Question

4. (25 point) Prove, by using a reduction argument, that the problem Context-Free「M = {(M): L(M) is context free} is undecidable. Hint: Study Section 5.1 in our text and mimic the technique which shows RegulartM is undecidable.

Answer 1

We reduce from Post's Correspondence Problem. Suppose we can, in fact, decide the language l(G)IG a CFG and L(G) ambiguous Given αί , . . . , α,n, A , . . . , β". Construct the following CFG G = (V, Σ, R, S): V-{ S, Si, Ss). (where ơi are new characters added to the alphabet, e.g., σί--) If the language is ambiguous, then there is a derivation of some string w in two different ways. Supposing, wlog, that the derivations both start with the rule S → S·reading the new characters backwards until they end makes sure there can only be one derivation, so that's not possible. Hence, we see that the only ambiguity can come from one Si and one S2 'start. But then, taking the substring of w up to the beginning of the new characters, we have a solution to the PCP (since the strings of indices used after those points match). Similarly, if there is no ambiguity, then the PCP cannot be solved, since a solution would imply an ambiguity that just follows S .Si * ασ and S-> S2 * A, where α-β are strings of matching α's and β's (since the d's match). Hence, we've reduced to PCP, and since that's undecidable, we're done.