quiz3-solutions-Spring-2007

# Quiz3-solutions-Spri - CS 360 Introduction to the Theory of Computing John Watrous University of Waterloo Solutions to Quiz 3 Question 1 Dene a

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CS 360: Introduction to the Theory of Computing John Watrous, University of Waterloo Solutions to Quiz 3 Question 1. DeFne a language A ⊆ { 0 , 1 , # } * as follows: A = b x # yx R : x,y ∈ { 0 , 1 } * B . Give the state transition diagram of a PDA that recognizes A . Solution. Here is one suitable PDA: q 0 q 1 q 2 q f 0 ,Z 0 / 0 Z 0 1 ,Z 0 / 1 Z 0 0 , 0 / 00 1 , 0 / 10 0 , 1 / 01 1 , 1 / 11 # ,Z 0 /Z 0 # , 0 / 0 # , 1 / 1 0 , 0 1 , 1 0 ,Z 0 /Z 0 1 ,Z 0 /Z 0 0 , 0 / 0 1 , 0 / 0 0 , 1 / 1 1 , 1 / 1 ε,Z 0 /Z 0 ε, 0 / 0 ε, 1 / 1 ε,Z 0 Question 2. Short answer questions. a. Draw the state transition diagram for a 1DTM that runs forever on all inputs. Assume the input alphabet is { 0 , 1 } . Solution. Here is one suitable answer: q 0 0 /B 1 /B B/B b. Consider the following language, which is not context-free: L = { r 1 s 1 t : r,s,t ∈ { 0 , 1 } * , | r | = | s | = | t |} . If we were to prove that L is not context-free using the Pumping Lemma for Context-±ree Lan- guages, we might start like this: Assume toward contradiction that

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## This note was uploaded on 09/22/2011 for the course CS 360 taught by Professor Johnwatrous during the Spring '08 term at Waterloo.

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Quiz3-solutions-Spri - CS 360 Introduction to the Theory of Computing John Watrous University of Waterloo Solutions to Quiz 3 Question 1 Dene a

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