The Post’s Correspondence Problem (PCP) is a classic undecidable problem in computer science and mathematics, introduced by Emil Post in 1946. It involves a set of dominoes, each containing two strings, and seeks to determine whether a sequence of dominoes can be selected and arranged to form identical strings when concatenated.

Post’s Correspondence Problem (PCP):

Problem Statement: Given a finite set of dominoes, where each domino contains two strings over a finite alphabet, determine whether there exists a sequence of dominoes such that the top string of each selected domino concatenates to the same string as the bottom string.

Example: Consider the set of dominoes:

1. $\{A,AB\}$
2. $\{B,BA\}$

The PCP asks whether there exists a sequence of these dominoes that, when arranged in that sequence, produces the same string from top to bottom. In this case, the answer is “yes” because selecting the first domino followed by the second one results in the sequence

ABBA, where the top and bottom strings are identical.

Modified Post’s Correspondence Problem:

The Modified Post’s Correspondence Problem introduces additional constraints to the original PCP, making it even more challenging.

Modification: In the modified version of the problem, an extra requirement is added that the lengths of the selected strings must match. This means that not only should the top and bottom strings match when concatenated, but they should also have the same length.

Example: Consider the same set of dominoes as before:

1. $\{A,AB\}$
2. $\{B,BA\}$

In the modified PCP, we must find a sequence of dominoes where not only do the top and bottom strings match but also have the same length. In this case, there is no such sequence because the lengths of the strings “A” and “AB” are different.

Significance:

1. Undecidability: Both the original PCP and the modified version are undecidable problems, meaning that there is no algorithm that can always correctly determine whether a solution exists.
2. Theoretical Implications: The undecidability of PCP and its variations demonstrate the existence of limitations in algorithmic computation and have important consequences for theoretical computer science and mathematics.
3. Applications: Although undecidable in the general case, PCP and its variations have applications in cryptography, formal language theory, and the study of algorithmic complexity. Despite being undecidable in the general case, specific instances of PCP can sometimes be solved efficiently using heuristic methods or specialized algorithms.