On Termination of One Rule Rewrite Systems

The undecidability of the termination of rewrite systems is usually proved by reduction to the halting of Turing machines. In particular, Dauchet proves the unde-cidability of the termination of one rule rewrite systems by coding Turing machines into one rule rewrite systems. Rewrite systems are a very simple model of computation and one may expect proofs in this model to be more straightforward than those referring to the more complex model of Turing machines. In this paper we reduce the problem of termination of one rule rewrite systems to problems somewhat more related to rewrite systems namely to Post correspondence problems and to termination of semi-Thue systems. Proofs we obtain this way are shorter and we expect other interesting applications from these codings. In particular, the second part proposes a simulation of semi-Thue systems by one rule systems. A correspondence system is a nite subset P of ordered pairs for some alphabet , determine, given a system, whether this system has a match. It is known Pos47, LP81] that modiied Post's correspondence system is undecidable even when has two elements. In this paper, we speak about strings on 0, 1 and $. Term rewrite system usually consider terms, therefore when we write the string a 1 a 2 : : :a n it should be understood as the term cons(a 1 ; cons(a 2 ; : : :; cons(a n ; nil) : : :)); and when we write the string a 1 a 2 : : :a n x where x is a variable, should be understood as the term