File:MonusTuringMachine.pdf

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English: Turing machine to compute the truncated subtraction ("monus"), after John E. Hopcroft and Jeffrey D. Ullman (1979) Introduction to Automata Theory, Languages, and Computation, Reading/MA: Addison-Wesley ISBN: 0-201-02988-X. Example 7.2, page 151-153. To enhance understandibility, Hopcroft's and Ullman's input symbols "0", "1", and "B" (blank symbol) have been renamed to "I", "#", and "_", repectively; likewise, their states "q0", ..., "q6" have been renamed to "a", ..., "g".


The machine is depicted in its start state, "a", and with its head in its start position on a tape representing the pair ⟨5,3⟩ in the unary numeral system.
According to the book, in a computation of "m∸n", the intuitive meaning of the states is:
a: Begin, replace the leading "I" by "_".
b: Search right, looking for the first "#".
c: Search right past "#"s until encountering an "I", change that "I" to "#".
d: Move left to a "_", then enter state "a" again to repeat the cycle.
e: In state "c", a "_" was encountered before an "I", each "I" of n was changed to "#", and n+1 "I"s of m were changed to "_". Now change all "#"s to "_"s until encountering a "_", change this "_" back to "I", and enter the stop state "g".
f: In state "a", a "#" was encountered instead of an "I", that is m≤n, and each "I" of m was already changed to "_". Now erase the rest of the tape, and enter the stop state "g".
g: Stop state. The computation is finished, the tape contains m∸n in unary notation, no further state changes are possible.

See File:MonusTuringMachine ExampleRuns.gif for some example runs and a C simulator of this machine.
Date
Source Own work
Author Jochen Burghardt
Other versions File:MonusTuringMachine.pdf — File:MonusTuringMachine_svg.svg — File:MonusTuringMachine ExampleRuns.gif
LaTeX source code
\documentclass[12pt]{article}
\setlength{\unitlength}{1mm}
\usepackage[pdftex]{color}
\usepackage{amssymb}
\usepackage[paperwidth=175\unitlength,paperheight=75\unitlength]{geometry}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{175\unitlength}
\setlength{\textheight}{75\unitlength}
\setlength{\oddsidemargin}{-23mm}
\setlength{\parindent}{0cm}
\pagestyle{empty}

% colors
\definecolor{fState}    {rgb}{0.50,0.00,0.00}
\definecolor{fSym}      {rgb}{0.00,0.00,0.50}
\definecolor{fDir}      {rgb}{0.00,0.50,0.00}
\definecolor{bCtrl}     {rgb}{0.99,0.90,0.90}
\definecolor{bTape}     {rgb}{0.90,0.90,0.99}
\definecolor{bHead}     {rgb}{0.90,0.99,0.90}

\newcommand{\0}{\textcolor{fState}{a}}
\newcommand{\1}{\textcolor{fState}{b}}
\newcommand{\2}{\textcolor{fState}{c}}
\newcommand{\3}{\textcolor{fState}{d}}
\newcommand{\4}{\textcolor{fState}{e}}
\newcommand{\5}{\textcolor{fState}{f}}
\newcommand{\6}{\textcolor{fState}{g}}
\newcommand{\I}{\textcolor{fSym}{\tt\#}}
\renewcommand{\O}{\textcolor{fSym}{\tt I}}
\newcommand{\B}{\textcolor{fSym}{\tt\_}}
\renewcommand{\L}{\textcolor{fDir}{L}}
\newcommand{\R}{\textcolor{fDir}{R}}

\begin{document}
\begin{picture}(170,70)
% ctrl
\textcolor{bCtrl}{\put(45,35){\makebox(0,0)[bl]{\rule{92mm}{35mm}}}}%
\textcolor{white}{\put(60,57){\makebox(0,0)[bl]{\rule{5mm}{4mm}}}}%
\textcolor{fState}{\put(45.000,35.000){\framebox(92.000,35.000)[bl]{}}}%
\put(50.000,40.000){\makebox(0,0)[bl]{%
        \begin{tabular}{|c|*{7}{|c@{}c@{}c}|}%
        \multicolumn{22}{c}{\bf Finite control} \\
        \hline
                &\multicolumn{3}{c|}{\0}
                &\multicolumn{3}{c|}{\1}
                &\multicolumn{3}{c|}{\2}
                &\multicolumn{3}{c|}{\3}
                &\multicolumn{3}{c|}{\4}
                &\multicolumn{3}{c|}{\5}
                &\multicolumn{3}{c|}{\6}
                \\%
        \hline
        \hline
        \O &\1&\B&\R &\1&\O&\R &\3&\I&\L &\3&\O&\L &\4&\O&\L &\5&\B&\R&&&\\%
        \hline
        \I &\5&\B&\R &\2&\I&\R &\2&\I&\R &\3&\I&\L &\4&\B&\L &\5&\B&\R&&&\\%
        \hline
        \B &  &  &   &  &  &   &\4&\B&\L &\0&\B&\R &\6&\O&\R &\6&\B&\R&&&\\%
        \hline
        \end{tabular}%
}}

% tape
\textcolor{bTape}{\put(1,0){\makebox(0,0)[bl]{\rule{168mm}{10mm}}}}%
\textcolor{fSym}{\put(0,0){\line(1,0){170}}}%
\textcolor{fSym}{\put(0,10){\line(1,0){170}}}%
\textcolor{fSym}{\multiput(5,0)(10,0){17}{\line(0,1){10}}}%
\put(165,11){\makebox(0,0)[br]{\bf Tape}}%
% tape contents
\textcolor{fSym}{\put(  9,4){\makebox(0,0)[bl]{\B}}}%
\textcolor{fSym}{\put( 19,4){\makebox(0,0)[bl]{\B}}}%
\textcolor{fSym}{\put( 29,4){\makebox(0,0)[bl]{\B}}}%
\textcolor{fSym}{\put( 39,4){\makebox(0,0)[bl]{\O}}}%
\textcolor{fSym}{\put( 49,4){\makebox(0,0)[bl]{\O}}}%
\textcolor{fSym}{\put( 59,4){\makebox(0,0)[bl]{\O}}}%
\textcolor{fSym}{\put( 69,4){\makebox(0,0)[bl]{\O}}}%
\textcolor{fSym}{\put( 79,4){\makebox(0,0)[bl]{\O}}}%
\textcolor{fSym}{\put( 89,4){\makebox(0,0)[bl]{\I}}}%
\textcolor{fSym}{\put( 99,4){\makebox(0,0)[bl]{\O}}}%
\textcolor{fSym}{\put(109,4){\makebox(0,0)[bl]{\O}}}%
\textcolor{fSym}{\put(119,4){\makebox(0,0)[bl]{\O}}}%
\textcolor{fSym}{\put(129,4){\makebox(0,0)[bl]{\B}}}%
\textcolor{fSym}{\put(139,4){\makebox(0,0)[bl]{\B}}}%
\textcolor{fSym}{\put(149,4){\makebox(0,0)[bl]{\B}}}%
\textcolor{fSym}{\put(159,4){\makebox(0,0)[bl]{\B}}}%

% head
\put(40,18){\makebox(0,0)[br]{\bf R/W Head}}%
\put(34,14){\makebox(0,0)[r]{$\leftarrow$}}%
\put(46,14){\makebox(0,0)[l]{$\rightarrow$}}%
\textcolor{bHead}{\put(37,10.5){\makebox(0,0)[bl]{\rule{6mm}{2.5mm}}}}%
\textcolor{bHead}{\put(35,13){\makebox(0,0)[bl]{\rule{10mm}{4mm}}}}%
\thicklines%
\textcolor{bHead}{\put(40,13){\oval(7.0,2)[b]}}%
\textcolor{bHead}{\put(40,13){\oval(7.5,2)[b]}}%
\textcolor{bHead}{\put(40,13){\oval(8.0,3)[b]}}%
\textcolor{bHead}{\put(40,13){\oval(8.5,3)[b]}}%
\textcolor{bHead}{\put(40,13){\oval(9.0,4)[b]}}%
\textcolor{bHead}{\put(40,13){\oval(9.5,4)[b]}}%
\thinlines%
\textcolor{fDir}{\put(35,17){\line(1,0){10}}}%
\textcolor{fDir}{\put(35,13){\line(0,1){4}}}%
\textcolor{fDir}{\put(45,13){\line(0,1){4}}}%
\textcolor{fDir}{\put(40,13){\oval(10,5)[b]}}%
% cable fixed part
\thicklines%
\textcolor{fDir}{\put(90.000,36.000){\circle*{1}}}%
\textcolor{fDir}{\put(40.000,16.000){\circle*{1}}}%
\textcolor{fDir}{\put(90.000,36.000){\line(0,-1){5}}}%
% cable varying part
%\textcolor{fDir}{\put(90,31){\Line{40,16}}}%
\textcolor{fDir}{\put(90.000,31.000){\line(-3,-1){30.000}}%...
 \put(60.000,21.000){\line(-4,-1){20.000}}}%

\end{picture}
\end{document}

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I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
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Under the following conditions:
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current15:14, 20 March 2019Thumbnail for version as of 15:14, 20 March 20191,033 × 441 (37 KB)Jochen Burghardt (talk | contribs)User created page with UploadWizard

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