Undecidability Results for Monoids with Linear-Time Decidable Word Problems

Abstract

Using a particularly adopted simulation of Turing machines by finite string-rewriting systems we show that all strong Markov properties are undecidable for the class Clin of finitely presented monoids with linear-time decidable word problems. Expanding on this construction it is then shown that also many other properties are undecidable for Clin, among them the property of having a context-free (or a regular) cross-section, the existence of a finite convergent presentation, and the homological and homotopical finiteness conditions left- and right-FPn (n ≥ 3), left- and right-FP∞, FDT and FHT.