I-terms in ordered resolution and superposition calculi: retrieving lost completeness

Authors:
Hicham Bensaid;Ricardo Caferra;Nicolas Peltier
Affiliations:
INPT, LIG, Rabat, Morocco and LIG, Grenoble INP, CNRS, Saint Martin d'Hères, France;LIG, Grenoble INP, CNRS, Saint Martin d'Hères, France;LIG, Grenoble INP, CNRS, Saint Martin d'Hères, France
Venue:
AISC'10/MKM'10/Calculemus'10 Proceedings of the 10th ASIC and 9th MKM international conference, and 17th Calculemus conference on Intelligent computer mathematics
Year:
2010

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Abstract

Ordered resolution and superposition are the state-of-the-art proof procedures used in saturation-based theorem proving, for non equational and equational clause sets respectively. In this paper, we present extensions of these calculi that permit one to reason about formulae built from terms with integer exponents (or I-terms), a schematisation language allowing one to denote infinite sequences of iterated terms [8]. We prove that the ordered resolution calculus is still refutationally complete when applied on (non equational) clauses containing I-terms. In the equational case, we prove that the superposition calculus is not complete in the presence of I-terms and we devise a new inference rule, called H-superposition, that restores completeness.