Article

From proofs to focused proofs: a modular proof of focalization in linear logic

Authors:

Alexis SaurinAuthors Info & Claims

CSL'07/EACSL'07: Proceedings of the 21st international conference, and Proceedings of the 16th annuall conference on Computer Science Logic

Pages 405 - 419

Published: 11 September 2007 Publication History

Abstract

Probably the most significant result concerning cut-free sequent calculus proofs in linear logic is the completeness of focused proofs. This completeness theorem has a number of proof theoretic applications -- e.g. in game semantics, Ludics, and proof search -- and more computer science applications -- e.g. logic programming, call-by-name/value evaluation. Andreoli proved this theorem for first-order linear logic 15 years ago. In the present paper, we give a new proof of the completeness of focused proofs in terms of proof transformation. The proof of this theorem is simple and modular: it is first proved for MALL and then is extended to full linear logic. Given its modular structure, we show how the proof can be extended to larger systems, such as logics with induction. Our analysis of focused proofs will employ a proof transformation method that leads us to study how focusing and cut elimination interact. A key component of our proof is the construction of a focalization graph which provides an abstraction over how focusing can be organized within a given cut-free proof. Using this graph abstraction allows us to provide a detailed study of atomic bias assignment in a way more refined that is given in Andreoli's original proof. Permitting more flexible assignment of bias will allow this completeness theorem to help establish the completeness of a number of other automated deduction procedures. Focalization graphs can be used to justify the introduction of an inference rule for multifocus derivation: a rule that should help us better understand the relations between sequentiality and concurrency in linear logic.

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Cited By

Chihani ZMiller DRenaud F(2017)A Semantic Framework for Proof EvidenceJournal of Automated Reasoning10.1007/s10817-016-9380-659:3(287-330)Online publication date: 1-Oct-2017
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Olarte CChiarugi DFalaschi MHermith D(2016)A proof theoretic view of spatial and temporal dependencies in biochemical systemsTheoretical Computer Science10.1016/j.tcs.2016.03.029641:C(25-42)Online publication date: 16-Aug-2016
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Scherer GRémy D(2015)Which simple types have a unique inhabitant?ACM SIGPLAN Notices10.1145/2858949.278475750:9(243-255)Online publication date: 29-Aug-2015
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cover image Guide Proceedings

CSL'07/EACSL'07: Proceedings of the 21st international conference, and Proceedings of the 16th annuall conference on Computer Science Logic

September 2007

599 pages

ISBN:3540749144

Editors:
Jacques Duparc
Université de Lausanne, Ecole des HEC, Lausanne, Switzerland
,
Thomas A. Henzinger
EPFL, Lausanne, Switzerland

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Logitech S.A.: Logitech S.A.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 11 September 2007

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Cited By

Chihani ZMiller DRenaud F(2017)A Semantic Framework for Proof EvidenceJournal of Automated Reasoning10.1007/s10817-016-9380-659:3(287-330)Online publication date: 1-Oct-2017
https://dl.acm.org/doi/10.1007/s10817-016-9380-6
Olarte CChiarugi DFalaschi MHermith D(2016)A proof theoretic view of spatial and temporal dependencies in biochemical systemsTheoretical Computer Science10.1016/j.tcs.2016.03.029641:C(25-42)Online publication date: 16-Aug-2016
https://dl.acm.org/doi/10.1016/j.tcs.2016.03.029
Scherer GRémy D(2015)Which simple types have a unique inhabitant?ACM SIGPLAN Notices10.1145/2858949.278475750:9(243-255)Online publication date: 29-Aug-2015
https://dl.acm.org/doi/10.1145/2858949.2784757
Scherer GRémy DFisher KReppy J(2015)Which simple types have a unique inhabitant?Proceedings of the 20th ACM SIGPLAN International Conference on Functional Programming10.1145/2784731.2784757(243-255)Online publication date: 29-Aug-2015
https://dl.acm.org/doi/10.1145/2784731.2784757
Baelde D(2012)Least and Greatest Fixed Points in Linear LogicACM Transactions on Computational Logic10.1145/2071368.207137013:1(1-44)Online publication date: 1-Jan-2012
https://dl.acm.org/doi/10.1145/2071368.2071370
Chaudhuri K(2012)Compact proof certificates for linear logicProceedings of the Second international conference on Certified Programs and Proofs10.1007/978-3-642-35308-6_17(208-223)Online publication date: 13-Dec-2012
https://dl.acm.org/doi/10.1007/978-3-642-35308-6_17
Basaldella MSaurin ATerui K(2011)On the meaning of focalizationLudics, dialogue and interaction10.5555/1980762.1980767(78-87)Online publication date: 1-Jan-2011
https://dl.acm.org/doi/10.5555/1980762.1980767
Chaudhuri K(2010)Magically constraining the inverse method using dynamic polarity assignmentProceedings of the 17th international conference on Logic for programming, artificial intelligence, and reasoning10.5555/1928380.1928395(202-216)Online publication date: 10-Oct-2010
https://dl.acm.org/doi/10.5555/1928380.1928395
Nigam VMiller DPorto ALópez-Fraguas F(2009)Algorithmic specifications in linear logic with subexponentialsProceedings of the 11th ACM SIGPLAN conference on Principles and practice of declarative programming10.1145/1599410.1599427(129-140)Online publication date: 7-Sep-2009
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Liang CMiller D(2009)Focusing and polarization in linear, intuitionistic, and classical logicsTheoretical Computer Science10.1016/j.tcs.2009.07.041410:46(4747-4768)Online publication date: 1-Nov-2009
https://dl.acm.org/doi/10.1016/j.tcs.2009.07.041
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