File:NP-hardness of pattern language membership.pdf
Size of this JPG preview of this PDF file: 799 × 182 pixels. Other resolutions: 320 × 73 pixels | 1,285 × 293 pixels.
Original file (1,285 × 293 pixels, file size: 17 KB, MIME type: application/pdf)
 | This is a file from the Wikimedia Commons. Information from its description page there is shown below. Commons is a freely licensed media file repository. You can help. |
Summary
| 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
|
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 21:24, 5 November 2013 |  | 1,285 × 293 (17 KB) | Jochen Burghardt | added rightmost 0es |
| 20:39, 5 November 2013 |  | 1,227 × 293 (17 KB) | Jochen Burghardt | User created page with UploadWizard |
File usage
No pages on the English Wikipedia use this file (pages on other projects are not listed).