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Least and Greatest Fixed Points in Linear Logic

Least and Greatest Fixed Points in Linear Logic Least and Greatest Fixed Points in Linear Logic DAVID BAELDE, University of Minnesota The rst-order theory of MALL (multiplicative, additive linear logic) over only equalities is a well-structured but weak logic since it cannot capture unbounded (in nite) behavior. Instead of accounting for unbounded behavior via the addition of the exponentials (! and ?), we add least and greatest xed point operators. The resulting logic, which we call MALL, satis es two fundamental proof theoretic properties: we establish weak normalization for it, and we design a focused proof system that we prove complete with respect to the initial system. That second result provides a strong normal form for cut-free proof structures that can be used, for example, to help automate proof search. We show how these foundations can be applied to intuitionistic logic. Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic ”Proof theory; F.3.1 [Logics and Meanings of Programs]: Specifying and Verifying and Reasoning about Programs ”Speci cation techniques; F.3.3 [Logics and Meanings of Programs]: Studies of Program Constructs ”Program and recursion schemes General Terms: Design, Theory, Veri cation Additional Key Words and Phrases: Fixed points, linear logic, (co)induction, recursive de nitions, cut elimination, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Computational Logic (TOCL) Association for Computing Machinery

Least and Greatest Fixed Points in Linear Logic

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References (59)

Publisher
Association for Computing Machinery
Copyright
Copyright © 2012 by ACM Inc.
ISSN
1529-3785
DOI
10.1145/2071368.2071370
Publisher site
See Article on Publisher Site

Abstract

Least and Greatest Fixed Points in Linear Logic DAVID BAELDE, University of Minnesota The rst-order theory of MALL (multiplicative, additive linear logic) over only equalities is a well-structured but weak logic since it cannot capture unbounded (in nite) behavior. Instead of accounting for unbounded behavior via the addition of the exponentials (! and ?), we add least and greatest xed point operators. The resulting logic, which we call MALL, satis es two fundamental proof theoretic properties: we establish weak normalization for it, and we design a focused proof system that we prove complete with respect to the initial system. That second result provides a strong normal form for cut-free proof structures that can be used, for example, to help automate proof search. We show how these foundations can be applied to intuitionistic logic. Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic ”Proof theory; F.3.1 [Logics and Meanings of Programs]: Specifying and Verifying and Reasoning about Programs ”Speci cation techniques; F.3.3 [Logics and Meanings of Programs]: Studies of Program Constructs ”Program and recursion schemes General Terms: Design, Theory, Veri cation Additional Key Words and Phrases: Fixed points, linear logic, (co)induction, recursive de nitions, cut elimination,

Journal

ACM Transactions on Computational Logic (TOCL)Association for Computing Machinery

Published: Jan 1, 2012

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