Normalization properties for Shallow TRS and Innermost Rewriting

Abstract

Computation with a term rewrite system (TRS) consists in the application of rules from a given starting term until a normal form is reached. Two natural questions arise from this definition: whether all terms can reach at least one normal form (weak normalization property), and whether all terms can reach at most one normal form (unique normalization property). Innermost rewriting is one of the most studied and used strategies for rewriting, since it corresponds to the “call by value” computation of programming languages. Henceforth, it is meaningful to study whether weak and unique normalization are indeed decidable for a significant class of TRS with the use of the innermost strategy. In this paper we study weak and unique normalization for shallow TRS and innermost rewriting. We show decidability of unique normalization in polynomial time, and decidability of weak normalization in doubly exponential time. We obtain an exptime-hardness proof for the latter case. Nevertheless, when a TRS satisfies the unique normalization property, we conclude the decidability of the weak normalization property in polynomial time, too. Hence, whether all terms reach one and only one normal form is decidable in polynomial time. All of these results are obtained assuming that the maximum arity of the signature is fixed for the problem, which is a usual assumption.