### Flatwords and Post Correspondence Problem

Tero Harju, Marjo Lipponen, Alexandru Mateescu
1996 Theoretical Computer Science
We investigate properties of flatwords and k-flatwords. In particular, these words are studied in connection with Post Correspondence Problem (PCP). An open problem occurs: where is the borderline between the decidability and the undecidability of k-flat PCP over an alphabet with n symbols? Our main results concern the related new types of prime solutions of PCP. SSDI 0304-3975 (95) 00092-5 94 T Harju et al. I Theoretical Computer Science 161 (1996) 93-108 with re-entrant routines is always a
more » ... t of flatwords. For more details concerning this problem, see [7] . In this paper we introduce k-flatwords and investigate some of their properties. We also study the relationship between k-flatwords and the Post Correspondence Problem. This is done in two ways: we define a new type of Post Correspondence Problem, k-flat PCP, and new types of primitive solutions for the instance (g,h) = PCP. The suitability of the Post Correspondence Problem for reduction arguments is due to the fact that in some sense the essence of computations is captured by PCP. Thus simple solutions of PCP mean simplifications of computations and the results contribute on an abstract level to the understanding of computations. Definition 1.1. Assume that C = {ai, a2,. . . , a,} and let k 3 1 be a fixed number. A word w E ,E* is a k-flatword iff [WI < 1 or w = xix2 . ..x." where ma2 and x, E C, ln, then FZk(C) is a finite set. For instance, if C = {a, b}, then FEZ(C) = (2, a, b, ab, ba}. Moreover, for any k, k>n, FZk(C) = FZ,(C).