## 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
<

uwe@mpi-inf.mpg.de>,
1998-07-14.