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| The '''superposition calculus''' is a calculus for ] in equational ]. It has been developed in the early 1990s and combines concepts from ] with ordering-based equality handling as developed in the context of (unfailing) ]. 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 ], i.e. it performs proofs by ]. Superposition is refutation-complete — given unlimited resources and a ''fair'' derivation strategy, every unsatisfiable (logic)|unsatisfiable clause set can eventually be proved to be unsatisfiable. | The '''superposition calculus''' is a calculus for ] in equational ]. It has been developed in the early 1990s and combines concepts from ] with ordering-based equality handling as developed in the context of (unfailing) ]. 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 ], i.e. it performs proofs by ]. Superposition is refutation-complete — given unlimited resources and a ''fair'' derivation strategy, every ] clause set can eventually be proved to be unsatisfiable. | ||
| As of 2007, most of the (state-of-the-art) ]s for ] are based on superposition (e.g. the ]), although only a few implement the pure calculus. | As of 2007, most of the (state-of-the-art) ]s for ] are based on superposition (e.g. the ]), although only a few implement the pure calculus. | ||
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| * ] | * ] | ||
| * ] | * ] | ||
| * Waldmeister theorem prover|Waldmeister | * ] | ||
| * Ayane theorem prover|Ayane at | * ] at | ||
| == References == | == References == | ||
Revision as of 17:25, 3 February 2013
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
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.
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