Decidable approximations of term rewriting systems

Abstract

A linear term rewriting system $\mathcal{R}$is growing when, for every rule l→r ∈ $\mathcal{R}$, each variable which is shared by l and r occurs at depth one in l. We show that the set of ground terms having a normal form w.r.t. a growing rewrite system is recognized by a finite tree automaton. This implies in particular that reachability and sequentiality of growing rewrite systems are decidable. Moreover, the word problem is decidable for related equational theories. We prove that our conditions are actually necessary: relaxing them yields undecidability of reachability.

Concerning sequentiality, the result may be restated in terms of approximations of term rewriting systems. An approximation of a system $\mathcal{R}$is a renaming of the variables in the right hand sides which yields a growing rewrite system. This gives the decidability of a new sufficient condition for sequentiality of left-linear rewrite systems, which encompasses known decidable properties such as strong, NV and NVNF sequentiality.