Elementary Theory of One - Step

We settle in the negative the following Problem 51 (Comon{Dauchet, RTA'93 3], RTA'95 4]). Given an arbitrary nite term rewriting system R, is the rst-order theory of one-step rewriting (!R) decidable? Decidability would imply the decid-ability of the rst-order theory of encompassment (that is, being an instance of a subterm) 1], as well as several known decidability results in rewriting. (It is well known that the theory of ! R is in general undecid-able.) u t The positive solution to the problem would have given a powerful uniied decision method for many problems in rewriting. The decidability of the problem was widely believed and conjectured. We present a nite term rewriting system R such that the dyadic word concatenation 1 is expressible via the one-step rewriting relation in R. Consequently (Quine 5], Smullyan 6]), the elementary theory of one-step rewriting is undecidable, in general. 1 Dyadic Concatenation Let f0; 1g be the set of nite words over a two-letter alphabet f0; 1g. Consider the rst-order model hf0; 1g ; Conci of signature fC oncg with f0; 1g as a carrier , and the ternary predicate Conc interpreted as follows: Conc(x; y; z) is true for x; y; z 2 f0; 1g if and only if x y = z, i.e., the concatenation of x and y equals z. Recall that the elementary theory of a model M of signature is the set of all rst-order sentences of signature true in M. Quine 5] and Smullyan 6] showed that the recursion theory can be developed independently of arithmetic, in terms of dyadic concatenation. As a consequence, Theorem1 ((5, 6]). The elementary theory of the model hf0; 1g ; Conci is un-decidable. u t We will present a nite term rewriting system R such that the predicate \to be a dyadic word" and the dyadic concatenation predicate are expressible via the one-step rewriting relation generated by R. By Theorem 1, this is suucient to prove the claim in the title of the note. 1 = Concatenation of words in a two-letter alphabet. Dyadic = twofold, having two components.