(Generalized) Post Correspondence Problem and semi-Thue systems

Let PCP(k) denote the following restriction of the well-known Post Correspondence Problem [9]: given alphabet Σ of cardinality k and two morphisms σ, τ : Σ → {0, 1}, decide whether there exists w ∈ Σ+ such that σ(w) = τ(w). Let Accessibility(k) denote the following restriction of the accessibility problem for semi-Thue systems: given a k-rule semi-Thue system T and two words u and v, decide whether v is derivable from u modulo T . In 1980, Claus showed that if Accessibility(k) is undecidable then PCP(k + 4) is also undecidable [2]. The aim of the paper is to present a detailed proof of the statement. We proceed in two steps, using the Generalized Post Correspondence Problem (GPCP) [4] as an auxiliary. Let GPCP(k) denote the following restriction of GPCP: given an alphabet Σ of cardinality k, two morphisms σ, τ : Σ → {0, 1} and four words s1, s2, t1, t2 ∈ {0, 1} , decide whether there exists w ∈ Σ such that s1σ(w)s2 = t1τ(w)t2. First, we prove that if Accessibility(k) is undecidable then GPCP(k+2) is also undecidable. Then, we prove that if GPCP(k) is undecidable then PCP(k+2) is also undecidable. (The latter result can also be found in [7].) To date, the sharpest undecidability bounds for both PCP and GPCP have been deduced from Claus’s result: since Matiyasevich and Sénizergues showed that Accessibility(3) is undecidable [8], GPCP(5) and PCP(7) are undecidable.