hw2sol - CSE 105: Introduction to the Theory of...

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CSE 105: Introduction to the Theory of Computation, Winter 2003 A. Hevia and J. Mao Solution to Problem Set 2 February 14, 2003 Solution to Problem Set 2 2.1 Given the grammar G : E E + T | T T T × F | F F ( E ) | a Give the parse trees and derivations for each string. a. The derivation is E T F a . The parse tree is shown in Figure 1. b. The derivation is E E + T T + T F + T a + T a + F a + a . The parse tree is shown in Figure 1. c. The derivation is E E + T E + T + T T + T + T F + T + T a + T + T a + F + T a + a + T a + a + F a + a + a . The parse tree is shown in Figure 1. d. The derivation is E T F ( E ) ( T ) ( F ) (( E )) (( T )) (( F )) (( a )). The parse tree is shown in Figure 1. E E T F a + T F a E T F a T F a E T F E T F E T F E T F a E T F a + E + a ( ) ( ) Figure 1: 2.1 (a), (b), (c), and (d) 2.3 The given context-free gramar G is 1
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CSE 105, Solution to Problem Set 2 2 R 7→ XRX | S S 7→ aTb | bTa T 7→ XTX | X | ² X 7→ a | b a. The variables of G are { R,X,S,T } . The terminals of G are { a,b } . The start variable is R . b. Strings in L ( G ) are ab , ba , and aaaabbaaa . c. Strings not in L ( G ) are aaa , aba , and bb . d. T aba : false. e. T * aba : true. f. T T : false. g. T * T : true. h. XXX * aba : true. i. X * aba : false. j. T * XX : true. k. T * : false. l. S * ² : false m. The language generated by L is the language of all strings w over { } such that w is not palindrome, that is, w 6 = w R . 2.6 b. L is the complement of the language { a n b n : n 0 } . First, let’s see what the complement of L looks like: L = { a n b m : n 6 = m } ∪ { ( a b ) * ba ( a b ) * } Let’s call the leftmost language L 1 and the rightmost L 2 . The context-free grammar that generate L 1 is S 1 aS 1 b | T | U T aT | a U Ub | b The context-free grammar that generate L 2 is S 2 RbaR R RR | a | b | ² Therefore, the context-free grammar G that generate L = L 1 L 2 is S S 1 | S 2 S 1 aS 1 b | T | U S 2 RbaR T aT | a U | b R RR | a | b | ²
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CSE 105, Solution to Problem Set 2 3 c. L = { w # x : w R is a substring of x for w,x ∈ { 0 , 1 } * } . The context-free grammar G that generate L is S TR T 0 T 0 | 1 T 1 | # R R RR | 0 | 1 | ² 2.7 a). The language of strings over the alphabet { a,b } with twice as many a ’s as b ’s. The PDA that recognizes this language is shown in Figure 2. a , h ² ²,² h $ ², $ ² q 0 q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 b , $ $ b h h ², $ ² a , $ $ a b , a ² ², a ² a a a a b h Figure 2: 2.7 (a) d). L = { x 1 # x 2 ··· # x k | k 1, each x i ∈ { } and for some i and j , x i = x R j } .
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hw2sol - CSE 105: Introduction to the Theory of...

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