Cancellative Abelian Monoids in Refutational Theorem Proving

Uwe Waldmann

PhD thesis, Universitšt 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 <>, 2000-01-17.