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Summary

DescriptionCookLevin.pdf	Schematized computation of a non-deterministic Turing machine M, as used in the proof of the Cook-Levin theorem. Each line corresponds to a computation step. Initially, in step 0, the tape (green area) contains a given input word I = I₀...I_n, surrounded by blank cells ("‿"), the tape read/write head is at position 0 (red column, also circled), and the state (blue column) is the machine's initial state s. In each step, M performs an action dependent on its current state and the current tape symbol under the head, possibly changing the state and the tape symbol, and moving the head one cell to the left or to the right. The proof assumes that M finishes its computation after at most p(n) steps, where p is some polynomial, by entering an "accept" state from a set F. Hence it is sufficient to consider the tape cells -p(n) ... +p(n). In the picture, M happens to finish exactly in the last step.
Date	17 October 2014
Source	Own work; inspired by Fig.7.8 in Sect.7.4 (p.255) of Michael Sipser (1997) Introduction to the Theory of Computation, Boston/MA: PWS Publishing Co. ; notation adapted to Cook-Levin theorem#Proof
Author	Jochen Burghardt
Other versions	CookLevin.pdf CookLevin svg.svg

LaTeX source code

\documentclass[12pt]{article}
\usepackage[pdftex]{color}
\usepackage[paperwidth=292mm,paperheight=145mm]{geometry}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{292mm}
\setlength{\textheight}{145mm}
\setlength{\evensidemargin}{-2.7cm}
\setlength{\oddsidemargin}{-2.5cm}
\setlength{\parindent}{0cm}
\setlength{\parskip}{1ex}
\setlength{\unitlength}{1mm}
\sloppy

\definecolor{bSt}       {rgb}{0.90,0.90,0.99}   % state column
\definecolor{fSt}       {rgb}{0.00,0.00,0.70}   % state

\definecolor{bHd}       {rgb}{0.99,0.90,0.90}   % head column
\definecolor{fHd}       {rgb}{0.70,0.00,0.00}   % head

\definecolor{bPo}       {rgb}{0.90,0.99,0.90}   % reachable head pos
\definecolor{bP}        {rgb}{0.92,0.99,0.90}   % (reachable head pos)

\definecolor{bTp}       {rgb}{0.95,0.99,0.90}   % tape columns
\definecolor{bIn}       {rgb}{0.80,0.99,0.80}   % input word
\definecolor{fTp}       {rgb}{0.00,0.70,0.00}   % tape content

\newcommand{\SP}{\scriptstyle\smile}    % empty tape

\begin{document}

\begin{picture}(290,140)%
        %\put(0,0){\makebox(0,0){+}}%
        %\put(290,140){\makebox(0,0){+}}%
\thicklines%
% step
\put(5.000,135.000){\makebox(0.000,0.000){Step}}%
\put(5,115){\makebox(0,0){$0$}}%
\put(5,105){\makebox(0,0){$1$}}%
\put(5,95){\makebox(0,0){$2$}}%
\put(5,85){\makebox(0,0){$3$}}%
\put(5,5){\makebox(0,0){$p(n)$}}%
% Q (state)
\textcolor{bSt}{\put(20,0){\makebox(0,0)[bl]{\rule{10mm}{120mm}}}}%
\textcolor{fSt}{\put(25.000,135.000){\makebox(0.000,0.000){State}}}%
\textcolor{fSt}{\put(25.000,115.000){\makebox(0,0){$s$}}}%
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\put(30.000,0.000){\line(0,1){120.000}}%
\put(20.000,0.000){\line(1,0){10.000}}%
\put(20.000,10.000){\line(1,0){10.000}}%
\put(20.000,120.000){\line(1,0){10.000}}%
\put(20.000,110.000){\line(1,0){10.000}}%
\put(20.000,100.000){\line(1,0){10.000}}%
\put(20.000,90.000){\line(1,0){10.000}}%
\put(20.000,80.000){\line(1,0){10.000}}%
% H (head position)
\textcolor{bHd}{\put(40,0){\makebox(0,0)[bl]{\rule{10mm}{120mm}}}}%
\textcolor{fHd}{\put(45.000,135.000){\makebox(0.000,0.000){Head}}}%
\textcolor{fHd}{\put(45.000,115.000){\makebox(0,0){$0$}}}%
\textcolor{fHd}{\put(45,45){\makebox(0,0){$\vdots$}}}%
\put(40.000,0.000){\line(0,1){120.000}}%
\put(50.000,0.000){\line(0,1){120.000}}%
\put(40.000,0.000){\line(1,0){10.000}}%
\put(40.000,10.000){\line(1,0){10.000}}%
\put(40.000,120.000){\line(1,0){10.000}}%
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\put(40.000,80.000){\line(1,0){10.000}}%
% T (tape)
\textcolor{bTp}{\put(60,0){\makebox(0,0)[bl]{\rule{230mm}{120mm}}}}%
%
\textcolor{bPo}{\put(160,100){\makebox(0,0)[bl]{\rule{30mm}{10mm}}}}%
\textcolor{bPo}{\put(150,90){\makebox(0,0)[bl]{\rule{50mm}{10mm}}}}%
\textcolor{bPo}{\put(140,80){\makebox(0,0)[bl]{\rule{70mm}{10mm}}}}%
\textcolor{bPo}{\put(130,70){\makebox(0,0)[bl]{\rule{90mm}{10mm}}}}%
\textcolor{bP}{\put(120,60){\makebox(0,0)[bl]{\rule{110mm}{10mm}}}}%
\textcolor{bP}{\put(110,50){\makebox(0,0)[bl]{\rule{130mm}{10mm}}}}%
\textcolor{bP}{\put(100,40){\makebox(0,0)[bl]{\rule{150mm}{10mm}}}}%
\textcolor{bP}{\put(90,30){\makebox(0,0)[bl]{\rule{170mm}{10mm}}}}%
\textcolor{bP}{\put(80,20){\makebox(0,0)[bl]{\rule{190mm}{10mm}}}}%
\textcolor{bPo}{\put(70,10){\makebox(0,0)[bl]{\rule{210mm}{10mm}}}}%
\textcolor{bPo}{\put(60,00){\makebox(0,0)[bl]{\rule{230mm}{10mm}}}}%
%
\textcolor{bIn}{\put(170,110){\makebox(0,0)[bl]{\rule{70mm}{10mm}}}}%
\textcolor{fTp}{\put(175.000,135.000){\makebox(0.000,0.000){Tape}}}%
\textcolor{fHd}{\put(175.000,115.000){\circle{9.1}}}%
%
\put(65.000,125.000){\makebox(0.000,0.000){$-p(n)$}}%
\put(145.000,125.000){\makebox(0.000,0.000){$-3$}}%
\put(155.000,125.000){\makebox(0.000,0.000){$-2$}}%
\put(165.000,125.000){\makebox(0.000,0.000){$-1$}}%
\put(175.000,125.000){\makebox(0.000,0.000){$0$}}%
\put(185.000,125.000){\makebox(0.000,0.000){$+1$}}%
\put(195.000,125.000){\makebox(0.000,0.000){$+2$}}%
\put(205.000,125.000){\makebox(0.000,0.000){$+3$}}%
\put(285.000,125.000){\makebox(0.000,0.000){$+p(n)$}}%
%
\textcolor{fTp}{\put(65.000,115.000){\makebox(0.000,0.000){$\SP$}}}%
\textcolor{fTp}{\put(105.000,115.000){\makebox(0.000,0.000){$\ldots$}}}%
\textcolor{fTp}{\put(145.000,115.000){\makebox(0.000,0.000){$\SP$}}}%
\textcolor{fTp}{\put(155.000,115.000){\makebox(0.000,0.000){$\SP$}}}%
\textcolor{fTp}{\put(165.000,115.000){\makebox(0.000,0.000){$\SP$}}}%
\textcolor{fTp}{\put(175.000,115.000){\makebox(0.000,0.000){$I_0$}}}%
\textcolor{fTp}{\put(185.000,115.000){\makebox(0.000,0.000){$I_1$}}}%
\textcolor{fTp}{\put(195.000,115.000){\makebox(0.000,0.000){$I_2$}}}%
\textcolor{fTp}{\put(205.000,115.000){\makebox(0.000,0.000){$I_3$}}}%
\textcolor{fTp}{\put(220.000,115.000){\makebox(0.000,0.000){$\ldots$}}}%
\textcolor{fTp}{\put(235.000,115.000){\makebox(0.000,0.000){$I_n$}}}%
\textcolor{fTp}{\put(245.000,115.000){\makebox(0.000,0.000){$\SP$}}}%
\textcolor{fTp}{\put(265.000,115.000){\makebox(0.000,0.000){$\ldots$}}}%
\textcolor{fTp}{\put(285.000,115.000){\makebox(0.000,0.000){$\SP$}}}%
%
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%
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%
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\put(60.000,120.000){\line(1,0){230.000}}%
\end{picture}
\end{document}

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