File:RegSubsetNP.pdf
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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 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
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\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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%
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%
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%
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% disjoint \a c \d
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%
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%
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% disjoint b \c \d
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%
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% disjoint a \c d
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%
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%
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% eps-transitions from disjoints
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% final state
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% full language
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%
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%
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%
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\end{document}
|
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 00:31, 7 January 2016 |  | 825 × 1,150 (11 KB) | wikimediacommons>Jochen Burghardt | User created page with UploadWizard |
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