On the Normalization and Unique Normalization Properties of Term Rewrite Systems

Abstract

Computation with a term rewrite system (TRS) consists of the application of rules from a given starting term until a normal form is reached. Two natural questions arise from this the definition: whether all terms can reach at least one normal form (normalization property), and whether all terms can reach at most one normal form (unique normalization property). We study the decidability of these properties for two syntactically restricted classes of TRS: for (i) shallow right-linear TRS, and for (ii) linear right-shallow TRS. We show that the normalization property is decidable for both cases (i) and (ii), and that the unique normalization property is undecidable for case (ii), whereas for case (i) remains unknown. Nevertheless, for case (i), if the normalization property is satisfied, then the unique normalization property becomes decidable. Hence, whether all terms reach exactly one normal form for TRS of kind (i) is decidable. These results are based on known constructions of tree automata with constraints and rewrite closure, and on reducing the normalization property to normalization from a concrete finite set of terms.