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, every unsatisfiable clause set can eventually be proved to be unsatisfiable.

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

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.