We study decision problems of the form: given a regular or linear context-free language L, is there a word of a given fixed form in L, where given fixed forms are based on word operations copy, marked copy, shuffle and their combinations.

Denote by ш the operation of interleaving, or shuffling, of words. We prove that, given a regular language R and a letter-to-letter morphism φ, it is undecidable whether or not there exists a word ω such that ω ш φ(ω) ∩ R ≠ ø.

We give a new simplified proof for undecidability of the Bi-Infinite Post Correspondence Problem (ℤPCP). We reduce the special case of the word problem of semi-Thue systems to ℤPCP.