Improved Matrix Interpretation

Abstract

Abstract. We present a new technique to prove termination of Term Rewriting Systems, with full automation. A crucial task in this context is to find suitable well-founded orderings. A popular approach consists in interpreting terms into a domain equipped with an adequate well-founded ordering. In addition to the usual interpretations: natural numbers or polynomials over integer/rational numbers, the recently introduced matrix based interpretations have proved to be very efficient regarding termination of string rewriting and of term rewriting. In this spirit we propose to interpret terms as polynomials over integer matrices. Designed for term rewriting, our generalisation subsumes previous approaches allowing for more orderings without increasing the search space. Thus it performs better than the original version. Another advantage is that, interpreting terms to actual polynomials of matrices, it opens the way to matrix non linear interpretations. This result is implemented in the CiME3 rewriting toolkit.

Formalizing Monotone Algebras for Certification of Termination and Complexity Proofs

“…Since similar tasks have already been conducted CoLoR [3] and Coccinelle [6] (using the approach of CiME [7]), we just shortly mention two distinguishing features of our work.…”

“…To support matrix interpretations, we basically follow the ideas of Courtieu et al [7], i.e., we integrate matrix interpretations as linear polynomial interpretations where the carrier consists of matrices. To this end, we first developed a list-based and executable formalization of matrices [29] within Isabelle/HOL and afterwards connected it in the theory Linear Poly Order to obtain matrix interpretations in IsaFoR.…”

“…To this end, we first developed a list-based and executable formalization of matrices [29] within Isabelle/HOL and afterwards connected it in the theory Linear Poly Order to obtain matrix interpretations in IsaFoR. Again, one of the distinguishing features of our work is the parametric carrier -for example in [7] the carrier is fixed to N. Note that the demand for other carriers like Q and R was clearly shown by Neurauter and Middeldorp [23]: matrix interpretations over R are strictly more powerful than those over Q which in turn are strictly more powerful than those over N.…”

Satisfiability of Non-linear (Ir)rational Arithmetic

“…Hence our contribution for the first time gives evidence that matrix interpretations over the non-negative reals do really extend the power of termination criteria in practice. Recently non-linear matrix interpretations have been introduced [9] by considering matrix domains instead of vector domains. We would like to investigate if this more general setting can also benefit from rational domains.…”

A3PAT, an approach for certified automated termination proofs

“…We give in particular the first full formalisation in COQ of the powerful subterm criterion [29] for termination of rewriting systems 2 , and we propose an extension (Theorem 4). Another example is our extended notion of matrix interpretations [16].…”