# ch3 - ELEMENTS OF COMPUTATION THEORY Chapter 3...

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Undergraduate Course ELEMENTS OF COMPUTATION THEORY College of Computer Science Chapter 3 ZHEJIANG UNIVERSITY Fall-Winter, 2007 P 120-122 3.1.3 Construct CFGs that generate each of these languages. ( c ) { w ∈ { a,b } * : w = w R } . Solution: ( c ) G = ( V, Σ ,R,S ) , where: V = { a,b,S } Σ = { a,b } R = { S aSa, S bSb, S a, S b, S e } . 3.1.9 Show that the following languages are context-free by exhibiting CFGs generating each. ( a ) { a m b n : m n } ( b ) { a m b n c p d q : m + n = p + q } Solution: ( a ) G = ( V, Σ ,R,S ) , where: V = { a,b,S } Σ = { a,b } R = { S aSb, S aS, S e } .

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( b ) Let m + n = p + q = N , then n = N - m , p = N - q . a m b n c p d q = a m b N - m c N - q d q In case of m q , a m b n c p d q = a q a m - q b N - m c N - m c m - q d q Therefore, we can obtain CFG G = ( V, Σ ,R,S ) , where: V = { a,b,c,d,S,A,B } Σ = { a,b } R = { S aSd , S A , A aAc , A B , B bBc, B e } In case of m < q , we can obtain the similar results. P 135 3.3.2 Construct pushdown automata that accept each of the following. ( c ) The language { w ∈ { a,b } * : w = w R } . ( d ) The language { w ∈ { a,b } * : w has twice as many b ’s as a ’s } . Solution: ( c ) M = ( K, Σ , Γ , Δ ,s,F ) , where: K = { q,r } Σ = { a,b } Γ = { a,b } F = { r } Δ = { (( q,a,e ) , ( q,a )) (( q,b,e ) , ( q,b )) (( q,e,e ) , ( r,e )) (( q,a,e ) , ( r,e )) (( q,b,e ) , ( r,e )) (( q,a,a ) , ( r,e )) (( q,b,b ) , ( r,e )) }
( d ) M = ( K, Σ , Γ , Δ ,s,F ) , where: K = { q } Σ = { a,b } Γ = { A,a,b } F = { q } Δ = { (( q,a,e ) , ( q,A )) (( q,b,e ) , ( q,b )) (( q,a,b ) , ( q,a )) (( q,b,A ) , ( q,a )) (( q,b,a ) , ( q,e )) } P142 3.4.1 Carry out the construction of Lemma 3.4.1 for the grammar of Example

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## This note was uploaded on 01/23/2012 for the course CS 220 taught by Professor Xiaogangjin during the Spring '09 term at Zhejiang University.

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ch3 - ELEMENTS OF COMPUTATION THEORY Chapter 3...

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