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Proofnets: The parallel syntax for prooftheory
 Logic and Algebra
, 1996
"... The paper is mainly concerned with the extension of proofnets to additives, for which the best known solution is presented. It proposes two cutelimination procedures, the lazy one being in linear time. The solution is shown to be compatible with quantifiers, and the structural rules of exponential ..."
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Cited by 128 (1 self)
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The paper is mainly concerned with the extension of proofnets to additives, for which the best known solution is presented. It proposes two cutelimination procedures, the lazy one being in linear time. The solution is shown to be compatible with quantifiers, and the structural rules of exponentials are also accommodated. Traditional prooftheory deals with cutelimination; these results are usually obtained by means of sequent calculi, with the consequence that 75 % of a cutelimination proof is devoted to endless commutations of rules. It is hard to be happy with this, mainly because: ◮ the structure of the proof is blurred by all these cases; ◮ whole forests have been destroyed in order to print the same routine lemmas; ◮ this is not extremely elegant. However oldfashioned prooftheory, which is concerned with the ritual question: “isthattheoryconsistent? ” never really cared. The situation changed when subtle algorithmic aspects of cutelimination became prominent: typically
A jump from parallel to sequential proofs. multiplicatives
 of Lecture Notes in Computer Science
, 2006
"... Abstract. We introduce a new class of multiplicative proof nets, Jproof nets, which are a typed version of Faggian and Maurel’s multiplicative Lnets. In Jproof nets, we can characterize nets with different degrees of sequentiality, by gradual insertion of sequentiality constraints. As a byproduct ..."
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Abstract. We introduce a new class of multiplicative proof nets, Jproof nets, which are a typed version of Faggian and Maurel’s multiplicative Lnets. In Jproof nets, we can characterize nets with different degrees of sequentiality, by gradual insertion of sequentiality constraints. As a byproduct, we obtain a simple proof of the sequentialisation theorem. 1
Obsessional cliques: a semantic characterization of bounded time complexity
 In Proceedings of LICS’06
, 2006
"... We give a semantic characterization of bounded complexity proofs. We introduce the notion of obsessional clique in the relational model of linear logic and show that restricting the morphisms of the category REL to obsessional cliques yields models of ELL and SLL. Conversely, we prove that these m ..."
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Cited by 2 (0 self)
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We give a semantic characterization of bounded complexity proofs. We introduce the notion of obsessional clique in the relational model of linear logic and show that restricting the morphisms of the category REL to obsessional cliques yields models of ELL and SLL. Conversely, we prove that these models are relatively complete: an LL proof whose interpretation is an obsessional clique is always an ELL/SLL proof. These results are achieved by introducing a system of ELL/SLL untyped proofnets, which is both correct and complete with respect to elementary/polynomial time complexity. 1.
Jump from parallel to sequential proofs: exponentials,” 2011, to appear in the special number Di?erential Linear Logic, Nets, and other quantitative approaches to ProofTheory of MSCS
"... In previous works, by importing ideas from game semantics (notably FaggianMaurelCurien’s ludics nets), we defined a new class of multiplicative/additive polarized proof nets, called Jproof nets. The distinctive feature of Jproof nets with respect to other proof net syntaxes, is the possibility o ..."
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In previous works, by importing ideas from game semantics (notably FaggianMaurelCurien’s ludics nets), we defined a new class of multiplicative/additive polarized proof nets, called Jproof nets. The distinctive feature of Jproof nets with respect to other proof net syntaxes, is the possibility of representing proof nets which are partially sequentialized, by using jumps (that is, untyped extra edges) as sequentiality constraints. Starting from this result, in the present work we extend Jproof nets to the multiplicative/exponential fragment, in order to take into account structural rules: more precisely, we replace the familiar linear logic notion of exponential box with a less restricting one (called cone) defined by means of jumps. As a consequence, we get a syntax for polarized nets where, instead of a structure of boxes nested one into the other, we have one of cones which can be partially overlapping. Moreover, we define cutelimination for exponential Jproof nets, proving, by a variant of Gandy’s method, that even in case of “superposed ” cones, reduction enjoys confluence and strong normalization.
Jump from Parallel to Sequential Proofs: Additives. ⋆
"... Abstract. In previous work, we introduced a framework for proof nets of the multiplicative fragment of Linear Logic, where partially sequentialised nets are allowed. In this paper we extend this result to include additives, using a definition of proof net, called Jproof net, which is the typed vers ..."
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Abstract. In previous work, we introduced a framework for proof nets of the multiplicative fragment of Linear Logic, where partially sequentialised nets are allowed. In this paper we extend this result to include additives, using a definition of proof net, called Jproof net, which is the typed version of an Lnet of Faggian and Maurel. In Jproof nets, we can characterize nets with different degrees of sequentiality, by gradual insertion of sequentiality constraints (jumps). As a byproduct, we obtain a simple proof of the sequentialisation theorem.
Reversible, Irreversible and Optimal
 in Electronic Notes in Theoretical Computer Science
, 1996
"... There are two quite different possibilities for implementing linear head reduction in calculus. Two ways which we are going to explain briefly here in the introduction and in details in the body of the paper. The paper itself is concerned with showing an unexpectedly simple relation between these ..."
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There are two quite different possibilities for implementing linear head reduction in calculus. Two ways which we are going to explain briefly here in the introduction and in details in the body of the paper. The paper itself is concerned with showing an unexpectedly simple relation between these two ways, which we term reversible and irreversible, namely that the latter may be obtained as a natural optimization of the former. Keywords: calculus, abstract machines, geometry of interaction, reversible computations. 1 Introduction Notation. We denote the application of U to V by (U )V , e.g., the Church integer 2 will be fx (f)(f)x. Linear head reduction. But what is exactly linear head reduction, to begin with. It is a variant of head reduction, where one substitutes at each step the leftmost occurrence of variable whenever it is engaged into a redex, as in: (f (f )(f)x)y y ! (f(y y)(f)x)y y ! (f(y (f)x)(f)x)y y ! (f(y (y y)x)(f)x)y y ! (f(y (y x)x)(f)x)y y where the succ...
Annals of Pure and Applied Logic 155 (2008) 173–182 Contents lists available at ScienceDirect Annals of Pure and Applied Logic
"... journal homepage: www.elsevier.com/locate/apal ..."
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"... 2 P. Baillot and M. Pedicini / Geometry of interaction and complexity 1. Introduction Geometry of interaction (GOI) was introduced by Girard ([6, 8]) as a semantics of computation which: ffl on the one hand, in contrast to denotational semantics interprets explicitly the dynamics of computation and ..."
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2 P. Baillot and M. Pedicini / Geometry of interaction and complexity 1. Introduction Geometry of interaction (GOI) was introduced by Girard ([6, 8]) as a semantics of computation which: ffl on the one hand, in contrast to denotational semantics interprets explicitly the dynamics of computation and handles finite objects,