http:^^www.cs.washington.edu^education^courses^322^96w^hw9soln^hw9soln.html

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Date: Wed, 08 Jan 1997 20:36:50 GMTServer: NCSA/1.4.2Content-type: text/html<!DOCTYPE HTML PUBLIC "-//W3O//DTD W3 HTML 2.0//EN"><!Converted with LaTeX2HTML 95 (Thu Jan 19 1995) by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds ><HEAD><TITLE>CSE 322: Assignment 9 Solution Set</TITLE></HEAD><BODY><meta name="description" value="CSE 322: Assignment 9 Solution Set"><meta name="keywords" value="hw9soln"><meta name="resource-type" value="document"><meta name="distribution" value="global"><P> <BR> <HR><A NAME=tex2html1 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="next" SRC="http://www.cs.washington.edu/general/latex2html_icons//next_motif.gif"></A>   <IMG ALIGN=BOTTOM ALT="up" SRC="http://www.cs.washington.edu/general/latex2html_icons//up_motif_gr.gif">   <IMG ALIGN=BOTTOM ALT="previous" SRC="http://www.cs.washington.edu/general/latex2html_icons//previous_motif_gr.gif">         <BR><B> Next:</B> <A NAME=tex2html2 HREF="node1.html">   About this document </A><BR> <HR> <P> CSE 322 Winter 1996: Assignment 9 Solution Set<P><OL><LI>Let <IMG  ALIGN=MIDDLE ALT="" SRC="img1.gif"> and<IMG  ALIGN=MIDDLE ALT="" SRC="img2.gif"> be NFAs.Construct a new NFA <b>M</b> that accepts the language<P><IMG  ALIGN=BOTTOM ALT="" SRC="img3.gif"><P><P>Let <IMG  ALIGN=MIDDLE ALT="" SRC="img4.gif"> wherefor all <IMG  ALIGN=MIDDLE ALT="" SRC="img5.gif">, <IMG  ALIGN=MIDDLE ALT="" SRC="img6.gif">, <IMG  ALIGN=MIDDLE ALT="" SRC="img7.gif"><P><IMG  ALIGN=BOTTOM ALT="" SRC="img8.gif"><P><P><b> Behavioral Lemma:</b> For all <IMG  ALIGN=MIDDLE ALT="" SRC="img9.gif">, <IMG  ALIGN=MIDDLE ALT="" SRC="img10.gif">,<IMG  ALIGN=MIDDLE ALT="" SRC="img11.gif">, <IMG  ALIGN=MIDDLE ALT="" SRC="img12.gif">, and<IMG  ALIGN=MIDDLE ALT="" SRC="img13.gif">, <IMG  ALIGN=MIDDLE ALT="" SRC="img14.gif">if and only if<IMG  ALIGN=MIDDLE ALT="" SRC="img15.gif">and<IMG  ALIGN=MIDDLE ALT="" SRC="img16.gif">.<P><LI>Suppose <IMG  ALIGN=MIDDLE ALT="" SRC="img17.gif"> is regular.If we intersect <b>L</b> with any regular language, the resultinglanguage is also regular.For instance, <IMG  ALIGN=MIDDLE ALT="" SRC="img18.gif"> is regular.But <IMG  ALIGN=MIDDLE ALT="" SRC="img19.gif">, and we know thatthis language is not regular from class.  This is a contradictionso <b>L</b> must not be regular.<P><LI>Show that the language <IMG  ALIGN=MIDDLE ALT="" SRC="img20.gif"> is not regular.<P>Assume <b>L</b> is regular, and let <IMG  ALIGN=MIDDLE ALT="" SRC="img21.gif"> be a DFA that accepts <b>L</b>.  Suppose <b>Q</b> contains <b>n</b> states.Consider the computation of <b>M</b> on any string of length <b>n</b> or more.Because there are only <b>n</b> total states, some state will be visited morethan once during the computation.  For example, when <b>M</b> processes thestring <IMG  ALIGN=MIDDLE ALT="" SRC="img22.gif">, the computation looks like this:<P><IMG  ALIGN=BOTTOM ALT="" SRC="img23.gif"><P>for some <IMG  ALIGN=MIDDLE ALT="" SRC="img24.gif">, <IMG  ALIGN=MIDDLE ALT="" SRC="img25.gif">, and <IMG  ALIGN=MIDDLE ALT="" SRC="img26.gif">.  Because we visit the same state twice, we can remove <IMG  ALIGN=BOTTOM ALT="" SRC="img27.gif"> from the string and still end in the same state:<P><IMG  ALIGN=BOTTOM ALT="" SRC="img28.gif"><P>This means that <IMG  ALIGN=MIDDLE ALT="" SRC="img29.gif">.  However,<IMG  ALIGN=MIDDLE ALT="" SRC="img30.gif">.  Also, note thatsince <IMG  ALIGN=MIDDLE ALT="" SRC="img31.gif"> and <IMG  ALIGN=MIDDLE ALT="" SRC="img32.gif">, <b>2n+1</b> must be greater than <b>j-i</b>.  This means that<IMG  ALIGN=MIDDLE ALT="" SRC="img33.gif">.  So <IMG  ALIGN=MIDDLE ALT="" SRC="img34.gif"> falls strictly between<IMG  ALIGN=BOTTOM ALT="" SRC="img35.gif"> and <IMG  ALIGN=MIDDLE ALT="" SRC="img36.gif">, two consecutive perfect squares.Therefore, <IMG  ALIGN=MIDDLE ALT="" SRC="img37.gif"> is not a perfect square and<IMG  ALIGN=MIDDLE ALT="" SRC="img38.gif">.  This is a contradiction, so <b>L</b> must not be regular.<P></OL><BR> <HR><UL> <LI> <A NAME=tex2html3 HREF="node1.html#SECTION00010000000000000000">   About this document ... </A></UL><BR> <HR><P><ADDRESS><I>James Fix <BR>Wed Mar  6 11:11:44 PST 1996</I></ADDRESS></BODY>

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