117b – Unsolvable problems – Lecture 2

We briefly discuss Church’s and Trakhtenbrot’s theorems: The set of logically valid sentences and the set of sentences valid in all finite models are undecidable. More precisely: If L is a language that contains either a binary symbol or the language of arithmetic, then the set of theorems of Q reduces to the set of validities of L, so this set is (r.e. and) undecidable. For the same L, we can axiomatize the behavior of a given Turing machine on a given input (by a similar argument to the association of a semi-Thue system to each Turing machine) in such a way that this axiomatization has a finite model iff the machine converges on this input, thus the halting problem reduces to the complement of the set of finite validities, so this last set is (co-r.e. and) undecidable.

We define Post correspondence systems and show that the problem of determining whether such a system admits a solution is unsolvable. Two applications to the theory of formal languages follow: There is no algorithm to test whether 2 given context-free grammars have disjoint languages, and there is no algorithm to test whether a given context-free grammar is ambiguous.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: