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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.
Applications of Linear Logic to Computation: An Overview
, 1993
"... This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and objectoriented programming and some other applications of LL, li ..."
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Cited by 41 (3 self)
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This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and objectoriented programming and some other applications of LL, like semantics of negation in LP, nonmonotonic issues in AI planning, etc. Although the overview covers pretty much the stateoftheart in this area, by necessity many of the works are only mentioned and referenced, but not discussed in any considerable detail. The paper does not presuppose any previous exposition to LL, and is addressed more to computer scientists (probably with a theoretical inclination) than to logicians. The paper contains over 140 references, of which some 80 are about applications of LL. 1 Linear Logic Linear Logic (LL) was introduced in 1987 by Girard [62]. From the very beginning it was recognized as relevant to issues of computation (especially concurrency and stat...
Interaction Combinators
 Information and Computation
, 1995
"... This paper is the continuation of the author 's work on interaction nets, inspired by Girard's proof nets for linear logic, but no preliminary knowledge of these topics is required for its reading. Introduction ..."
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Cited by 31 (2 self)
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This paper is the continuation of the author 's work on interaction nets, inspired by Girard's proof nets for linear logic, but no preliminary knowledge of these topics is required for its reading. Introduction
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 ..."
Correctness of Multiplicative Proof Nets is Linear
 In Fourteenth Annual IEEE Symposium on Logic in Computer Science
, 1999
"... We reformulate Danos contractibility criterion in terms of a sort of unification. As for term unification, a direct implementation of the unification criterion leads to a quasilinear algorithm. Linearity is obtained after observing that the disjointset unionfind at the core of the unification cri ..."
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Cited by 18 (1 self)
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We reformulate Danos contractibility criterion in terms of a sort of unification. As for term unification, a direct implementation of the unification criterion leads to a quasilinear algorithm. Linearity is obtained after observing that the disjointset unionfind at the core of the unification criterion is a special case of unionfind with a real linear time solution. Keywords: linear logic, data structures. 1 Introduction A multiplicative proof net is a graph representation of a multiplicative linear logic derivation [5]. 1 The proof net N corresponding to the derivation P is a (directed) hypergraph with a link (i.e., an hyperedge) for each rule and a vertex for each formula occurrence s.t. every link of N connects the active formulas of the corresponding rule of P. For instance, let P be a cutfree derivation using atomic axioms only and ending with the sequent ` A; the corresponding proof net N is the syntax tree of the formula A, plus a set of connections between pairs of oc...
On the axiomatisation of boolean categories with and without medial
 THEORY APPL. CATEG
, 2007
"... ..."
Interaction Nets and Term Rewriting Systems
, 1998
"... Term rewriting systems provide a framework in which it is possible to specify and program in a traditional syntax (oriented equations). Interaction nets, on the other hand, provide a graphical syntax for the same purpose, but can be regarded as being closer to an implementation since the reductio ..."
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Cited by 13 (7 self)
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Term rewriting systems provide a framework in which it is possible to specify and program in a traditional syntax (oriented equations). Interaction nets, on the other hand, provide a graphical syntax for the same purpose, but can be regarded as being closer to an implementation since the reduction process is local and asynchronous, and all the operations are made explicit, including discarding and copying of data. Our aim is to bridge the gap between the above formalisms by showing how to understand interaction nets in a term rewriting framework. This allows us to transfer results from one paradigm to the other, deriving syntactical properties of interaction nets from the (wellstudied) properties of term rewriting systems; in particular concerning termination and modularity. Keywords: term rewriting, interaction nets, termination, modularity. 1 Introduction Term rewriting systems provide a general framework for specifying and reasoning about computation. They can be regarde...
Embedding the finitary picalculus in differential interaction nets
 In Proceedings of the Higher Order Rewriting workshop (HOR 2006
, 2006
"... Abstract. We propose a translation of a finitary (that is, replicationfree) version of the monadic localised picalculus into the purely exponential part of promotionfree differential interaction nets. This embedding is a simulation of reduction. Since the introduction of Linear Logic by Girard in ..."
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Cited by 11 (5 self)
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Abstract. We propose a translation of a finitary (that is, replicationfree) version of the monadic localised picalculus into the purely exponential part of promotionfree differential interaction nets. This embedding is a simulation of reduction. Since the introduction of Linear Logic by Girard in 1986, it was clear to many logicians and computer scientists that some deep connection between this new logical setting and concurrency should show up. This impression has been enforced by the introduction of interaction nets by Lafont [1], where reduction is given by a purely local and asynchronous interaction. There is an apparent contradiction between nondeterminism and the CurryHoward approach to computation. Indeed, one of the main properties that one expects from a wellbehaved proof system is not only that it possesses a cutelimination procedure, but also that this procedure enjoys a confluence property similar to the ChurchRosser property of the λcalculus. But confluence is a way of expressing determinism in a rewriting setting, so that being able to represent
Types, potency, and idempotency: why nonlinearity and amnesia make a type system work
 In ICFP ’04: Proceedings of the ninth ACM SIGPLAN international conference on Functional programming, 138–149, ACM
, 2004
"... Useful type inference must be faster than normalization. Otherwise, you could check safety conditions by running the program. We analyze the relationship between bounds on normalization and type inference. We show how the success of type inference is fundamentally related to the amnesia of the type ..."
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Cited by 8 (1 self)
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Useful type inference must be faster than normalization. Otherwise, you could check safety conditions by running the program. We analyze the relationship between bounds on normalization and type inference. We show how the success of type inference is fundamentally related to the amnesia of the type system: the nonlinearity by which all instances of a variable are constrained to have the same type. Recent work on intersection types has advocated their usefulness for static analysis and modular compilation. We analyze SystemI (and some instances of its descendant, System E), an intersection type system with a type inference algorithm. Because SystemI lacks idempotency, each occurrence of a variable requires a distinct type. Consequently, type inference is equivalent to normalization in every single case, and time bounds on type inference and normalization are identical. Similar relationships hold for other intersection type systems without idempotency. The analysis is founded on an investigation of the relationship between linear logic and intersection types. We show a lockstep correspondence between normalization and type inference. The latter shows the promise of intersection types to facilitate static analyses of varied granularity, but also belies an immense challenge: to add amnesia to such analysis without losing all of its benefits.