The superposition calculus is a

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

for

reasoning Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, lang ...

in equational

first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

. It was developed in the early 1990s and combines concepts from

first-order resolution In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation complete theorem-proving technique for sentences in propositional logic and first-order logic. For propositional logic, systematically ...

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 In language, a clause is a constituent that comprises a semantic predicand (expressed or not) and a semantic predicate. A typical clause consists of a subject and a syntactic predicate, the latter typically a verb phrase composed of a verb with ...

, i.e. it performs proofs by

refutation In argumentation, an objection is a reason arguing against a premise, argument, or conclusion. Definitions of objection vary in whether an objection is always an argument (or counterargument) or may include other moves such as questioning. An ...

. Superposition is refutation-complete—given unlimited resources and a ''fair'' derivation strategy, from any

unsatisfiable In mathematical logic, a formula is ''satisfiable'' if it is true under some assignment of values to its variables. For example, the formula x+3=y is satisfiable because it is true when x=3 and y=6, while the formula x+1=x is not satisfiable over ...

clause set a contradiction will eventually be derived. , 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 SPASS is an automated theorem prover for first-order logic with equality developed at the Max Planck Institute for Computer Science and using the superposition calculus. The name originally stood for ''Synergetic Prover Augmenting Superposition wi ...

Vampire A vampire is a mythical creature that subsists by feeding on the Vitalism, vital essence (generally in the form of blood) of the living. In European folklore, vampires are undead, undead creatures that often visited loved ones and caused mi ...

* Waldmeisterbr>(official web page)

References

*
Rewrite-Based Equational Theorem Proving with Selection and Simplification
', Leo Bachmair and

Harald Ganzinger Harald Ganzinger (31 October 1950, Werneck – 3 June 2004, Saarbrücken) was a German computer scientist who together with Leo Bachmair developed the superposition calculus, which is (as of 2007) used in most of the state-of-the-art automated ...

, Journal of Logic and Computation 3(4), 1994. * ''Paramodulation-Based Theorem Proving'', Robert Nieuwenhuis and Alberto Rubio,

Handbook of Automated Reasoning The ''Handbook of Automated Reasoning'' (, 2128 pages) is a collection of survey articles on the field of automated reasoning. Published in June 2001 by MIT Press, it is edited by John Alan Robinson and Andrei Voronkov. Volume 1 describes methods ...

I(7),

Elsevier Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', th ...

Science and

MIT Press The MIT Press is a university press affiliated with the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts (United States). It was established in 1962. History The MIT Press traces its origins back to 1926 when MIT publish ...

, 2001. Mathematical logic Logical calculi {{mathlogic-stub