A Unified Sequent Calculus for Focused Proofs

Communicated by P.J. Scott MSC: 03F52 03B47 03B20 Keywords: Focused proof systems Unity of logic Linear logic a b s t r a c tWe present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicative-additive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structural rules to polarity information. We identify a general set of criteria under which cut-elimination holds in such fragments. From cut-elimination we derive a unified proof of the completeness of focusing. Furthermore, each sublogic can interact with other fragments through cut. We examine certain circumstances, for example, in which a classical lemma can be used in an intuitionistic proof while preserving intuitionistic provability. We also examine the possibility of defining classical-linear hybrid logics.

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“…Finally, in Section 10 we discuss some future work and we briefly conclude in Section 11. This paper is an extended version of [17].…”

Section: Introductionmentioning

confidence: 99%

A focused approach to combining logics

Liang

Miller

2011

Annals of Pure and Applied Logic

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“…Recent analysis shows that the duality between patterns and terms reflects the duality between phases in focused proof systems [15]. Finally we demonstrated [7,8] that superdeduction systems share strong similarities with focused proof systems such as LKF [9,10], a focused sequent calculus for classical logic. Answers should naturally arise from the study of the computational content of such focused systems [11,5].…”

Section: Extending λ µ μmentioning

confidence: 62%

Superdeduction in Lambda-Bar-Mu-Mu-Tilde

Houtmann

2011

Electron. Proc. Theor. Comput. Sci.

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Superdeduction is a method specially designed to ease the use of first-order theories in predicate logic. The theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way. A proof-term language and a cut-elimination reduction already exist for superdeduction, both based on Christian Urban's work on classical sequent calculus. However the computational content of Christian Urban's calculus is not directly related to the (lambda-calculus based) Curry-Howard correspondence. In contrast the Lambda bar mu mu tilde calculus is a lambda-calculus for classical sequent calculus. This short paper is a first step towards a further exploration of the computational content of superdeduction proofs, for we extend the Lambda bar mu mu tilde calculus in order to obtain a proofterm langage together with a cut-elimination reduction for superdeduction. We also prove strong normalisation for this extension of the Lambda bar mu mu tilde calculus

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“…However, this encoding is only globally adequate [16]. It is possible to refine this encoding to obtain a fully adequate encoding [12] in an enriched classical linear logic which is not apparently an instance of classical subexponential logic. Corollary 18 further improves our undertanding of encodings of intuitionistic implicication by permuting !…”

Section: Theorem 12mentioning

confidence: 99%

Classical and Intuitionistic Subexponential Logics Are Equally Expressive

Chaudhuri

2010

Computer Science Logic

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Abstract. It is standard to regard the intuitionistic restriction of a classical logic as increasing the expressivity of the logic because the classical logic can be adequately represented in the intuitionistic logic by double-negation, while the other direction has no truth-preserving propositional encodings. We show here that subexponential logic, which is a family of substructural refinements of classical logic, each parametric over a preorder over the subexponential connectives, does not suffer from this asymmetry if the preorder is systematically modified as part of the encoding. Precisely, we show a bijection between synthetic (i.e., focused) partial sequent derivations modulo a given encoding. Particular instances of our encoding for particular subexponential preorders give rise to both known and novel adequacy theorems for substructural logics.

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A Unified Sequent Calculus for Focused Proofs

Cited by 6 publications

References 12 publications

A focused approach to combining logics

A focused approach to combining logics

Superdeduction in Lambda-Bar-Mu-Mu-Tilde

Classical and Intuitionistic Subexponential Logics Are Equally Expressive

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