Translations Between Gentzen–Prawitz and Jaśkowski–Fitch Natural Deduction Proofs

Standefer, Shawn

doi:10.1007/s11225-018-9828-2

Cited by 4 publications

(4 citation statements)

References 37 publications

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“…This gives a block structure to texts, as is well-known from programming languages such as Algol. For an overview of how this notation is used in systems of natural deduction, see [11].…”

Section: Context Of Proof Of Part (A)mentioning

confidence: 99%

Characteristics of de Bruijn's early proof checker Automath

Geuvers,

Nederpelt

2022

Preprint

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The 'mathematical language' Automath, conceived by N.G. de Bruijn in 1968, was the first theorem prover actually working and was used for checking many specimina of mathematical content. Its goals and syntactic ideas inspired Th. Coquand and G. Huet to develop the calculus of constructions, CC ([1]), which was one of the first widely used interactive theorem provers and forms the basis for the widely used Coq system ([2]).The original syntax of Automath ([3, 4]) is not easy to grasp. Yet, it is essentially based on a derivation system that is similar to the Calculus of Constructions ('CC'). The relation between the Automath syntax and CC has not yet been sufficiently described, although there are many references in the type theory community to Automath.In this paper we focus on the backgrounds and on some uncommon aspects of the syntax of Automath. We expose the fundamental aspects of a 'generic' Automath system, encapsulating the most common versions of Automath. We present this generic Automath system in a modern syntactic frame. The obtained system makes use of λD, a direct extension of CC with definitions described in [5].

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“…This gives a block structure to texts, as is well-known from programming languages such as Algol. For an overview of how this notation is used in systems of natural deduction, see [11].…”

Section: Context Of Proof Of Part (A)mentioning

confidence: 99%

Characteristics of de Bruijn's early proof checker Automath

Geuvers,

Nederpelt

2022

Preprint

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“…There are two ways one can proceed for giving a tree natural deduction system for E. The first is to use one of the systems discussed by Standefer & Brady (2019) as the basis for a tree system. The second is to use the tree system of Standefer (2018). These offer different views of natural deduction for E.…”

Section: A∪bmentioning

confidence: 99%

“…The tree-to-cascade translation is substantially different. The cascade-to-tree translation of Standefer ( 2019) is roughly the same as the one used here, but the tree-to-cascade translation used here is substantially different from that of Standefer (2019). The present translation is an improvement in not requiring the shuffling around and subsequent identification of assumptions, instead using the extra information contained in the indices to identify assumptions at the outset.…”

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confidence: 99%

Translations Between Linear and Tree Natural Deduction Systems for Relevant Logics

Standefer

2021

The Review of Symbolic Logic

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Anderson and Belnap presented indexed Fitch-style natural deduction systems for the relevant logics R, E, and T. This work was extended by Brady to cover a range of relevant logics. In this paper I present indexed tree natural deduction systems for the Anderson–Belnap–Brady systems and show how to translate proofs in one format into proofs in the other, which establishes the adequacy of the tree systems.

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“…Jaśkowski e Gentzen conduziram pesquisas paralelas e independentes em lógica, coincidentemente, publicaram em 1934 duas abordagens para a dedução natural. Apesar da não interação entre as pesquisas, os dois métodos de representação de provas em dedução natural resultantes das duas abordagens são equivalentes para a LPI e a LPC (22) e possuem traduções entre si (23). Cada representação possui vantagens e desvantagens em relação a outra, no entanto, a expressividade de ambas as representações é a mesma, sendo melhores denominadas como estilos de representação de provas em dedução natural.…”

Section: Estilos De Provas Em Dedução Naturalunclassified

Uma Abordagem Experimental Sobre a Compressão De Provas Em Dedução Natural Minimal Implicacional

BARROS¹

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AgradecimentosÀ minha família, que me apoiou desde o início na decisão de cursar esse mestrado. Toda a ajuda de vocês foi determinante no sucesso dessa jornada, nunca vou poder retribuir o que fizeram por mim nesse momento da minha vida.Ao meu orientador, professor Edward Hermann Haeusler, pela disponibilidade, sugestões, comentários, atenção e compreensão durante todo o mestrado. Por todas as conversas descontraídas, que sempre ajudaram a tranquilizar nos momentos difíceis. Espero que nossos times de futebol tenham mais sucesso no futuro! Aos membros da bancas das defesas da proposta de dissertação e da dissertação, Bruno Lopes e Luiz Carlos Pereira. Obrigado por todos os comentários, dicas e sugestões que elevaram a qualidade do trabalho.Aos meus amigos, por todas as conversas e conselhos que sempre renovaram minhas forças nos bons e nos maus momentos.Aos meus colegas do TecMF que contribuíram direta ou indiretamente no trabalho.Ao CNPq pelo apoio financeiro à essa pesquisa.The size of formal proofs has some important theoretical implications in computational complexity theory. The problem of determining if some formula of Intuitionistic Propositional Logic and the purely implicational fragment of Minimal Logic (M⊃) is a tautology is PSPACE-Complete. Any propositional logic with a natural deduction system that satisfies the subformula principle is PSPACE. To know if any tautology in M⊃ admits polynomially sized proof is related to whether NP = PSPACE. Proof compression techniques reported in literature use two main approaches to proof compressing: generating already compressed proofs; compressing an already generated proof. Proposed by Gordeev and Haeusler (6), the Horizontal Compression is a natural deduction proof compression technique that uses directed graphs to represent proofs. Given a tautology proof in M⊃, which may have an exponential size in relation to conclusion length, the goal of Horizontal Compression is that the compressed proof has a polynomially limited size in relation to conclusion length. Our work presents the first implementation of Horizontal Compression, together with the first results of the execution of the technique on proofs of M⊃ tautologies.

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Translations Between Gentzen–Prawitz and Jaśkowski–Fitch Natural Deduction Proofs

Cited by 4 publications

References 37 publications

Characteristics of de Bruijn's early proof checker Automath

Characteristics of de Bruijn's early proof checker Automath

Translations Between Linear and Tree Natural Deduction Systems for Relevant Logics

Uma Abordagem Experimental Sobre a Compressão De Provas Em Dedução Natural Minimal Implicacional

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