(1,0){29.500}}
\put(205,95){\line(1,2){25}}
\put(230,145){\line(1,-2){14.750}}
\put(234.500,95){\line(0,1){20.500}}
\put(234.500,115.500){\line(1,0){10.250}}
% matching subst
\color{fSbs}
\put(229.750,104.500){\makebox(0,0){\huge $\sigma$}}
\put(239.750,104.500){\makebox(0,0){\huge $\sigma$}}
\put(215.250,114.500){\line(1,0){30}}
\put(215.250,114.500){\line(-1,-2){9.500}}
\put(245.250,114.500){\line(1,-2){9.500}}
\put(205.750,95.500){\line(1,0){49}}
% path
\thinlines
\color{fPath}
\put(233,155){\makebox(0,0){\Large $p$}}
\put(235,175){\line(-1,-6){2}}
\put(233,163){\line(2,-5){4}}
\put(237,153){\line(-5,-3){5}}
\put(232,150){\line(-2,-5){2}}
\thicklines
%%%%%%%%%%%%%%%% rewrite result l->r %%%%%%%%%%%%%%%%%%%
\color{bS}
\multiput(355,65)(0.050,0.100){400}{\line(1,0){10}}
\multiput(385,125)(0.050,-0.100){480}{\line(-1,-2){10}}
\multiput(385,45)(0,0.050){400}{\line(1,0){30}}
\multiput(425,45)(-0.050,0.100){320}{\line(-1,0){20}}
\color{bSR}
\multiput(365,65)(0.050,0.100){200}{\line(1,0){10}}
\multiput(380,95)(0.050,-0.100){200}{\line(-1,-2){5}}
\multiput(395,65)(-0.050,0.100){100}{\line(-1,0){20}}
\color{bR}
\multiput(355,45)(0.050,0.100){200}{\line(1,0){20}}
\multiput(365,45)(0,0.100){200}{\line(1,0){20}}
% s
\color{fS}
%\put(385,125){\circle*{1}} % tip
\put(390,95){\makebox(0,0){\huge $s$}}
\put(355,65){\line(1,0){30}}
\put(385,45){\line(0,1){20}}
\put(385,45){\line(1,0){40}}
\put(355,65){\line(1,2){30}}
\put(385,125){\line(1,-2){40}}
% r
\color{fL}
%\put(380,95){\circle*{1}} % tip
\put(380,75){\makebox(0,0){\huge $r$}}
\put(355,45){\line(1,0){29.500}}
\put(355,45){\line(1,2){25}}
\put(380,95){\line(1,-2){14.750}}
\put(384.500,45){\line(0,1){20.500}}
\put(384.500,65.500){\line(1,0){10.250}}
% matching subst
\color{fSbs}
\put(390.500,54.500){\makebox(0,0){\huge $\sigma$}}
\put(385.250,64.500){\line(1,0){10}}
%\put(365.250,64.500){\line(-1,-2){9.500}}
\put(395.250,64.500){\line(1,-2){9.500}}
\put(385.250,45.500){\line(1,0){19.500}}
% path
\thinlines
\color{fPath}
\put(383,105){\makebox(0,0){\Large $p$}}
\put(385,125){\line(-1,-6){2}}
\put(383,113){\line(2,-5){4}}
\put(387,103){\line(-5,-3){5}}
\put(382,100){\line(-2,-5){2}}
\thicklines
\end{picture}
\end{document}
|
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