A Coinductive Approach to Proof Search through Typed Lambda-Calculi

Autor: Luís Pinto, José Espírito Santo, Ralph Matthes
Přispěvatelé: University of Minho [Braga], Assistance à la Certification d’Applications DIstribuées et Embarquées (IRIT-ACADIE), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Fundação para a Ciência e a Tecnologia UIDB/00013/2020 and UIDP/00013/2020, ANR-11-BS02-0016,CLIMT,Méthodes catégoriques et logiques en transformations de modèles(2011), European Project: COST Action CA15123 ,COST - European Cooperation in Science and Technology,EUTYPES(2016), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Universidade do Minho
Jazyk: angličtina
Rok vydání: 2016
Předmět:
Computer Science - Logic in Computer Science
Logic
coinduction / corecursion
Proof search
0102 computer and information sciences
Fixed point
Mathematical proof
01 natural sciences
Curry–Howard correspondence
[INFO.INFO-FL]Computer Science [cs]/Formal Languages and Automata Theory [cs.FL]
Solution spaces
Computer Science::Logic in Computer Science
Finitary
Rule of inference
Contraction (operator theory)
Mathematics
[MATH.MATH-CT]Mathematics [math]/Category Theory [math.CT]
Science & Technology
inhabitation problem
[INFO.INFO-PL]Computer Science [cs]/Programming Languages [cs.PL]
Coinduction
Sequent calculus
[INFO.INFO-LO]Computer Science [cs]/Logic in Computer Science [cs.LO]
Ciências Naturais::Ciências da Computação e da Informação
Mathematics - Logic
16. Peace & justice
proof search
solution space
Algebra
[MATH.MATH-LO]Mathematics [math]/Logic [math.LO]
TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES
010201 computation theory & mathematics
TheoryofComputation_LOGICSANDMEANINGSOFPROGRAMS
Computer Science::Programming Languages
F.4.1
Ciências da Computação e da Informação [Ciências Naturais]
Curry-Howard isomorphism
Coinductive methods
Zdroj: Annals of Pure and Applied Logic
Annals of Pure and Applied Logic, Elsevier Masson, 2021, 172 (10), ⟨10.1016/j.apal.2021.103026⟩
Annals of Pure and Applied Logic, 2021, 172 (10), ⟨10.1016/j.apal.2021.103026⟩
Repositório Científico de Acesso Aberto de Portugal
Repositório Científico de Acesso Aberto de Portugal (RCAAP)
instacron:RCAAP
ISSN: 0168-0072
DOI: 10.1016/j.apal.2021.103026⟩
Popis: In reductive proof search, proofs are naturally generalized by solutions, comprising all (possibly infinite) structures generated by locally correct, bottom-up application of inference rules. We propose a rather natural extension of the Curry-Howard paradigm of representation, from proofs to solutions: to represent solutions by (possibly infinite) terms of the coinductive variant of the typed lambda-calculus that represents proofs. We take this as a starting point for a new, comprehensive approach to proof search; our case study is proof search in the sequent calculus LIT for intuitionistic implication logic. A second, finitary representation is proposed, comprising a syntax of lambda-terms extended with a formal greatest fixed point, and a type system that can be seen as a logic of coinductive proofs. In the finitary system, fixed-point variables enjoy a relaxed form of binding that allows the detection of cycles through the type system. Formal sums are used in both representations to express alternatives in the search process, so that not only individual solutions but actually solution spaces are expressed. Moreover, formal sums are used in the coinductive syntax to define "decontraction" (contraction bottom-up)-an operation whose theory we initiate in this paper. A semantics is defined assigning a coinductive lambda-term to each finitary term, making use of decontraction as a semantical match to the relaxed form of binding of fixed-point variables present in the finitary system. The main result is the existence of an equivalent finitary representation for any full solution space expressed coinductively. This result is the main ingredient in the proof that our logic of coinductive proofs is sound and complete with respect to the coinductive semantics. These results are the foundation for an original approach to proof search, where the search builds the finitary representation of the full solution space, and the a posteriori analysis typically consisting in applying a syntax-directed procedure or function. The paper illustrates the potential of the methodology to the study of proof search and inhabitation problems in the simply-typed lambda-calculus, reviewing results detailed elsewhere, and including new results that obtain extensive generalizations of the so-called monatomic theorem.
Jose Espirito Santo and Luis Pinto are both funded by Portuguese Funds through FCT -Fundacao para a Ciencia e a Tecnologia, within the Projects UIDB/00013/2020 and UIDP/00013/2020. Ralph Matthes had been funded by the Climtproject (ANR-11-BS02-016 of the French Agence Nationale de la Recherche). All authors had been partially funded by COST action CA15123 EUTYPES.
Databáze: OpenAIRE
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