Post correspondence problem

Many word problems are undecidable based on the Post correspondence problem.

The Post correspondence problem.

The unsolvability of his Post correspondence problem turned out to be exactly what was needed to obtain unsolvability results in the theory of formal languages.

The Post correspondence problem is an undecidable decision problem that was introduced by Emil Post in 1946.

The circular Post correspondence problem asks whether indexes can be found such that and are conjugate words, i.e., they are equal modulo rotation.

The general decision problem of whether a grammar is ambiguous is undecidable because it can be shown that it is equivalent to the Post correspondence problem.

Post correspondence problem: Given a finite set of strings, determine if there is a string that can be factored into a composition of the strings (allowing repeats) in two different ways.

The undecidability of this problem follows from the fact that if such an algorithm to determine it existed, we would be able to solve the Post correspondence problem, which is known to be undecidable.

One of the most important variants of PCP is the bounded Post correspondence problem, which asks if we can find a match using no more than k tiles, including repeated tiles.

Moreover, they showed that if this requirement is slightly loosened so that only one of the first two characters need to differ (the so-called 2-marked Post Correspondence Problem), the problem becomes undecidable again.

Another variant of PCP is called the marked Post Correspondence Problem, in which each u must begin with a different symbol, and each v must also begin with a different symbol.