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Focusing and Proof-Nets in Linear and Non-commutative Logic

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Part of the Lecture Notes in Computer Science book series (LNAI,volume 1705)

Abstract

Linear Logic [4] has raised a lot of interest in computer research, especially because of its resource sensitive nature. One line of research studies proof construction procedures and their interpretation as computational models, in the “Logic Programming” tradition. An efficient proof search procedure, based on a proof normalization result called “Focusing”, has been described in [2]. Focusing is described in terms of the sequent system of commutative Linear Logic, which it refines in two steps. It is shown here that Focusing can also be interpreted in the proof-net formalism, where it appears, at least in the multiplicative fragment, to be a simple refinement of the “Splitting lemma” for proof-nets. This change of perspective allows to generalize the Focusing result to (the multiplicative fragment of) any logic where the “Splitting lemma” holds. This is, in particular, the case of the Non-Commutative logic of [1], and all the computational exploitation of Focusing which has been performed in the commutative case can thus be revised and adapted to the non commutative case.

Keywords

  • Atomic Formula
  • Sequent System
  • Linear Logic
  • Commutative Case
  • Proof Search

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This work was performed while the second author was visiting XRCE; this visit was supported by the European TMR (Training and Mobility for Researchers) Network “Linear Logic in Computer Science” (esp. the Rome and Marseille sites, XRCE being attached to the latter).

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© 1999 Springer-Verlag Berlin Heidelberg

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Andreoli, JM., Maieli, R. (1999). Focusing and Proof-Nets in Linear and Non-commutative Logic. In: Ganzinger, H., McAllester, D., Voronkov, A. (eds) Logic for Programming and Automated Reasoning. LPAR 1999. Lecture Notes in Computer Science(), vol 1705. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48242-3_20

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  • DOI: https://doi.org/10.1007/3-540-48242-3_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66492-5

  • Online ISBN: 978-3-540-48242-0

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