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73
Light Linear Logic
"... The abuse of structural rules may have damaging complexity effects. ..."
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Cited by 619 (3 self)
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The abuse of structural rules may have damaging complexity effects.
Concurrent Games and Full Completeness
, 1998
"... A new concurrent form of game semantics is introduced. This overcomes the problems which had arisen with previous, sequential forms of game semantics in modelling Linear Logic. It also admits an elegant and robust formalization. A Full Completeness Theorem for MultiplicativeAdditive Linear Logic is ..."
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Cited by 49 (16 self)
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A new concurrent form of game semantics is introduced. This overcomes the problems which had arisen with previous, sequential forms of game semantics in modelling Linear Logic. It also admits an elegant and robust formalization. A Full Completeness Theorem for MultiplicativeAdditive Linear Logic is proved for this semantics. 1 Introduction This paper contains two main contributions: ffl the introduction of a new form of game semantics, which we call concurrent games. ffl a proof of full completeness of this semantics for MultiplicativeAdditive Linear Logic. We explain the significance of each of these in turn. Concurrent games Traditional forms of game semantics which have appeared in logic and computer science have been sequential in format: a play of the game is formalized as a sequence of moves. The key feature of this sequential format is the existence of a global schedule (or polarization) : in each (finite) position, it is (exactly) one player's turn to move 1 . This seq...
Proofs nets for unitfree multiplicativeadditive linear logic
 18th IEEE Intl. Symp. Logic in Computer Science (LICS’03
, 2003
"... A cornerstone of the theory of proof nets for unitfree multiplicative linear logic (MLL) is the abstract representation of cutfree proofs modulo inessential commutations of rules. The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cutfr ..."
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Cited by 39 (4 self)
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A cornerstone of the theory of proof nets for unitfree multiplicative linear logic (MLL) is the abstract representation of cutfree proofs modulo inessential commutations of rules. The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cutfree monomial proof nets can correspond to the same cutfree proof. Thus the problem of finding a satisfactory notion of proof net for unitfree multiplicativeadditive linear logic (MALL) has remained open since the inception of linear logic in 1986. We present a new definition of MALL proof net which remains faithful to the cornerstone of the MLL theory. 1
Orderenriched categorical models of the classical sequent calculus
 LECTURE AT INTERNATIONAL CENTRE FOR MATHEMATICAL SCIENCES, WORKSHOP ON PROOF THEORY AND ALGORITHMS
, 2003
"... It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the wellknown categorical semantics of linear classical sequent proofs, we give models of weakening and contra ..."
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Cited by 21 (2 self)
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It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the wellknown categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cutreduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish nondeterministic choices of cutelimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.
From proof nets to the free * autonomous category
 Logical Methods in Computer Science, 2(4:3):1–44, 2006. Available from: http://arxiv.org/abs/cs/0605054. [McK05] Richard McKinley. Classical categories and deep inference. In Structures and Deduction 2005 (Satellite Workshop of ICALP’05
, 2005
"... Vol. 2 (4:3) 2006, pp. 1–44 www.lmcsonline.org ..."
Elementary Structures in Process Theory (1) Sets with Renaming
, 1997
"... We study a general algebraic framework which underlies a wide range of computational formalisms... ..."
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Cited by 19 (6 self)
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We study a general algebraic framework which underlies a wide range of computational formalisms...
Canonical sequent proofs via multifocusing
 Fifth IFIP International Conference on Theoretical Computer Science, volume 273 of IFIP International Federation for Information Processing
, 2008
"... Abstract The sequent calculus admits many proofs of the same conclusion that differ only by trivial permutations of inference rules. In order to eliminate this “bureaucracy” from sequent proofs, deductive formalisms such as proof nets or natural deduction are usually used instead of the sequent calc ..."
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Cited by 18 (10 self)
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Abstract The sequent calculus admits many proofs of the same conclusion that differ only by trivial permutations of inference rules. In order to eliminate this “bureaucracy” from sequent proofs, deductive formalisms such as proof nets or natural deduction are usually used instead of the sequent calculus, for they identify proofs more abstractly and geometrically. In this paper we recover permutative canonicity directly in the cutfree sequent calculus by generalizing focused sequent proofs to admit multiple foci, and then considering the restricted class of maximally multifocused proofs. We validate this definition by proving a bijection to the wellknown proofnets for the unitfree multiplicative linear logic, and discuss the possibility of a similar correspondence for larger fragments. 1
Polarized ProofNets: ProofNets for LC (Extended Abstract)
 Typed Lambda Calculi and Applications '99
, 1999
"... ) Olivier Laurent Institut de Math'ematiques de Luminy CNRSMarseille, France olaurent@iml.univmrs.fr Abstract. We define a notion of polarization in linear logic (LL) coming from the polarities of JeanYves Girard's classical sequent calculus LC [4]. This allows us to define a translation b ..."
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Cited by 16 (3 self)
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) Olivier Laurent Institut de Math'ematiques de Luminy CNRSMarseille, France olaurent@iml.univmrs.fr Abstract. We define a notion of polarization in linear logic (LL) coming from the polarities of JeanYves Girard's classical sequent calculus LC [4]. This allows us to define a translation between the two systems. Then we study the application of this polarization constraint to proofnets for full linear logic described in [7]. This yields an important simplification of the correctness criterion for polarized proofnets. In this way we obtain a system of proofnets for LC. The study of cutelimination takes an important place in prooftheory. Much work is spent to deal with commutation of rules for cutelimination in sequent calculi. The introduction of proofnets (see [7] for instance) solves commutation problems and allows us to define a clear notion of reduction and complexity. In [4], JeanYves Girard defines the sequent calculus LC using polarities. LC is a refinement...
Elementary Complexity and Geometry of Interaction
, 2000
"... We introduce a geometry of interaction model given by an algebra of clauses equipped with resolution (following [Gir95]) into which proofs of Elementary Linear Logic can be interpreted. In order to extend geometry of interaction computation (the so called execution formula) to a wider class of prog ..."
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Cited by 14 (4 self)
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We introduce a geometry of interaction model given by an algebra of clauses equipped with resolution (following [Gir95]) into which proofs of Elementary Linear Logic can be interpreted. In order to extend geometry of interaction computation (the so called execution formula) to a wider class of programs in the algebra than just those coming from proofs, we define a variant of execution (called weak execution). Its application to any program of clauses is shown to terminate with a bound on the number of steps which is elementary in the size of the program. We establish that weak execution coincides with standard execution on programs coming from proofs. Keywords: Elementary Linear Logic, Geometry of interaction, Complexity, Semantics.
ConnectionBased Proof Construction in Linear Logic
 14 th Conference on Automated Deduction, Lecture Notes in Artificial Intelligence 1249
, 1997
"... Abstract. We present a matrix characterization of logical validity in the multiplicative fragment of linear logic. On this basis we develop a matrixbased proof search procedure for this fragment and a procedure which translates the machinefound proofs back into the usual sequent calculus for linea ..."
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Cited by 13 (7 self)
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Abstract. We present a matrix characterization of logical validity in the multiplicative fragment of linear logic. On this basis we develop a matrixbased proof search procedure for this fragment and a procedure which translates the machinefound proofs back into the usual sequent calculus for linear logic. Both procedures are straightforward extensions of methods which originally were developed for a uniform treatment of classical, intuitionistic and modal logics. They can be extended to further fragments of linear logic once a matrix characterization has been found. 1