Extending the 2D Dependency Pair Framework for Conditional Term Rewriting Systems

Abstract

Recently, a new dependency pair framework for proving operational termination of Conditional Term Rewriting Systems (CTRSs) has been introduced. We call it 2D Dependency Pair (DP) Framework for CTRSs because it makes explicit and exploits the bidimensional nature of the termination behavior of conditional rewriting, where rewriting steps $s\rightarrow t$ and rewritings $s\rightarrow ^*t$ (in zero or more steps) are defined for specific terms $s$ and $t$ by using an inference system where appropriate proof trees should be exhibited for such particular goals. In this setting, the horizontal component of the termination behavior concerns the existence of infinite sequences of rewriting steps, and the vertical component captures infinitely many climbs during the development of a proof tree for a single rewriting step. In this paper we extend the 2D DP Framework for CTRSs with several powerful processors for proving and disproving operational termination that specifically exploit the structure of conditional rules. We provide the first implementation of the 2D DP Framework as part of the termination tool mu-term. Our benchmarks suggest that, with our new processors, the 2D DP Framework is currently the most powerful technique for proving operational termination of CTRSs.

Partially supported by NSF grant CNS 13-19109, EU (FEDER), Spanish MINECO projects TIN2010-21062-C02-02 and TIN 2013-45732-C4-1-P, and GV project PROMETEO/2011/052. Salvador Lucas’ research was developed during a sabbatical year at the UIUC and was also supported by GV grant BEST/2014/026. Raúl Gutiérrez is also supported by Juan de la Cierva Fellowship JCI-2012-13528.