File:CritPair.pdf
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Summary[edit]
| 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(135.000,230.000){\vector(-3,-1){60.000}}
\put(335.000,230.000){\vector(3,-1){60.000}}
%\put(105.000,225.000){\makebox(0.000,0.000)[r]{\bf Rule 1}}
%\put(365.000,225.000){\makebox(0.000,0.000)[l]{\bf Rule 2}}
% rewrites
\put(185.000,105.000){\vector(-3,-1){60.000}}
\put(285.000,105.000){\vector(3,-1){60.000}}
% local confluence?
\multiput(125.000,35.000)(30.000,-10.000){3}{\vector(3,-1){20.000}}
\multiput(345.000,35.000)(-30.000,-10.000){3}{\vector(-3,-1){20.000}}
\put(235.000,5.000){\makebox(0.000,0.000){\Huge\bf ?}}
%%%%%%%%%%%%%%%%% rule: t %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\color{bT}
\multiput(5.000,190.000)(0.050,0.100){300}{\line(1,0){30.000}}
\multiput(35.000,250.000)(0.050,-0.100){480}{\line(-1,-2){15.000}}
\multiput(35.000,170.000)(0.000,0.050){400}{\line(1,0){30.000}}
\multiput(75.000,170.000)(-0.050,0.100){320}{\line(-1,0){30.000}}
\color{fT}
%\put(35.000,250.000){\circle*{1.000}} % tip
\put(40.000,220.000){\makebox(0.000,0.000){$t$}}
\put(5.000,190.000){\line(1,0){30.000}}
\put(35.000,170.000){\line(0,1){20.000}}
\put(35.000,170.000){\line(1,0){40.000}}
\put(5.000,190.000){\line(1,2){30.000}}
\put(35.000,250.000){\line(1,-2){40.000}}
%%%%%%%%%%%%%%%%% rule: s %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\color{bS}
\multiput(155.000,240.000)(0.050,0.100){300}{\line(1,0){30.000}}
\multiput(185.000,300.000)(0.050,-0.100){480}{\line(-1,-2){15.000}}
\multiput(185.000,220.000)(0.000,0.050){400}{\line(1,0){30.000}}
\multiput(225.000,220.000)(-0.050,0.100){320}{\line(-1,0){30.000}}
\color{fS}
%\put(185.000,300.000){\circle*{1.000}} % tip
\put(190.000,270.000){\makebox(0.000,0.000){$s$}}
\put(155.000,240.000){\line(1,0){30.000}}
\put(185.000,220.000){\line(0,1){20.000}}
\put(185.000,220.000){\line(1,0){40.000}}
\put(155.000,240.000){\line(1,2){30.000}}
\put(185.000,300.000){\line(1,-2){40.000}}
%%%%%%%%%%%%%%%% rule: l %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\color{bL}
\multiput(265.000,240.000)(0.050,0.100){200}{\line(1,0){10.000}}
\multiput(280.000,270.000)(0.050,-0.100){200}{\line(-1,-2){5.000}}
\multiput(295.000,240.000)(-0.050,0.100){100}{\line(-1,0){20.000}}
\multiput(255.000,220.000)(0.050,0.100){200}{\line(1,0){20.000}}
\multiput(265.000,220.000)(0.000,0.100){200}{\line(1,0){20.000}}
\color{fL}
%\put(280.000,270.000){\circle*{1.000}} % tip
\put(280.000,250.000){\makebox(0.000,0.000){$l$}}
\put(255.000,220.000){\line(1,0){29.500}}
\put(255.000,220.000){\line(1,2){25.000}}
\put(280.000,270.000){\line(1,-2){14.750}}
\put(284.500,220.000){\line(0,1){20.500}}
\put(284.500,240.500){\line(1,0){10.250}}
%%%%%%%%%%%%%%%% rule: r %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\color{bR}
\multiput(415.000,190.000)(0.050,0.100){200}{\line(1,0){10.000}}
\multiput(430.000,220.000)(0.050,-0.100){200}{\line(-1,-2){5.000}}
\multiput(445.000,190.000)(-0.050,0.100){100}{\line(-1,0){20.000}}
\multiput(405.000,170.000)(0.050,0.100){200}{\line(1,0){20.000}}
\multiput(415.000,170.000)(0.000,0.100){200}{\line(1,0){20.000}}
\color{fR}
%\put(430.000,220.000){\circle*{1.000}} % tip
\put(430.000,200.000){\makebox(0.000,0.000){$r$}}
\put(405.000,170.000){\line(1,0){29.500}}
\put(405.000,170.000){\line(1,2){25.000}}
\put(430.000,220.000){\line(1,-2){14.750}}
\put(434.500,170.000){\line(0,1){20.500}}
\put(434.500,190.500){\line(1,0){10.250}}
%%%%%%%%%%%%%%%% rewrite result s->t %%%%%%%%%%%%%%%%%%%
\color{bT}
\multiput(55.000,65.000)(0.050,0.100){400}{\line(1,0){10.000}}
\multiput(85.000,125.000)(0.050,-0.100){480}{\line(-1,-2){10.000}}
\multiput(85.000,45.000)(0.000,0.050){400}{\line(1,0){30.000}}
\multiput(125.000,45.000)(-0.050,0.100){320}{\line(-1,0){20.000}}
\color{bT}
\multiput(65.000,65.000)(0.050,0.100){200}{\line(1,0){10.000}}
\multiput(80.000,95.000)(0.050,-0.100){200}{\line(-1,-2){5.000}}
\multiput(95.000,65.000)(-0.050,0.100){100}{\line(-1,0){20.000}}
\color{bL}
\multiput(55.000,45.000)(0.050,0.100){200}{\line(1,0){20.000}}
\multiput(65.000,45.000)(0.000,0.100){200}{\line(1,0){20.000}}
\color{fT}
%\put(85.000,125.000){\circle*{1.000}} % tip
\put(90.000,95.000){\makebox(0.000,0.000){$t$}}
\put(55.000,65.000){\line(1,0){30.000}}
\put(85.000,45.000){\line(0,1){20.000}}
\put(85.000,45.000){\line(1,0){40.000}}
\put(55.000,65.000){\line(1,2){30.000}}
\put(85.000,125.000){\line(1,-2){40.000}}
% matching subst
\color{fSbs}
\put(79.750,54.500){\makebox(0.000,0.000){$\sigma$}}
\put(65.250,64.500){\line(-1,-2){9.500}}
\put(55.750,45.500){\line(1,0){29.000}}
%%%%%%%%%%%%%%%% redex overlap %%%%%%%%%%%%%%%%%%%%%%%%%
\color{bS}
\multiput(205.000,115.000)(0.050,0.100){400}{\line(1,0){10.000}}
\multiput(235.000,175.000)(0.050,-0.100){480}{\line(-1,-2){10.000}}
\multiput(235.000,95.000)(0.000,0.050){400}{\line(1,0){30.000}}
\multiput(275.000,95.000)(-0.050,0.100){320}{\line(-1,0){20.000}}
\color{bSL}
\multiput(215.000,115.000)(0.050,0.100){200}{\line(1,0){10.000}}
\multiput(230.000,145.000)(0.050,-0.100){200}{\line(-1,-2){5.000}}
\multiput(245.000,115.000)(-0.050,0.100){100}{\line(-1,0){20.000}}
\color{bL}
\multiput(205.000,95.000)(0.050,0.100){200}{\line(1,0){20.000}}
\multiput(215.000,95.000)(0.000,0.100){200}{\line(1,0){20.000}}
% s
\color{fS}
%\put(235.000,175.000){\circle*{1.000}} % tip
\put(240.000,145.000){\makebox(0.000,0.000){$s$}}
\put(205.000,115.000){\line(1,0){30.000}}
\put(235.000,95.000){\line(0,1){20.000}}
\put(235.000,95.000){\line(1,0){40.000}}
\put(205.000,115.000){\line(1,2){30.000}}
\put(235.000,175.000){\line(1,-2){40.000}}
% l
\color{fL}
%\put(230.000,145.000){\circle*{1.000}} % tip
\put(230.000,125.000){\makebox(0.000,0.000){$l$}}
\put(205.000,95.000){\line(1,0){29.500}}
\put(205.000,95.000){\line(1,2){25.000}}
\put(230.000,145.000){\line(1,-2){14.750}}
\put(234.500,95.000){\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(0.000,0.000){$\sigma$}}
\put(239.750,104.500){\makebox(0.000,0.000){$\sigma$}}
\put(215.250,114.500){\line(1,0){30.000}}
\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){49.000}}
% path
\thinlines
\color{fPath}
\put(233.000,155.000){\makebox(0.000,0.000){$p$}}
\put(235.000,175.000){\line(-1,-6){2.000}}
\put(233.000,163.000){\line(2,-5){4.000}}
\put(237.000,153.000){\line(-5,-3){5.000}}
\put(232.000,150.000){\line(-2,-5){2.000}}
\thicklines
%%%%%%%%%%%%%%%% rewrite result l->r %%%%%%%%%%%%%%%%%%%
\color{bS}
\multiput(355.000,65.000)(0.050,0.100){400}{\line(1,0){10.000}}
\multiput(385.000,125.000)(0.050,-0.100){480}{\line(-1,-2){10.000}}
\multiput(385.000,45.000)(0.000,0.050){400}{\line(1,0){30.000}}
\multiput(425.000,45.000)(-0.050,0.100){320}{\line(-1,0){20.000}}
\color{bSR}
\multiput(365.000,65.000)(0.050,0.100){200}{\line(1,0){10.000}}
\multiput(380.000,95.000)(0.050,-0.100){200}{\line(-1,-2){5.000}}
\multiput(395.000,65.000)(-0.050,0.100){100}{\line(-1,0){20.000}}
\color{bR}
\multiput(355.000,45.000)(0.050,0.100){200}{\line(1,0){20.000}}
\multiput(365.000,45.000)(0.000,0.100){200}{\line(1,0){20.000}}
% s
\color{fS}
%\put(385.000,125.000){\circle*{1.000}} % tip
\put(390.000,95.000){\makebox(0.000,0.000){$s$}}
\put(355.000,65.000){\line(1,0){30.000}}
\put(385.000,45.000){\line(0,1){20.000}}
\put(385.000,45.000){\line(1,0){40.000}}
\put(355.000,65.000){\line(1,2){30.000}}
\put(385.000,125.000){\line(1,-2){40.000}}
% r
\color{fL}
%\put(380.000,95.000){\circle*{1.000}} % tip
\put(380.000,75.000){\makebox(0.000,0.000){$r$}}
\put(355.000,45.000){\line(1,0){29.500}}
\put(355.000,45.000){\line(1,2){25.000}}
\put(380.000,95.000){\line(1,-2){14.750}}
\put(384.500,45.000){\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(0.000,0.000){$\sigma$}}
\put(385.250,64.500){\line(1,0){10.000}}
%\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(383.000,105.000){\makebox(0.000,0.000){$p$}}
\put(385.000,125.000){\line(-1,-6){2.000}}
\put(383.000,113.000){\line(2,-5){4.000}}
\put(387.000,103.000){\line(-5,-3){5.000}}
\put(382.000,100.000){\line(-2,-5){2.000}}
\thicklines
\end{picture}
\end{document}
|
Licensing[edit]
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 11:21, 8 November 2019 |  | 2,685 × 1,800 (94 KB) | Jochen Burghardt (talk | contribs) | rm picture delimiter marks |
| 11:18, 8 November 2019 | No thumbnail | 0 × 0 (95 KB) | Jochen Burghardt (talk | contribs) | stronger colors; larger distance rules / applications | |
| 20:14, 2 January 2014 | No thumbnail | 0 × 0 (80 KB) | Jochen Burghardt (talk | contribs) | cropped | |
| 20:01, 2 January 2014 | No thumbnail | 0 × 0 (81 KB) | Jochen Burghardt (talk | contribs) | decent bg colors | |
| 20:00, 2 January 2014 | No thumbnail | 0 × 0 (80 KB) | Jochen Burghardt (talk | contribs) | User created page with UploadWizard |
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