COMS W3261 CS Theory
Lecture 18: Undecidable Problems

1. PCP and Modified PCP are Undecidable

An instance of Post's Correspondence Problem consists of two lists of strings over some alphabet where the two lists have the same number of strings. For example, the pair (A, B) where A = (w₁, w₂, ... , w_k) and B = (x₁, x₂, ... , x_k) is an instance of PCP.
A solution to this instance of PCP, if one exists, is a sequence of one or more integers i₁, i₂, ... , i_m such that w_i₁ w_i₂ ... w_{i_m} = x_i₁ x_i₂ ... x_{i_m}. An integer index can be repeated many times in a solution.

An instance of MPCP consists of two lists of strings over some alphabet where the two lists have the same number of strings. For example, the pair (A, B) where A = (w₁, w₂, ... , w_k) and B = (x₁, x₂, ... , x_k) is an instance of MPCP.
However, a solution to this instance of the MPCP is a sequence of zero or more integers i₁, i₂, ... , i_m such that w₁w_i₁ w_i₂ ... w_{i_m} = x₁x_i₁ x_i₂ ... x_{i_m}. Note that the first string in each of the lists has to be the first string in the solution.

We can show PCP is undecidable by reducing the universal language L_u to MPCP and then reducing MPCP to PCP.

A language L is undecidable if (1) L is not recursively enumerable or (2) L is recursively enumerable but not recursive.
We have shown the diagonal language L_d is not recursively enumerable and the universal language L_u is recursively enumerable but not recursive. We can use the undecidability of these two languages to show a number of problems about Turing machines are undecidable:

Define L_e to be { M | L(M) = ∅ }. We can show that L_e is not recursively enumerable. See HMU Section 9.3.2 for a proof.
Define L_ne to be { M | L(M ) ≠ ∅ }. L_ne is the complement of L_e. We can show that L_ne is recursively enumerable but not recursive. Again, see HMU Section 9.3.2 for a proof.
Note that we have just stated that L_ne is not recursive. If L_e were recursively enumerable, both it and L_ne would be recursive because L_e and L_ne are complements of each other. This is the reason why L_e cannot be recursively enumerable.
The Halting Problem

In Alan Turing’s original formulation of Turing machines acceptance was just by halting not necessarily by halting in a final state.
We can define H(M) for a Turing machine M to be the set of input strings w such that M halts on w in either a final or a nonfinal state.
The famous halting problem is the set of pairs {(M, w) | w is in H(M)}.
We can show the halting problem is recursively enumerable but not recursive.

There is a useful result called "Rice's Theorem" that states that any nontrivial property of Turing machines that depends only on the language that the TM accepts must be undecidable. A property of the recursively enumerable languages is just a set of recursively enumerable languages. A property is trivial if it either is the empty language or is the set of all recursively enumerable languages; otherwise, the property is said to be nontrivial. One example of a nontrivial property is whether a language is context free. Thus, by Rice's Theorem determining whether the language accepted by a Turing machine is context free is undecidable.

We have already shown that it is undecidable whether a context-free grammar is ambiguous. The proof technique we used was to reduce Post’s Correspondence Problem to the ambiguity problem for CFG’s.
We can also use this proof technique to show a number of important problems about context-free languages are undecidable.
Let G and H be CFG’s, and let R be a regular expression. The following problems are undecidable:

Show that if the complement of a language L is undecidable, then L itself is undecidable.
Show that the Halting Problem is recursively enumerable but not recursive.
Show that it is undecidable whether the language accepted by a TM is infinite.
Show that the complement of the list language L_A [HMU, p. 413] is context free.
Show that it is undecidable whether the language generated by one context-free grammar is contained within the language generated by another context-free grammar.
Let L be the set of CFG’s G such that L(G) contains at least one palindrome. Show that L is undecidable.
Give an example of a language L such that both L and the complement of L are not recursively enumerable.