Soundness of Unravelings for Conditional Term Rewriting Systems via Ultra-Properties Related to Linearity

TL;DR

This paper investigates conditions under which unravelings of conditional term rewriting systems are sound, focusing on properties like linearity and non-erasing, and compares different unraveling variants for deterministic CTRSs.

Contribution

It establishes that an optimized unraveling variant is sound under certain linearity conditions and relates its soundness to other existing unravelings, clarifying their relative strengths.

Findings

Optimized unraveling is sound if the unraveled TRS is left-linear or both right-linear and non-erasing.

Soundness of the optimized unraveling implies soundness of Ohlebusch's unraveling.

Soundness of Ohlebusch's unraveling is the weakest among several unraveling methods.

Abstract

Unravelings are transformations from a conditional term rewriting system (CTRS, for short) over an original signature into an unconditional term rewriting systems (TRS, for short) over an extended signature. They are not sound w.r.t. reduction for every CTRS, while they are complete w.r.t. reduction. Here, soundness w.r.t. reduction means that every reduction sequence of the corresponding unraveled TRS, of which the initial and end terms are over the original signature, can be simulated by the reduction of the original CTRS. In this paper, we show that an optimized variant of Ohlebusch's unraveling for a deterministic CTRS is sound w.r.t. reduction if the corresponding unraveled TRS is left-linear or both right-linear and non-erasing. We also show that soundness of the variant implies that of Ohlebusch's unraveling. Finally, we show that soundness of Ohlebusch's unraveling is the weakest in soundness of the other unravelings and a transformation, proposed by Serbanuta and Rosu, for (normal) deterministic CTRSs, i.e., soundness of them respectively implies that of Ohlebusch's unraveling.