Explain the Post Correspondence Problem in TOC

The Post Correspondence Problem (PCP) was introduced by Emil Post in 1946 and is an undecidable decision problem.

The PCP problem over an alphabet Σ is state. Given the following two lists, M and N of non-empty strings over Σ−

M = (x1, x2, x3,………, xn)
N = (y1, y2, y3,………, yn)

We can say that there is a Post Correspondence Solution, if for some i1,i2,………… ik,

Where 1≤ ij ≤ n, the condition xi1 …….xik = yi1 …….yik satisfies.

Example 1

Find whether the lists M = (abb, aa, aaa) and N = (bba, aaa, aa) have a Post Correspondence Solution.




x2x1x3 = ‘aaabbaaa’

and y2y1y3 = ‘aaabbaaa’

We can see that

x2x1x3 = y2y1y3

Hence, the solution is i = 2, j = 1, and k = 3.

Consider another example

Let us consider ,in PCP problem we have N number of Dominos (tiles). The aim is to arrange tiles in such order that string made by Numerators is same as string made by Denominators.

In simple words, lets assume we have two lists both containing N words, aim is to find out concatenation of these words in some sequence such that both lists yield same result.

Let’s try understanding this by taking two lists A and B

A=[aa, bb, abb] and B=[aab, ba, b]

Now for sequence 1, 2, 1, 3 first list will yield aabbaaabb and second list will yield same string aabbaaabb.

So the solution to this PCP becomes 1, 2, 1, 3.

Post Correspondence Problems can be represented in two ways

  • Domino’s Form.

  • Table form