Proof of Undecidability of Post Correspondence Principle

ChatGPT4:

The Post Correspondence Problem (PCP) is a classic decision problem in the field of computability theory and theoretical computer science, formulated by Emil Post in 1946. The essence of why PCP is undecidable lies in its connection to the concept of algorithmic undecidability, which means there is no general algorithm that can solve all instances of the problem.

The problem can be described as follows:

Given two lists of strings, A = [a1, a2, ... , an] and B = [b1, b2, ... , bn], over some alphabet, the question is whether there exists a finite sequence of indices [i1, i2, ... , ik] with 1 ≤ ij ≤ n for each j, such that the concatenation of the strings from list A using this sequence of indices is equal to the concatenation of the strings from list B using the same sequence of indices. That is, we want to know if [ai1, ai2, ... , aik] and B = [bi1, bi2, ... , bik],

The undecidability of the Post Correspondence Problem comes from its ability to simulate the behavior of any Turing machine, which is a theoretical model for computation. Here's why it's undecidable:

1. **Simulation of Turing Machines**: The PCP can be used to simulate the behavior of any Turing machine. By constructing a specific instance of PCP that corresponds to the computation of a Turing machine on an input, solving the PCP would be equivalent to determining whether the Turing machine halts on that input. Since the Halting Problem (whether a Turing machine halts or runs forever on a given input) is undecidable, so too is the PCP.

2. **No General Algorithm**: Because PCP can encode the computation of Turing machines, and because there is no general algorithm that can decide the Halting Problem for all Turing machines and inputs, there is also no general algorithm that can solve all instances of the PCP.

3. **Reduction from the Halting Problem**: The formal proof of undecidability involves reducing the Halting Problem to the PCP. If there were an algorithm that could solve the PCP, then we could use it to solve the Halting Problem by translating instances of the Halting Problem into instances of the PCP. This would contradict the undecidability of the Halting Problem.

In summary, the undecidability of the Post Correspondence Problem arises from its ability to encode arbitrary computations, which means that solving it would entail solving the Halting Problem, a known undecidable problem. This foundational result has profound implications for the limits of computation and algorithmic solvability, echoing through the fields of computability theory, algorithm design, and beyond.

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