Conference Paper

A Unified Sequent Calculus for Focused Proofs

DOI: 10.1109/LICS.2009.47 Conference: Proceedings of the 24th Annual IEEE Symposium on Logic in Computer Science, LICS 2009, 11-14 August 2009, Los Angeles, CA, USA
Source: DBLP

ABSTRACT

We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical logic, intuitionistic logic, and multiplicative-additive linear logic are derived as fragments of LKU by increasing the sensitivity of specialized structural rules to polarity information. We develop a unified, streamlined framework for proving cut-elimination in the various fragments. Furthermore, each sublogic can interact with other fragments through cut. We also consider the possibility of introducing 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 [5] "
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    ABSTRACT: We 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.
    Full-text · Article · Sep 2011 · Annals of Pure and Applied Logic
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    ABSTRACT: Existing focused proof systems for classical and intuitionistic logic allow contraction for exactly those formulas chosen for focus. For proof-search applications, contraction is undesirable, as we risk traversing the same path multiple times. We present here a contraction-free focused sequent calculus for classical propositional logic, called LKF CF , which is a modification of the recently developed proof system LKF. We prove that our system is sound and complete with respect to LKF, and therefore it is also sound and complete with respect to propositional classical logic. LKF can be justified with a compilation into focused proofs for linear logic; in this work we show how to do a similar compilation for LKF CF , but into focused proofs for linear logic with subexponentials instead. We use two subexponentials, neither allowing contraction but one allowing weakening. We show how the focused proofs for linear logic can then simulate proofs in LKF CF . Returning to proof-search, we end this work with a small experimental study showing that a proof-search implementation based on LKF CF performs well compared to implementations based on leanT A P and several variants and optimizations on LK and LKF.
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    ABSTRACT: We introduce Polarized Intuitionistic Logic, which allows intuitionis-tic and classical logic to mix. The logic is based on a new analysis of the intuition-istic distinction between "left" and "right" as a form of polarity information. In contrast to double-negation translations, classical logic is transparently captured. The logic is given a Kripke-style semantics and is presented as a sequent calcu-lus that admits cut elimination. We discuss the impact of this logic on traditional intuitionistic concepts such as Glivenko's theorem and Markov's principle.
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