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a10 - CpSc 421 Homework 10 Solutions 1(Sipser problem 5.21...

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CpSc 421 Homework 10 Solutions 1. (Sipser problem 5.21, 20 points ) Let AMBIG CFG = { [ G ] | G is an ambiguis CFG } (where [ G ] denotes a string that represents the grammar). Show that AMBIG CFG is undecidable. Hint: You can use a reduction from PCP. Given an instance P = t 1 b 1 , t 2 b 2 , . . . t k b k , of the Post Correspondence Problem, construct a CFG G with the rules S T | B T t 1 T a 1 | . . . | t k T a k | t 1 a 1 | . . . | t k a k B b 1 B a 1 | . . . | b k B a k | b 1 a 1 | . . . | b k a k , where a 1 , . . . , a k are new terminal symbols. Prove that this reduction works. Solution: If P is solvable, then G is ambiguous. Let i 1 , i 2 , . . . i n be a solution to P . Let w = t i 1 t i 2 · · · t i n a i n · · · a i 2 a i 1 = b i 1 b i 2 · · · b i n a i n · · · a i 2 a i 1 The string w has two derivations: S S T -→ T T t i 1 T a i 1 -→ t i 1 T a i 1 T t i 2 T a i 2 -→ t i 1 t i 2 T a i 2 a i 1 T →··· -→ t i 1 t i 2 . . . T · · · a i 2 a i 1 T t in a in -→ t i 1 t i 2 · · · t i n a i n · · · a i 2 a i 1 and S S B -→ B B b i 1 B a i 1 -→ b i 1 B a i 1 B b i 2 B a i 2 -→ b i 1 b i 2 B a i 2 a i 1 B →··· -→ b i 1 b i 2 . . . B · · · a i 2 a i 1 B b in a in -→ b i 1 b i 2 · · · b i n a i n · · · a i 2 a i 1 Thus, G is ambibuous. If G is ambiguous, then P is solvable. Let G T be the same grammar as G except that it has T as the start variable and likewise for G B . G T is unambiguous. This is because the string of a i ’s at the end of any string derived from G T describes the sequence of steps taken in the derivation. Thus, two strings can have the same suffix of a i ’s iff they have the same derivation. This means that if G T generates x and y and x = y , then x and y have the same derivation. Therefore, G T is unambiguous.
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G B is unambiguous. The proof is equivalent to that for G T .
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