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.
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