Reducing relative termination to dependency pair problems

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-21401-6_11Relative termination, a generalized notion of termination, has been used in a number of different contexts like proving the confluence of rewrite systems or analyzing the termination of narrowing. In this paper, we introduce a new technique to prove relative termination by reducing it to dependency pair problems. To the best of our knowledge, this is the first significant contribution to Problem #106 of the RTA List of Open Problems. The practical significance of our method is illustrated by means of an experimental evaluation.Germán Vidal is partially supported by the EU (FEDER) and the Spanish Ministerio de Economía y Competitividad under grant TIN2013-44742-C4-R and by the Generalitat Valenciana under grant PROMETEOII201/013. Akihisa Yamadais supported by the Austrian Science Fund (FWF): Y757Iborra, J.; Nishida, N.; Vidal Oriola, GF.; Yamada, A. (2015). 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A Model-based Completeness Proof of Extended Narrowing And Resolution

We give a proof of refutational completeness for Extended Narrowing And Resolution (ENAR), a calculus introduced by Dowek, Hardin and Kirchner in the context of Theorem Proving Modulo. ENAR integrates narrowing with respect to a set of rewrite rules on propositions into automated first-order theorem proving by resolution. Our proof allows to impose ordering restriction- s on ENAR and provides general redundancy criteria, which are crucial for finding nontrivial proofs. On the other hand, it requires confluence and termination of the rewrite system, and in addition the existence of a well-founded ordering on propositions that is compatible with rewriting, compatible with ground inferences, total on ground clauses, and has some additional technical properties. Such orderings exist for hierarchical definitions of predicates. As an exampe we provide such an ordering for a fragment of set theory