On Natural Deduction for Herbrand Constructive Logics I: Curry-Howard Correspondence for Dummett's Logic LC

Aschieri, Federico

doi:10.2168/lmcs-12(3:13)2016

Cited by 6 publications

(7 citation statements)

References 26 publications

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“…All approaches explored so far to provide a precise formalization of G as a logic for parallelism, either sacrificed analyticity [1] or tried to devise forms of natural deduction whose structures mirror hypersequents -which are sequents operating in parallel [4]. Hypersequents were indeed successfully used in [3] to define an analytic calculus for G and were intuitively connected to parallel computations: the key rule introduced by Avron to capture the linearity axiom -called communicationenables sequents to exchange their information and hence to "communicate".…”

Section: Gödel Logic Avron's Conjecture and Previous Attemptsmentioning

confidence: 99%

Gödel logic: From natural deduction to parallel computation

Aschieri¹,

Ciabattoni²,

Genco³

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Abstract-Propositional Gödel logic G extends intuitionistic logic with the non-constructive principle of linearity (A → B) ∨ (B → A). We introduce a Curry-Howard correspondence for G and show that a simple natural deduction calculus can be used as a typing system. The resulting functional language extends the simply typed λ-calculus via a synchronous communication mechanism between parallel processes, which increases its expressive power. The normalization proof employs original termination arguments and proof transformations implementing forms of code mobility. Our results provide a computational interpretation of G, thus proving A. Avron's 1991 thesis.

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Section: Gödel Logic Avron's Conjecture and Previous Attemptsmentioning

confidence: 99%

Gödel logic: From natural deduction to parallel computation

Aschieri¹,

Ciabattoni²,

Genco³

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…This is the third in a series of papers about natural deduction for Herbrand constructive logics, which we defined to be intermediate logics satisfying Herbrand's Theorem for every existential statement [1,2]. Indeed, intermediate logics prove intuitionistically as well as classically valid theorems, yet they often possess a strong constructive flavour.…”

Section: Introductionmentioning

confidence: 99%

On Natural Deduction for Herbrand Constructive Logics III: The Strange Case of the Intuitionistic Logic of Constant Domains

Aschieri

2018

Electron. Proc. Theor. Comput. Sci.

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The logic of constant domains is intuitionistic logic extended with the so-called forall-shift axiom, a classically valid statement which implies the excluded middle over decidable formulas. Surprisingly, this logic is constructive and so far this has been proved by cut-elimination for ad-hoc sequent calculi. Here we use the methods of natural deduction and Curry-Howard correspondence to provide a simple computational interpretation of the logic.

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“…This is very interesting by itself, as it shows that the categorical modelling really captures all the essential features of the interpretation. But it also opens new possibilities for modelling of constructive set theories (in the style of Nemoto and Rathjan [19]) and of categorical modelling of intermediate logics (intuitionistic propositional logic plus (IP) or (MK), see [1,6]). This leads into applications both into the investigation of functional abstract machines [18,22], of reverse mathematics [19] and of quantified modal logic [25].…”

Section: Discussionmentioning

confidence: 99%

Dialectica Logical Principles

Trotta

Spadetto

Paiva

2021

Logical Foundations of Computer Science

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Gödel' s Dialectica interpretation was designed to obtain a relative consistency proof for Heyting arithmetic, to be used in conjunction with the double negation interpretation to obtain the consistency of Peano arithmetic. In recent years, proof theoretic transformations (socalled proof interpretations) that are based on Gödel's Dialectica interpretation have been used systematically to extract new content from proofs and so the interpretation has found relevant applications in several areas of mathematics and computer science. Following our previous work on 'Gödel fibrations', we present a (hyper)doctrine characterisation of the Dialectica which corresponds exactly to the logical description of the interpretation. To show that we derive the soundness of the interpretation of the implication connective, as expounded on by Spector and Troelstra, in the categorical model. This requires extra logical principles, going beyond intuitionistic logic, namely Markov's Principle (MP) and the Independence of Premise (IP) principle, as well as some choice. We show how these principles are satisfied in the categorical setting, establishing a tight (internal language) correspondence between the logical system and the categorical framework. This tight correspondence should come handy not only when discussing the applications of the Dialectica already known, like its use to extract computational content from (some) classical theorems (proof mining), its use to help to model specific abstract machines, etc. but also to help devise new applications.

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On Natural Deduction for Herbrand Constructive Logics I: Curry-Howard Correspondence for Dummett's Logic LC

Cited by 6 publications

References 26 publications

Gödel logic: From natural deduction to parallel computation

Gödel logic: From natural deduction to parallel computation

On Natural Deduction for Herbrand Constructive Logics III: The Strange Case of the Intuitionistic Logic of Constant Domains

Dialectica Logical Principles

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