Superposition calculus

The superposition calculus is a calculus for reasoning in equational first-order logic. It has been developed in the early 1990s and combines concepts from first-order resolution with ordering-based equality handling as developed in the context of (unfailing) Knuth–Bendix completion. It can be seen as a generalization of either resolution (to equational logic) or unfailing completion (to full clausal logic). As most first-order calculi, superposition tries to show the unsatisfiability of a set of first-order clauses, i.e. it performs proofs by refutation. Superposition is refutation-complete — given unlimited resources and a fair derivation strategy, from any unsatisfiable clause set a contradiction will eventually be derived.

As of 2007, most of the (state-of-the-art) theorem provers for first-order logic are based on superposition (e.g. the E equational theorem prover), although only a few implement the pure calculus.

Implementations

• E
• SPASS
• Vampire
• Waldmeister
• Ayane at Google Code

References

• Rewrite-Based Equational Theorem Proving with Selection and Simplification, Leo Bachmair and Harald Ganzinger, Journal of Logic and Computation 3(4), 1994.
• Paramodulation-Based Theorem Proving, Robert Nieuwenhuis and Alberto Rubio, Handbook of Automated Reasoning I(7), Elsevier Science and MIT Press, 2001.