COMS W3261
Computer Science Theory
Lecture 15: November 12, 2012
The Universal Language and
Post's Correspondence Problem

Outline

L_u, the universal language, is the set of binary strings that encode a pair (M, w) consisting of a Turing machine and an input string accepted by that Turing machine. That is,

L_u = { (M, w) | M is an encoding of a Turing machine, w is an encoding of a binary string, and w is in L(M) }.

We can construct a Turing machine U, called the universal Turing machine, to recognize L_u.
It is easiest to think of U as a multitape TM.

To simulate a move of M, U searches for a transition on the current state of M (stored on tape 3) and the current state tape symbol of M (stored at the position on tape 2 scanned by U.

Suppose L_u were recursive. Then there exists a TM M that accepts the complement of L_u.
Then we can transform M into a TM M' that accepts L_d as follows:

M' transforms its input string w into a pair (w,w).
M' simulates M on (w,w) assuming the first w is an encoding of a TM M_i and the second w is an encoding of a binary string w_i. Since M accepts the complement of L_u, it will accept (w,w) iff M_i does not accept w_i.

Thus, M' accepts w iff w is in L_d. But we have previously shown there does not exist a TM that defines L_d.
We conclude L_u is not recursive.

A reduction is the conversion of one problem A to another B in such a way that a solution to B can be used to solve A.
The reduction should turn positive instances of A into positive instances of B and negative instances of A into negative instances of B.
Formally, a reduction from A to B is a Turing machine that takes an instance of A written on its tape and halts with an instance of B on its tape.
We say a function f: Σ* → Σ* is a computable function if there is a TM M such that given w on the input tape, M makes a sequence of moves and halts with f(w) on the input tape.
A language A is mapping reducible to language B if there is a computable function f such that for every w in Σ*, w is in A iff f(w) is in B. The function f is called a reduction of A to B.
If we can reduce an undecidable problem A to problem B, then we can conclude that problem B is also undecidable.
If we can show the complement of a problem A is undecidable, then A itself is undecidable.
If we can reduce an non-RE problem A to problem B, then we can conclude that problem B is also non-RE.

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. Let A = (w₁, w₂, ... , w_k) and B = (x₁, x₂, ... , x_k) be the two lists.
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}.
Example: Let A = (a, b, ca, abc) and B = (ab, ca, a, c). The sequence 1, 2, 3, 1, 4 is a solution because the same string abcaaabc is obtained by concatenating the corresponding strings from either list A [(a)(b)(ca)(a)(abc)] or list B [(ab)(ca)(a)(ab)(c)].
Post's correspondence problem is to determine whether an instance has a solution.

The Modified PCP has the additional requirement that the first string from list A and the first string from list B has to be the first string in the solution. The example above has this property.
Formally, a solution to an 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}.
We can show that the modified PCP problem can be reduced to the PCP problem as follows:

We are given an instance of MPCP with lists A = (w₁, w₂, ... , w_k) and B = (x₁, x₂, ... , x_k).
Assume * and $ are new symbols.
From (A, B) we construct a PCP instance (C, D) with
C = (y₀, y₁, ... , y_k+1) and D = (z₀, z₁, ... , z_k+1) where
1. y_i is w_i with a * after each symbol in w_i, for i = 1, 2, ..., k.
2. z_i is x_i with a * before each symbol in x_i, for i = 1, 2, ..., k.
3. y₀ = *y₁ and y_k+1 = $.
4. z₀ = z₁ and z_k+1 = *$.
We can show i₁, i₂, ... , i_m is a solution to the given (A,B)-MPCP instance iff 0, i₁, i₂, ... , i_m, i_k+1 is a solution to this constructed (C,D)-PCP instance.

We can show that given (M, w), an instance of L_u, we can reduce this instance of L_u to an instance (A, B) of the MPCP such that M accepts w iff (A, B) has a solution. We do this by showing that (A,B) simulates the computation of M on w.
We will thus show that both the MPCP and the PCP problems are undecidable.

We can reduce an instance of the PCP problem to an instance of determining whether a CFG is ambiguous, thereby showing it is undecidable to determine whether a CFG is ambiguous.
We will illustrate the reduction with the following example. Let (A, B) be an instance of the PCP problem with A = (a, b, ca, abc) and B = (ab, ca, a, c). Let G be the CFG with the productions


S → A | B
A → aA1 | bA2 | caA3 | abcA4 | a1| b2 | ca3 | abc4
B → abB1 | caB2 | aB3 | cB4 | ab1 | ca2 | a3 | c4

There are two distinct leftmost derivations for the string abcaaabc41321 because this instance of the PCP problem has a solution.

If we can reduce a problem A to a decidable problem B, can we conclude A is decidable?
If we can reduce a decidable problem A to a problem B, can we conclude B is decidable?
If we can reduce a problem A to an undecidable problem B, can we conclude problem A is undecidable?
If we can reduce an undecidable problem A to a problem B, can we conclude problem B is undecidable?
Does the following instance of PCP have a solution: A = (ab, b, aba, aa) and B = (abab, a, b, a)?
Show that the equivalence problem for CFG's is undecidable.