Superposition for Divisible Torsion-Free Abelian Groups

Uwe Waldmann

In Claude Kirchner and Hélène Kirchner, editors, Automated Deduction - CADE-15, 15th International Conference on Automated Deduction, LNAI 1421, pages 144-159, Lindau, Germany, July 5-10, 1998. Springer-Verlag.

Extended abstract: A Superposition Calculus for Divisible Torsion-Free Abelian Groups, in Maria Paola Bonacina and Ulrich Furbach, editors, FTP'97: International Workshop First-Order Theorem Proving, pages 130-134, Schloß Hagenberg, Linz, Austria, October 27-28, 1997. RISC-Linz Report Series No. 97-50.

Abstract: Variable overlaps are one of the main sources for the inefficiency of AC or ACU theorem proving calculi. In the presence of the axioms of abelian groups or at least cancellative abelian monoids, ordering restrictions allow us to avoid some of these overlaps, but inferences with unshielded variables remain necessary. In divisible torsion-free abelian groups, for instance the rational numbers, every clause can be transformed into an equivalent clause without unshielded variables. We show how such a variable elimination algorithm can be integrated into the cancellative superposition calculus. The resulting calculus is refutationally complete with respect to the axioms of divisible torsion-free abelian groups and allows us to dispense with variable overlaps altogether. If abstractions are performed eagerly, the calculus makes it furthermore possible to avoid the computation of AC unifiers and AC orderings.

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Uwe Waldmann <>, 1998-07-14.