Automating the Dependency Pair Method

Abstract

Developing automatable methods for proving termination of term rewrite systems that resist traditional techniques based on simplification orders has become an active research area in the past few years. The dependency pair method of Arts and Giesl is one of the most popular such methods. However, there are several obstacles that hamper its automation. In this paper we present new ideas to overcome these obstacles. We provide ample numerical data supporting our ideas.

Certification of Termination Proofs Using CeTA

“…Nevertheless, we also feature one new technique that has not been certified so far. Whereas currently only the initial dependency graph estimation of [1] has been certified, we integrated the most powerful estimation which does not require tree automata techniques and is based on a combination of [9,12] where the function tcap is required. Initial problems in the formalization of tcap led to the development of etcap, an equivalent but more efficient version of tcap which is also beneficial for termination provers.…”

“…The estimation EDG * [12] does the same check and additionally uses the reversed TRS R −1 = {r → | → r ∈ R}, i.e., it uses the fact that tσ → * R uµ implies uµ → * R −1 tσ and checks whether cap(u) does not unify with t. Of course in the application of cap(u) we have to take the reversed rules into account (possibly changing the set of defined symbols) and it is not applicable if R contains a collapsing rule → x where x ∈ V.…”

Proving Non-looping Non-termination Automatically

“…For the desired narrowing, we have to instantiate the recursive pattern rule (8) such that the base term of its right-hand side contains the left-hand side of the rule gt(s(x), 0) → tt. To this end, we use inference rule (V).…”

Proving Termination of Integer Term Rewriting

“…The DP framework [1,12,13,16,19] is one of the most powerful and popular methods for automated termination analysis of TRSs and the DP technique is implemented in almost all current TRS termination tools. Our goal is to extend the DP framework in order to handle ITRSs.…”