Multi-Dimensional Interpretations for Termination of Term Rewriting

Abstract

Interpretation methods constitute a foundation of termination analysis for term rewriting. From time to time remarkable instances of interpretation methods appeared, such as polynomial interpretations, matrix interpretations, arctic interpretations, and their variants. In this paper we introduce a general framework, the multi-dimensional interpretation method, that subsumes these variants as well as many previously unknown interpretation methods as instances. Employing the notion of derivers, we prove the soundness of the proposed method in an elegant way. We implement the proposed method in the termination prover and verify its significance through experiments.

Term Orderings for Non-reachability of (Conditional) Rewriting

“…Just as rewrite orders generalize reduction orders, we first introduce the notion of "rewrite pairs" by removing the well-foundedness assumption of reduction pairs. Standard definitions of reduction pairs put less order-like assumptions than the above definition, but the above (more natural) assumptions do not lose the generality of previous definitions [34]. Due to these assumptions, our rewrite pair satisfies the assumption of discrimination pairs [1].…”

“…Corollary 2 is useful when models over non-totally ordered carriers are considered. There are important methods (for termination) that crucially rely on such carriers: the matrix interpretations [5], or more generally the tuple interpretations [15,34].…”

Certifying Higher-Order Polynomial Interpretations

“…Not only do many benchmarks contain hundreds of rules, modern termination tools make use of various proof methods that have been developed for decades. Indeed, a single termination proof might for instance make use of a combination of dependency pairs [3,9,20], recursive path orders [14,21], rule removal, and multiple kinds of interpretations [8,19,25,33]. This makes bugs very difficulty to find.…”