The **superposition calculus** is a calculus for reasoning in equational first-order logic. It was 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 (official web page)

## 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.