http:^^www.cs.washington.edu^education^courses^322^96w^hw4^hw4.html

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Date: Wed, 08 Jan 1997 20:43:56 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 4 Due Friday, January 26, 1996</TITLE></HEAD><BODY><meta name="description" value="CSE 322 Assignment 4 Due Friday, January 26, 1996"><meta name="keywords" value="hw4"><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><P><H1>CSE 322 Assignment 4 Due Friday, January 26, 1996</H1><P><STRONG></STRONG><P><P><STRONG></STRONG><P><P><OL><LI> Number 36 on page 69.  Better yet construct the derivation tree for<IMG  ALIGN=MIDDLE ALT="" SRC="img1.gif">.<LI>Consider the grammar: <IMG  ALIGN=MIDDLE ALT="" SRC="img2.gif">.  <OL><LI>By example show that this grammar is ambiguous.<LI>Design an unambiguous grammar which generates the same languageas this grammar.  Explain why your grammaris unambiguous.</OL><LI> Use the construction given in class for producing a regular grammarfrom a regular expression to construct a regular grammar which generatesthe language defined by the regular expression <IMG  ALIGN=MIDDLE ALT="" SRC="img3.gif">.<LI>Let <b>L</b> be the subset of <IMG  ALIGN=MIDDLE ALT="" SRC="img4.gif"> where the <b>a</b>'s and <b>b</b>'s are matchedas parentheses.  The grammar <b>G</b> withproductions  <IMG  ALIGN=MIDDLE ALT="" SRC="img5.gif">.  generates this language.  Prove by induction on the length of a derivation that if <IMG  ALIGN=BOTTOM ALT="" SRC="img6.gif"> and <IMG  ALIGN=MIDDLE ALT="" SRC="img7.gif">then <b>x</b> satisfies the twoproperties (i) every prefix of <b>x</b> has at least as many <b>a</b>'s as <b>b</b>'sand (ii) <b>x</b> has an equal number of <b>a</b>'s as <b>b</b>'s.<P>Note that there is only one derivation of length 1, namely, <IMG  ALIGN=BOTTOM ALT="" SRC="img8.gif">.For the inductive step, if <IMG  ALIGN=BOTTOM ALT="" SRC="img9.gif"> then there are twopossibilities <IMG  ALIGN=BOTTOM ALT="" SRC="img10.gif"> or<IMG  ALIGN=BOTTOM ALT="" SRC="img11.gif">.  Your job is to carefully completethe induction proof using these ideas.</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>Mon Jan 22 10:11:12 PST 1996</I></ADDRESS></BODY>

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