Efficiency and Completeness of the Set of Support Strategy in Theorem Proving

Wos, Larry; Robinson, George; Carson, Daniel F.

doi:10.1007/978-3-642-81952-0_29

Cited by 29 publications

(40 citation statements)

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“…It is originally based on the set-of-support strategy for the resolution [1]. Depending on which clauses are simplified, there exists (at least) two variants of this algorithm, the Otter and the Discount loops, named after the prover in which they appeared.…”

Section: Implementation Issuesmentioning

confidence: 99%

“…When proving a goal in a consistent theory by refutation, resolving the clauses of the theory is useless, since it will not bring out a contradiction. This idea defines the set-of-support strategy for resolution [1], where clauses generated by resolution must have at least one parent outside the theory. The completeness of this method can be proved provided the theory is consistent.…”

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confidence: 99%

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Embedding Deduction Modulo into a Prover

Burel¹

2010

Computer Science Logic

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Abstract. Deduction modulo consists in presenting a theory through rewrite rules to support automatic and interactive proof search. It induces proof search methods based on narrowing, such as the polarized resolution modulo. We show how to combine this method with more traditional ordering restrictions. Interestingly, no compatibility between the rewriting and the ordering is requested to ensure completeness. We also show that some simplification rules, such as strict subsumption eliminations and demodulations, preserve completeness. For this purpose, we use a new framework based on a proof ordering. These results show that polarized resolution modulo can be integrated into existing provers, where these restrictions and simplifications are present. We also discuss how this integration can actually be done by diverting the main algorithm of state-of-the-art provers.Whatever their applications, proofs are rarely searched for without context: mathematical proofs rely on set theory, or Euclidean geometry, or arithmetic, etc.; proofs of program correctness are done using e.g. pointer arithmetic and/or theories defining data structures (chained lists, trees, . . . ); concerning security, theories are used for instance to model properties of encryption algorithms. It is therefore essential to have theoretical foundations and practical methods that handle theories conveniently and efficiently. For this purpose, there are two directions: to develop methods that are really specific to a particular theory; or to develop a generic framework that can handle all theories. The first option is appealing for efficiency reasons: for instance, combining a SAT solver with the Simplex method leads to very powerful SMT solvers for linear arithmetic. However, developing methods for new theories is hard. Even the combination of such specific methods is not trivial, although there have been a lot of interesting results in that direction in the recent years. In this paper, we are more interested in the second option: having a generic way to handle theories efficiently. A naive way to do so would be to use an axiomatization of the theory, but in general, this approach would be really inefficient for automated proving. Somehow, we need to present the theory so as to take advantage of its properties.A first idea is to use the consistency of the theory. When proving a goal in a consistent theory by refutation, resolving the clauses of the theory is useless, since it will not bring out a contradiction. This idea defines the set-of-support

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Section: Implementation Issuesmentioning

confidence: 99%

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confidence: 99%

Embedding Deduction Modulo into a Prover

Burel¹

2010

Computer Science Logic

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“…Finally, we note that from the Theorem (and its manner of proof) the completeness of the set of support strategy of Was, Robinson, Carson [1965] is obtained. A refutation is a refutation of S with set of support T^S if every clause of the refutation of S which is a resolvent has at least one parent clause either a resolvent itself or a member of T.…”

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confidence: 90%

“…Papers of this type include Robinson [1965b], Wos, Robinson, Carson [1965], and Andrews [1968]. In this paper also a restricted format for resolution is shown to be a complete strategy.…”

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confidence: 96%

A Linear Format for Resolution

Loveland¹

1983

Automation of Reasoning

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“…Scott [17,19], Clin-S [4], Mgtp [6], Pttp+GLiDeS [1,2], and Scg [8], use semantics to guide (or restrict) the proof search. Those techniques are based on the idea that a resolution step between two clauses from the assumption part is not likely to contribute to the generation of the empty clause [21], a clause evaluated to true in a guiding model (or model set) is not likely to lead to an empty clause [17,19,1,2], or similarly a clause evaluated to true in some smaller subset of the guiding model set is more likely to generate an empty clause [8]. Typically, models are either generated incrementally during the proof search or supplied by the user.…”

Section: Introductionmentioning

confidence: 99%

Towards Semantic Goal-Directed Forward Reasoning in Resolution

Choi

2002

Artificial Intelligence: Methodology, Systems, and Applications

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Abstract. Semantic goal-directed forward reasoning is a three stage procedure. In the first stage a reference set of models is generated from the negated theorem. In the second stage the assumption clause set is refined to a set which has an as small set of models as possible in common with the negated theorem with respect to the reference set of models. In the last stage a refutation is generated in the space consisting of the original problem along with the refined assumption. In order to form a refined assumption, unlike traditional approaches like set of support, only clauses from assumptions are resolved with each other.

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Efficiency and Completeness of the Set of Support Strategy in Theorem Proving

Cited by 29 publications

References 1 publication

Embedding Deduction Modulo into a Prover

Embedding Deduction Modulo into a Prover

A Linear Format for Resolution

Towards Semantic Goal-Directed Forward Reasoning in Resolution

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