Complexity of Normal Form Properties and Reductions for Term Rewriting Problems Complexity of Normal Form Properties and Reductions for Term Rewriting Problems

Abstract

We present several new and some significantly improved polynomial-time reductions between basic decision problems of term rewriting systems. We prove two theorems that imply tighter upper bounds for deciding the uniqueness of normal forms (UN$^{=}$) and unique normalization (UN$^{→}$) properties under certain conditions. From these theorems we derive a new and simpler polynomial-time algorithm for the UN$^{=}$ property of ground rewrite systems, and explicit upper bounds for both UN$^{=}$ and UN$^{→}$ properties of left-linear right-ground systems. We also show that both properties are undecidable for right-ground systems. It was already known that these properties are undecidable for linear systems. Hence, in a sense the decidability results are close to optimal.