The First-Order Theory of Linear One-Step Rewriting is Undecidable

The theory of one-step rewriting for a given rewrite system R and signature σ is the first-order theory of the following structure: its universe consists of all σ-ground terms, and its only predicate is the relation “x rewrites to y in one step by R”. The structure contains no function symbols and no equality. We show that there is no algorithm deciding the ∃∗∀∗-fragment of this theory for an arbitrary finite, linear and non-erasing term-rewriting system. With the same technique we prove that the theory of encompassment plus one-step rewriting by the rule f(x) → g(x) and the modal theory of one-step rewriting are undecidable.