Automated Synthesis of a Finite Complexity Ordering for Saturation

TL;DR

This paper introduces a new saturation procedure for clause sets using a finite complexity atom ordering, ensuring decidability of entailment and improving inference redundancy handling in automated reasoning.

Contribution

It presents a novel on-the-fly construction method for finite atom orderings compatible with standard term orderings, enhancing saturation procedures.

Findings

Finite atom orderings enable decidable entailment.

Non-redundant inferences are preserved during saturation.

The approach generalizes existing ordered resolution techniques.

Abstract

We present in this paper a new procedure to saturate a set of clauses with respect to a well-founded ordering on ground atoms such that A < B implies Var(A) {\subseteq} Var(B) for every atoms A and B. This condition is satisfied by any atom ordering compatible with a lexicographic, recursive, or multiset path ordering on terms. Our saturation procedure is based on a priori ordered resolution and its main novelty is the on-the-fly construction of a finite complexity atom ordering. In contrast with the usual redundancy, we give a new redundancy notion and we prove that during the saturation a non-redundant inference by a priori ordered resolution is also an inference by a posteriori ordered resolution. We also prove that if a set S of clauses is saturated with respect to an atom ordering as described above then the problem of whether a clause C is entailed from S is decidable.