A Resolution Calculus for First-order Schemata

Aravantinos, Vincent; Echenim, Mnacho; Peltier, Nicolas

doi:10.3233/fi-2013-855

Cited by 17 publications

(11 citation statements)

References 18 publications

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“…For example in Fig. 1 we provide the output of Peltier's superposition induction prover [3] when ran on the clausified form of the cut structure of the successor eventually constant schema. The existence of this derivation implies that the proof analysis method of [14] may be applied to the successor eventually constant schema.…”

Section: A Motivational Examplementioning

confidence: 99%

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Schematic Refutations of Formula Schemata

2020

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Proof schemata are infinite sequences of proofs which are defined inductively. In this paper we present a general framework for schemata of terms, formulas and unifiers and define a resolution calculus for schemata of quantifier-free formulas. The new calculus generalizes and improves former approaches to schematic deduction. As an application of the method we present a schematic refutation formalizing a proof of a weak form of the pigeon hole principle.

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Section: A Motivational Examplementioning

confidence: 99%

“…The refutational completeness of RPL Ψ 0 is not an issue as already RPL 0 is refutationally complete for PL 0 formulas [3,14]. Note that this is not the case any more if parameters occur in formulas.…”

Section: Definition 21 (Essentially Disjoint) Letmentioning

confidence: 99%

Schematic Refutations of Formula Schemata

2020

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“…Concerning the schematic theorem prover used in [5], it represents another line of research, involving schematic representations [8]- [10]. In [10], inductive theorem proving over schematic clause sets is considered, however, schematic unification, as we discuss it, was not.…”

Section: Introductionmentioning

confidence: 99%

A Special Case of Schematic Syntactic Unification

Cerna¹

2022

Preprint

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We present a unification problem based on firstorder syntactic unification which ask whether every problem in a schematically-defined sequence of unification problems is unifiable, so called loop unification. Alternatively, our problem may be formulated as a recursive procedure calling first-order syntactic unification on certain bindings occurring in the solved form resulting from unification. Loop unification is closely related to Narrowing as the schematic constructions can be seen as a rewrite rule applied during unification, and primal grammars, as we deal with recursive term constructions. However, loop unification relaxes the restrictions put on variables as fresh as well as used extra variables may be introduced by rewriting. In this work we consider an important special case, so called semiloop unification. We provide a sufficient condition for unifiability of the entire sequence based on the structure of a sufficiently long initial segment. It remains an open question whether this condition is also necessary for semiloop unification and how it may be extended to loop unification.

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“…An extension to a special case of multiple parameters was also investigated by D. Cerna [13]. In a more recent work, V. Aravantinos et al [6] introduced a superposition resolution calculus for a clausal representation of indexed propositional logic. The calculus provided decidability results for an even larger fragment of the monadic fragment.…”

Section: Introductionmentioning

confidence: 99%

“…The schematic construction of objects that form the basis of proof schemata as described in this paper was introduced by V. Aravantinos et al [2,3,4,5,6,7]. Initially, they considered formulas of an indexed propositional logic with a single free numeric parameter and with two new logical connectors, i.e.…”

Section: Introductionmentioning

confidence: 99%

Integrating a Global Induction Mechanism into a Sequent Calculus

Cerna

Lettmann

2017

Lecture Notes in Computer Science

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Abstract. Most interesting proofs in mathematics contain an inductive argument which requires an extension of the LK-calculus to formalize. The most commonly used calculi for induction contain a separate rule or axiom which reduces the valid proof theoretic properties of the calculus. To the best of our knowledge, there are no such calculi which allow cut-elimination to a normal form with the subformula property, i.e. every formula occurring in the proof is a subformula of the end sequent. Proof schemata are a variant of LK-proofs able to simulate induction by linking proofs together. There exists a schematic normal form which has comparable proof theoretic behaviour to normal forms with the subformula property. However, a calculus for the construction of proof schemata does not exist. In this paper, we introduce a calculus for proof schemata and prove soundness and completeness with respect to a fragment of the inductive arguments formalizable in Peano arithmetic.

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A Resolution Calculus for First-order Schemata

Cited by 17 publications

References 18 publications

Schematic Refutations of Formula Schemata

Schematic Refutations of Formula Schemata

A Special Case of Schematic Syntactic Unification

Integrating a Global Induction Mechanism into a Sequent Calculus

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