## Cancellative Abelian Monoids in Refutational Theorem Proving

**Uwe Waldmann**
PhD thesis, Universität des Saarlandes, 1997.
[Postscript file, 509 kB]

*Revised version:*
Cancellative abelian monoids
and related structures in refutational theorem proving (Part I/II).
*Journal of Symbolic Computation*, 33(6):777-829/831-861, 2002.

**Abstract:**
We present a constraint superposition calculus in which the axioms
of cancellative abelian monoids and, optionally, further axioms
(e.g., torsion-freeness) are integrated. Cancellative abelian
monoids comprise abelian groups, but also such ubiquitous structures
as the natural numbers or multisets. Our calculus requires neither
extended clauses nor explicit inferences with the theory axioms.
The number of variable overlaps is significantly reduced by strong
ordering restrictions and powerful variable elimination techniques;
in divisible torsion-free abelian groups, variable overlaps can even
be avoided completely. Thanks to the equivalence of torsion-free
cancellative and totally ordered abelian monoids, our calculus allows
us to solve equational problems in totally ordered abelian monoids
without requiring a detour via ordering literals.

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Uwe Waldmann
<

uwe@mpi-inf.mpg.de>,
2000-01-17.