Mathematics in Computer Science

, Volume 6, Issue 4, pp 427–456 | Cite as

Superposition Decides the First-Order Logic Fragment Over Ground Theories

Article

Abstract

The hierarchic superposition calculus over a theory T, called SUP(T), enables sound reasoning on the hierarchic combination of a theory T with full first-order logic, FOL(T). If a FOL(T) clause set enjoys a sufficient completeness criterion, the calculus is even complete. Clause sets over the ground fragment of FOL(T) are not sufficiently complete, in general. In this paper we show that any clause set over the ground FOL(T) fragment can be transformed into a sufficiently complete one, and prove that SUP(T) terminates on the transformed clause set, hence constitutes a decision procedure provided the existential fragment of the theory T is decidable. Thanks to the hierarchic design of SUP(T), the decidability result can be extended beyond the ground case. We show SUP(T) is a decision procedure for the non-ground FOL fragment plus a theory T, if every non-constant function symbol from the underlying FOL signature ranges into the sort of the theory T, and every term of the theory sort is ground. Examples for T are in particular decidable fragments of arithmetic.

Keywords

Theorem proving Combination of theories Decision procedure Arithmetic 

Mathematics Subject Classification (2010)

Primary 68T15 Secondary 03B25 

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Universität des Saarlandes, Max-Planck-Institut für InformatikSaarbrückenGermany

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