# Here is how we can build a mapping reduction from PCP to CFG grammar ambiguity; please fill in…

Here is how we can build a mapping reduction from PCP to CFG grammar ambiguity; please fill in missing steps (if any) and argue that the mapping reduction actually works (achieves its purpose). Let A = w1,w2,…,wn and B = x1, x2,…, xn be two lists of words over a finite alphabet S. Let a1,a2,…,an be symbols that do not appear in any of the wi or xi . Let G be a CFG ({S,SA,SB}, S?{a1,…,an}, P, S), where P contains the productions S ? SA, S ? SB, For 1 = i = n, SA ? wiSA ai , For 1 = i = n, SA ? wiai , For 1 = i = n, SB ? xiSBai , and For 1 = i = n, SB ? xiai . Now, argue that G is

»Here is how we can build a mapping reduction from PCP to CFG grammar ambiguity; please fill in missing steps (if any) and argue that the mapping reduction actually works (achieves its purpose). Let A = w1,w2,…,wn and B = x1, x2,…, xn be two lists of words over a finite alphabet S. Let a1,a2,…,an be symbols that do not appear in any of the wi or xi . Let G be a CFG ({S,SA,SB}, S?{a1,…,an}, P, S), where P contains the productions S ? SA, S ? SB, For 1 = i = n, SA ? wiSA ai , For 1 = i = n, SA ? wiai , For 1 = i = n, SB ? xiSBai , and For 1 = i = n, SB ? xiai . Now, argue that G is ambiguous if and only if the PCP instance (A,B) has a solution (thus, we may view the process of going from (A,B) to G as a mapping reduction).

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