File:NP-hardness of pattern language membership.pdf
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Summary
[edit]| Description |
English: Shows a reduction of the 1-in-3-SAT problem to the problem of deciding whether a given string is a member of a given pattern language. The construction is inspired by Dana Angluin's original proof (Thm.3.6 in "Finding Patterns Common to a Set of Strings", JCSS 21/1980), but simplified by using 1-in-3- rather than plain 3-SAT. |
| Date | |
| Source | Own work |
| Author | Jochen Burghardt |
| Other versions | File:NP-hardness of pattern language membership svg.svg |
| LaTeX source code |
|---|
\documentclass[12pt]{article}
\usepackage[pdftex]{color}
\usepackage[paperwidth=218mm,paperheight=50mm]{geometry}
\setlength{\topmargin}{-45mm}
\setlength{\textwidth}{216mm}
\setlength{\textheight}{51mm}
\setlength{\oddsidemargin}{-25mm}
\pagestyle{empty}
\begin{document}
\definecolor{fSep} {rgb}{0.00,0.00,0.50} % separating 0
\definecolor{fPos} {rgb}{0.00,0.50,0.00} % true
\definecolor{fNeg} {rgb}{0.50,0.00,0.00} % false
\newcommand{\0}{\textcolor{fSep}{\bf 0}} % separating 0
\newcommand{\1}[1]{\overline{#1}} % negation
\newcommand{\2}[1]{\textcolor{fNeg}{#1}} % false
\newcommand{\3}[1]{\textcolor{fPos}{#1}} % true
$$\begin{array}{l*{32}{c}}
\bf CNF:
&&&&&&&&&&&&&&&&
& & \multicolumn{4}{c}{(\1w \!\lor\! x \!\lor\! \1y)}
& \land & \multicolumn{4}{c}{( w \!\lor\! y \!\lor\! z)}
& \land & \multicolumn{4}{c}{( x \!\lor\! \1y \!\lor\! \1z)}
\\
\\
\bf Pattern:
& \0 & \multicolumn{3}{c}{wW}
& \0 & \multicolumn{3}{c}{xX}
& \0 & \multicolumn{3}{c}{yY}
& \0 & \multicolumn{3}{c}{zZ}
& \0 & \multicolumn{4}{c}{W x Y}
& \0 & \multicolumn{4}{c}{w y z}
& \0 & \multicolumn{4}{c}{x Y Z}
& \0
\\
\bf String:
& \0 & 1 & 1 & 1
& \0 & 1 & 1 & 1
& \0 & 1 & 1 & 1
& \0 & 1 & 1 & 1
& \0 & 1 & 1 & 1 & 1
& \0 & 1 & 1 & 1 & 1
& \0 & 1 & 1 & 1 & 1
& \0
\\
\\
\bf Solution:
&& \multicolumn{3}{c}{\1w=1}
&& \multicolumn{3}{c}{\1x=1}
&& \multicolumn{3}{c}{ y=1}
&& \multicolumn{3}{c}{\1z=1}
& & \multicolumn{4}{c}{(1 \!\lor\! 0 \!\lor\! 0)}
& \land & \multicolumn{4}{c}{(0 \!\lor\! 1 \!\lor\! 0)}
& \land & \multicolumn{4}{c}{(0 \!\lor\! 0 \!\lor\! 1)}
\\
\\
\cline{3-5}
\cline{7-9}
\cline{11-13}
\cline{15-17}
\cline{19-22}
\cline{24-27}
\cline{29-32}
& \0 & \multicolumn{1}{|c|}{\2w} & \multicolumn{2}{|c|}{\3W}
& \0 & \multicolumn{1}{|c|}{\2x} & \multicolumn{2}{|c|}{\3X}
& \0 & \multicolumn{2}{|c|}{\3y} & \multicolumn{1}{|c|}{\2Y}
& \0 & \multicolumn{1}{|c|}{\2z} & \multicolumn{2}{|c|}{\3Z}
& \0 & \multicolumn{2}{|c|}{\3W} & \multicolumn{1}{|c|}{\2x} & \multicolumn{1}{|c|}{\2Y}
& \0 & \multicolumn{1}{|c|}{\2w} & \multicolumn{2}{|c|}{\3y} & \multicolumn{1}{|c|}{\2z}
& \0 & \multicolumn{1}{|c|}{\2x} & \multicolumn{1}{|c|}{\2Y} & \multicolumn{2}{|c|}{\3Z}
& \0
\\
& \0 & \multicolumn{1}{|c|}{\21} & \multicolumn{1}{|c }{\31} & \multicolumn{1}{ c|}{\31}
& \0 & \multicolumn{1}{|c|}{\21} & \multicolumn{1}{|c }{\31} & \multicolumn{1}{ c|}{\31}
& \0 & \multicolumn{1}{|c }{\31} & \multicolumn{1}{ c|}{\31} & \multicolumn{1}{|c|}{\21}
& \0 & \multicolumn{1}{|c|}{\21} & \multicolumn{1}{|c }{\31} & \multicolumn{1}{ c|}{\31}
& \0 & \multicolumn{1}{|c }{\31} & \multicolumn{1}{ c|}{\31} & \multicolumn{1}{|c|}{\21} & \multicolumn{1}{|c|}{\21}
& \0 & \multicolumn{1}{|c|}{\21} & \multicolumn{1}{|c }{\31} & \multicolumn{1}{ c|}{\31} & \multicolumn{1}{|c|}{\21}
& \0 & \multicolumn{1}{|c|}{\21} & \multicolumn{1}{|c|}{\21} & \multicolumn{1}{|c }{\31} & \multicolumn{1}{ c|}{\31}
& \0
\\
\cline{3-5}
\cline{7-9}
\cline{11-13}
\cline{15-17}
\cline{19-22}
\cline{24-27}
\cline{29-32}
\end{array}$$
\end{document}
% finding a 1-in-3-satisfiable CNF:
w x y z W X Y Z WxY wyz xYZ
--------------------------------------------------
%0 0 0 0 1 1 1 1 101 000 011
%0 0 0 1 1 1 1 0 101 001 010
0 0 1 0 1 1 0 1 100 010 001
%0 0 1 1 1 1 0 0 100 011 000
%0 1 0 0 1 0 1 1 111 000 111
%0 1 0 1 1 0 1 0 111 001 110
%0 1 1 0 1 0 0 1 110 010 101
%0 1 1 1 1 0 0 0 110 011 100
%1 0 0 0 0 1 1 1 001 100 011
%1 0 0 1 0 1 1 0 001 101 010
%1 0 1 0 0 1 0 1 000 110 001
%1 0 1 1 0 1 0 0 000 111 000
%1 1 0 0 0 0 1 1 011 100 111
%1 1 0 1 0 0 1 0 011 101 110
%1 1 1 0 0 0 0 1 010 110 101
%1 1 1 1 0 0 0 0 010 111 100
|
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[edit]I, the copyright holder of this work, hereby publish it under the following license:
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 21:24, 5 November 2013 |  | 1,285 × 293 (17 KB) | Jochen Burghardt (talk | contribs) | added rightmost 0es |
| 20:39, 5 November 2013 | No thumbnail | 0 × 0 (17 KB) | Jochen Burghardt (talk | contribs) | User created page with UploadWizard |
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