On some variants of Post's correspondence problem
The author proves the undecidability of several variants of Post's Correspondence Problem (PCP). The n-permutation PCP (n-PPCP) consists in finding out, for any two morphisms \(f,g:\Sigma^*_ 1\to \Sigma^*_ 2,\) whether or not \(f(A)=g(B)\) for some nonempty words A, B such that \(A=A_ 1...A_ n\) and \(B=A_{p(1)}...A_{p(n)},\) where \(A_ 1,...,A_ n\) are (possibly empty) words and p is an n-permutation; then PCP amounts to 1-PPCP. For each \(n\geq 1\), the n-PPCP is shown to be undecidable (by reducing the emptiness problem for deterministic context- sensitive languages to it via some morphisms on the instantaneous descriptions of deterministic linear bounded automata). A refinement of this proof leads to the undecidability of PCP for circular words, for doubly infinite words, and for doubly infinite powers of words. The author conjectures the undecidability of the following (n,m)-PPCP: for any morphisms f,g, find out whether or not there exists words A, B satisfying the afore-mentioned decomposition relation and such that f(A), g(B) also satisfy this relation with n replaced by m.
scientific article; zbMATH DE number 3854453
scientific article