A new algorithm for derivability in the constructive propositional calculus

Vorob’ëv, N. N.

doi:10.1090/trans2/094/02

Cited by 25 publications

(27 citation statements)

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“…A presentation is given in Dyckhoff (1992) -the idea itself appeared as early as Vorob'ev (1958). The idea is that sums and (positive) products do not need to be deconstructed twice, and thus need not be contracted on the left.…”

Section: Counting Inhabitants Broda and Damasmentioning

confidence: 99%

Which simple types have a unique inhabitant?

Scherer

Rémy

2015

Proceedings of the 20th ACM SIGPLAN International Conference on Functional Programming

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We study the question of whether a given type has a unique inhabitant modulo program equivalence. In the setting of simplytyped lambda-calculus with sums, equipped with the strong βη-equivalence, we show that uniqueness is decidable. We present a saturating focused logic that introduces irreducible cuts on positive types "as soon as possible". Backward search in this logic gives an effective algorithm that returns either zero, one or two distinct inhabitants for any given type. Preliminary application studies show that such a feature can be useful in strongly-typed programs, inferring the code of highly-polymorphic library functions, or "glue code" inside more complex terms.

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Section: Counting Inhabitants Broda and Damasmentioning

confidence: 99%

Which simple types have a unique inhabitant?

Scherer

Rémy

2015

Proceedings of the 20th ACM SIGPLAN International Conference on Functional Programming

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“…As the reader can easily check inspecting the rules, LSJ is a contraction-free calculus and, differently from other well-known contraction-free proposals [4,5,11,14,20], it has the subformula property: every formula occurring in a derivation is a subformula of the root sequent.…”

Section: The Calculus Lsjmentioning

confidence: 99%

“…The main difference among LSJ and the terminating calculi in [4,5,12,13,20] is that LSJ meets the subformula property. Paper [3] presents a sequent calculus with the subformula property whose termination is based on the rule a-fortiori; differently from LSJ the depth of its proofs is not linearly bounded.…”

Section: Related Workmentioning

confidence: 99%

“…The major drawback of this solution is that deductions may have infinite depth, hence some loop-checking mechanism is needed to guarantee termination. Vorob'ev [20] introduced rules to treat left-implicative formulas according to the main connective of the antecedent. See also [4,11,14], where calculi with analogous properties are given.…”

Section: Introductionmentioning

confidence: 99%

“…We remark that, even if termination can be easily achieved also for the calculi in [4,5,11,14,20], the subformula property provides a more elegant termination argument. Moreover, our rules better capture the original goal of Gentzen [6] to justify connectives via introduction rules acting on their subformulas.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Contraction-Free Linear Depth Sequent Calculi for Intuitionistic Propositional Logic with the Subformula Property and Minimal Depth Counter-Models

2012

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In this paper we present LSJ, a contraction-free sequent calculus for Intuitionistic propositional logic whose proofs are linearly bounded in the length of the formula to be proved and satisfy the subformula property. We also introduce a sequent calculus RJ for intuitionistic unprovability with the same properties of LSJ. We show that from a refutation of RJ of a sequent σ we can extract a Kripke countermodel for σ . Finally, we provide a procedure that given a sequent σ returns either a proof of σ in LSJ or a refutation in RJ such that the extracted counter-model is of minimal depth.

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An intuitionistic formula hierarchy based on high‐school identities

Brock-Nannestad

Ilik

2019

Mathematical Logic Qtrly

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To the memory of Kosta Došen and Roy DyckhoffWe revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e., sequent) isomorphisms corresponding to the highschool identities, we show that one can obtain a more compact variant of a proof system, consisting of noninvertible proof rules only, and where the invertible proof rules have been replaced by a formula normalization procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non-invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first-order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism.

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A new algorithm for derivability in the constructive propositional calculus

Cited by 25 publications

References 0 publications

Which simple types have a unique inhabitant?

Which simple types have a unique inhabitant?

Contraction-Free Linear Depth Sequent Calculi for Intuitionistic Propositional Logic with the Subformula Property and Minimal Depth Counter-Models

An intuitionistic formula hierarchy based on high‐school identities

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