Uniqueness of Normal Forms for Shallow Term Rewrite Systems

Abstract

Uniqueness of normal forms (UN = ) is an important property of term rewrite systems. UN = is decidable for ground (i.e., variable-free) systems and undecidable in general. Recently, it was shown to be decidable for linear, shallow systems. We generalize this previous result and show that this property is decidable for shallow rewrite systems, in contrast to confluence, reachability, and other related properties, which are all undecidable for flat systems. We also prove an upper bound on the complexity of our algorithm. Our decidability result is optimal in a sense, since we prove that the UN = property is undecidable for two classes of linear rewrite systems: left-flat systems in which right-hand sides are of height at most two and right-flat systems in which left-hand sides are of height at most two.