File:RegSubsetNP.pdf

Summary

Description
English: Example to demonstrate that the subset property for regular languages is NP-hard. Given a propositional formula in disjunctive normal form in n variables , a corresponding nondeterministic finite automaton A can be constructed such that it accepts the regular language {0,1}n if, and only if, the formula is true for all variable assignments. Therefore, the NP-hard problem of deciding the universal validity of a propositional formula in disjunctive normal form can be reduced to the problem of deciding whether {0,1}n ⊆ L(A).

The shown example uses the formula abc+abd+abc+abd+bcd+acd+bcd+acd in n=4 variables. The corresponding automaton A is shown in the upper part of the picture; unlabelled transitions are ε-transitions. Each summand in the formula corresponds to one "row" in the automaton A; e.g. the last one, acd, corresponds to the bottommost row of A; the summand is valid exactly for those variable assignments that correspond to a string from the row's accepted language, i.e. {1101,1001} for this row.

The language {0,1}n is regular since it is accepted by the the automaton B shown at the lower picture part.
Date
Source Own work
Author Jochen Burghardt
Other versions File:RegSubsetNP svg.svg
LaTeX source code
\documentclass[12pt]{article}
\setlength{\unitlength}{1mm}
\usepackage[pdftex]{color}
\usepackage[paperwidth=140\unitlength,paperheight=195\unitlength]{geometry}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{140\unitlength}
\setlength{\textheight}{195\unitlength}
\setlength{\oddsidemargin}{-23mm}
\setlength{\parindent}{0cm}
\pagestyle{empty}
% colors
\definecolor{cS}        {rgb}{0.00,0.00,0.40}   % state
\definecolor{cE}        {rgb}{0.40,0.40,0.40}   % epsilon transition
\definecolor{cO}        {rgb}{0.40,0.00,0.00}   % 0 transition
\definecolor{cI}        {rgb}{0.00,0.40,0.00}   % 1 transition

\begin{document}
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% start state
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% eps-transitions to disjoints
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\textcolor{cE}{\put(6.000,100.000){\vector(1,3){20.000}}}%
\textcolor{cE}{\put(6.000,100.000){\vector(1,2){20.000}}}%
\textcolor{cE}{\put(6.000,100.000){\vector(1,1){20.000}}}%
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\textcolor{cS}{\put(27.000,60.000){\circle*{2.000}}}%
\textcolor{cS}{\put(27.000,40.000){\circle*{2.000}}}%

% disjoint a b c
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%
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%
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\textcolor{cI}{\put(79.000,184.000){\makebox(0.000,0.000)[b]{$1$}}}%
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%
\textcolor{cI}{\put(90.000,180.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(100.000,183.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(100.000,184.000){\makebox(0.000,0.000)[b]{$1$}}}%
\textcolor{cO}{\put(90.000,179.500){\line(4,-1){10.000}}}%
\textcolor{cO}{\put(100.000,177.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(100.000,176.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(110.000,180.000){\circle*{2.000}}}%
% disjoint a \b \d
\textcolor{cI}{\put(27.000,160.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(37.000,163.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(37.000,164.000){\makebox(0.000,0.000)[b]{$1$}}}%
\textcolor{cS}{\put(47.000,160.000){\circle*{2.000}}}%
%
\textcolor{cO}{\put(48.000,159.500){\line(4,-1){10.000}}}%
\textcolor{cO}{\put(58.000,157.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(58.000,156.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(68.000,160.000){\circle*{2.000}}}%
%
\textcolor{cI}{\put(69.000,160.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(79.000,163.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(79.000,164.000){\makebox(0.000,0.000)[b]{$1$}}}%
\textcolor{cO}{\put(69.000,159.500){\line(4,-1){10.000}}}%
\textcolor{cO}{\put(79.000,157.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(79.000,156.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(89.000,160.000){\circle*{2.000}}}%
%
\textcolor{cO}{\put(90.000,159.500){\line(4,-1){10.000}}}%
\textcolor{cO}{\put(100.000,157.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(100.000,156.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(110.000,160.000){\circle*{2.000}}}%
% disjoint \a \b \c
\textcolor{cO}{\put(27.000,139.500){\line(4,-1){10.000}}}%
\textcolor{cO}{\put(37.000,137.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(37.000,136.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(47.000,140.000){\circle*{2.000}}}%
%
\textcolor{cO}{\put(48.000,139.500){\line(4,-1){10.000}}}%
\textcolor{cO}{\put(58.000,137.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(58.000,136.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(68.000,140.000){\circle*{2.000}}}%
%
\textcolor{cO}{\put(69.000,139.500){\line(4,-1){10.000}}}%
\textcolor{cO}{\put(79.000,137.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(79.000,136.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(89.000,140.000){\circle*{2.000}}}%
%
\textcolor{cI}{\put(90.000,140.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(100.000,143.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(100.000,144.000){\makebox(0.000,0.000)[b]{$1$}}}%
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\textcolor{cO}{\put(100.000,137.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(100.000,136.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(110.000,140.000){\circle*{2.000}}}%
% disjoint \a b d
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\textcolor{cO}{\put(37.000,116.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(47.000,120.000){\circle*{2.000}}}%
%
\textcolor{cI}{\put(48.000,120.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(58.000,123.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(58.000,124.000){\makebox(0.000,0.000)[b]{$1$}}}%
\textcolor{cS}{\put(68.000,120.000){\circle*{2.000}}}%
%
\textcolor{cI}{\put(69.000,120.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(79.000,123.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(79.000,124.000){\makebox(0.000,0.000)[b]{$1$}}}%
\textcolor{cO}{\put(69.000,119.500){\line(4,-1){10.000}}}%
\textcolor{cO}{\put(79.000,117.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(79.000,116.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(89.000,120.000){\circle*{2.000}}}%
%
\textcolor{cI}{\put(90.000,120.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(100.000,123.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(100.000,124.000){\makebox(0.000,0.000)[b]{$1$}}}%
\textcolor{cS}{\put(110.000,120.000){\circle*{2.000}}}%
% disjoint \b c d
\textcolor{cI}{\put(27.000,100.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(37.000,103.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(37.000,104.000){\makebox(0.000,0.000)[b]{$1$}}}%
\textcolor{cO}{\put(27.000,99.500){\line(4,-1){10.000}}}%
\textcolor{cO}{\put(37.000,97.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(37.000,96.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(47.000,100.000){\circle*{2.000}}}%
%
\textcolor{cO}{\put(48.000,99.500){\line(4,-1){10.000}}}%
\textcolor{cO}{\put(58.000,97.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(58.000,96.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(68.000,100.000){\circle*{2.000}}}%
%
\textcolor{cI}{\put(69.000,100.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(79.000,103.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(79.000,104.000){\makebox(0.000,0.000)[b]{$1$}}}%
\textcolor{cS}{\put(89.000,100.000){\circle*{2.000}}}%
%
\textcolor{cI}{\put(90.000,100.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(100.000,103.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(100.000,104.000){\makebox(0.000,0.000)[b]{$1$}}}%
\textcolor{cS}{\put(110.000,100.000){\circle*{2.000}}}%
% disjoint \a c \d
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\textcolor{cO}{\put(37.000,77.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(37.000,76.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(47.000,80.000){\circle*{2.000}}}%
%
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\textcolor{cI}{\put(58.000,84.000){\makebox(0.000,0.000)[b]{$1$}}}%
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\textcolor{cO}{\put(58.000,76.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(68.000,80.000){\circle*{2.000}}}%
%
\textcolor{cI}{\put(69.000,80.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(79.000,83.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(79.000,84.000){\makebox(0.000,0.000)[b]{$1$}}}%
\textcolor{cS}{\put(89.000,80.000){\circle*{2.000}}}%
%
\textcolor{cO}{\put(90.000,79.500){\line(4,-1){10.000}}}%
\textcolor{cO}{\put(100.000,77.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(100.000,76.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(110.000,80.000){\circle*{2.000}}}%
% disjoint b \c \d
\textcolor{cI}{\put(27.000,60.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(37.000,63.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(37.000,64.000){\makebox(0.000,0.000)[b]{$1$}}}%
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\textcolor{cO}{\put(37.000,57.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(37.000,56.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(47.000,60.000){\circle*{2.000}}}%
%
\textcolor{cI}{\put(48.000,60.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(58.000,63.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(58.000,64.000){\makebox(0.000,0.000)[b]{$1$}}}%
\textcolor{cS}{\put(68.000,60.000){\circle*{2.000}}}%
%
\textcolor{cO}{\put(69.000,59.500){\line(4,-1){10.000}}}%
\textcolor{cO}{\put(79.000,57.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(79.000,56.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(89.000,60.000){\circle*{2.000}}}%
%
\textcolor{cO}{\put(90.000,59.500){\line(4,-1){10.000}}}%
\textcolor{cO}{\put(100.000,57.000){\vector(4,1){10.000}}}%
\textcolor{cO}{\put(100.000,56.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(110.000,60.000){\circle*{2.000}}}%
% disjoint a \c d
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\textcolor{cI}{\put(37.000,43.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(37.000,44.000){\makebox(0.000,0.000)[b]{$1$}}}%
\textcolor{cS}{\put(47.000,40.000){\circle*{2.000}}}%
%
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\textcolor{cO}{\put(58.000,36.000){\makebox(0.000,0.000)[t]{$0$}}}%
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%
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\textcolor{cO}{\put(79.000,36.000){\makebox(0.000,0.000)[t]{$0$}}}%
\textcolor{cS}{\put(89.000,40.000){\circle*{2.000}}}%
%
\textcolor{cI}{\put(90.000,40.500){\line(4,1){10.000}}}%
\textcolor{cI}{\put(100.000,43.000){\vector(4,-1){10.000}}}%
\textcolor{cI}{\put(100.000,44.000){\makebox(0.000,0.000)[b]{$1$}}}%
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% eps-transitions from disjoints
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\textcolor{cE}{\put(111.000,160.000){\vector(1,-3){19.800}}}%
\textcolor{cE}{\put(111.000,140.000){\vector(1,-2){19.700}}}%
\textcolor{cE}{\put(111.000,120.000){\vector(1,-1){19.600}}}%
\textcolor{cE}{\put(111.000,100.000){\vector(1,0){19.500}}}%
\textcolor{cE}{\put(111.000,80.000){\vector(1,1){19.600}}}%
\textcolor{cE}{\put(111.000,60.000){\vector(1,2){19.700}}}%
\textcolor{cE}{\put(111.000,40.000){\vector(1,3){19.800}}}%
% final state
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% full language
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%
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%
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%
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\textcolor{cI}{\put(79.000,11.000){\makebox(0.000,0.000)[b]{$1$}}}%
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%
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\textcolor{cS}{\put(111.000,7.000){\circle{3.000}}}%
\end{picture}
\end{document}

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Category:CC-BY-SA-4.0#RegSubsetNP.pdf
Category:Self-published work Category:Files by User:Jochen Burghardt Category:Boolean satisfiability problem Category:Formal languages Category:Images with LaTeX source code Category:Subsets Category:Propositional formula
Category:Boolean satisfiability problem Category:CC-BY-SA-4.0 Category:Files by User:Jochen Burghardt Category:Formal languages Category:Images with LaTeX source code Category:Propositional formula Category:Self-published work Category:Subsets