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96
Weakly Distributive Categories
 Journal of Pure and Applied Algebra
, 1991
"... There are many situations in logic, theoretical computer science, and category theory where two binary operationsone thought of as a (tensor) "product", the other a "sum"play a key role. In distributive and autonomous categories these operations can be regarded as, respect ..."
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Cited by 140 (20 self)
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There are many situations in logic, theoretical computer science, and category theory where two binary operationsone thought of as a (tensor) "product", the other a "sum"play a key role. In distributive and autonomous categories these operations can be regarded as, respectively, the and/or of traditional logic and the times/par of (multiplicative) linear logic. In the latter logic, however, the distributivity of product over sum is conspicuously absent: this paper studies a "linearization" of that distributivity which is present in both case. Furthermore, we show that this weak distributivity is precisely what is needed to model Gentzen's cut rule (in the absence of other structural rules) and can be strengthened in a natural way to generate  autonomous categories. We also point out that this "linear" notion of distributivity is virtually orthogonal to the usual notion as formalized by distributive categories. 0 Introduction There are many situations in logic, theoretical co...
Natural Deduction and Coherence for Weakly Distributive Categories
 Journal of Pure and Applied Algebra
, 1991
"... This paper examines coherence for certain monoidal categories using techniques coming from the proof theory of linear logic, in particular making heavy use of the graphical techniques of proof nets. We define a two sided notion of proof net, suitable for categories like weakly distributive categorie ..."
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Cited by 82 (27 self)
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This paper examines coherence for certain monoidal categories using techniques coming from the proof theory of linear logic, in particular making heavy use of the graphical techniques of proof nets. We define a two sided notion of proof net, suitable for categories like weakly distributive categories which have the twotensor structure (times/par) of linear logic, but lack a negation operator. Representing morphisms in weakly distributive categories as such nets, we derive a coherence theorem for such categories. As part of this process, we develop a theory of expansionreduction systems with equalities and a term calculus for proof nets, each of which is of independent interest. In the symmetric case the expansion reduction system on the term calculus yields a decision procedure for the equality of maps for free weakly distributive categories. The main results of this paper are these. First we have proved coherence for the full theory of weakly distributive categories, extending simi...
A Brief Guide to Linear Logic
, 1993
"... An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation. ..."
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Cited by 56 (10 self)
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An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation.
A Generalization of Analytic Deduction via Labelled Deductive Systems I: Basic Substructural Logics
 Journal of Automated Reasoning
, 1995
"... In this series of papers we set out to generalize the notion of classical analytic deduction (i.e. deduction via elimination rules) by combining the methodology of Labelled Deductive Systems [Gab94] with the classical system KE [DM94]. LDS is a unifying framework for the study of logics and of their ..."
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Cited by 52 (8 self)
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In this series of papers we set out to generalize the notion of classical analytic deduction (i.e. deduction via elimination rules) by combining the methodology of Labelled Deductive Systems [Gab94] with the classical system KE [DM94]. LDS is a unifying framework for the study of logics and of their interactions. In the LDS approach the basic units of logical derivation are not just formulae but labelled formulae, where the labels belong to a given "labelling algebra". The derivation rules act on the labels as well as on the formulae, according to certain fixed rules of propagation. By virtue of the extra power of the labelling algebras, standard (classical or intuitionistic) proof systems can be extended to cover a much wider territory without modifying their structure. The system KE is a new tree method for classical analytic deduction based on "analytic cut". It is a refutation system, like analytic tableaux and resolution, but it is essentially more efficient than tableaux and, un...
Substructural Logics on Display
, 1998
"... Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek ca ..."
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Cited by 50 (16 self)
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Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponentialfree linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also have a "cyclic" counterpart. These logics have been studied extensively and are quite well understood. Generalising further, one can start with intuitionistic BiLambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bilinear, Birelevant, BiBCK and Biintuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and som...
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 42 (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...
Noncommutative logic I : the multiplicative fragment
, 1998
"... INTRODUCTION Unrestricted exchange rules of Girard's linear logic [8] force the commutativity of the multiplicative connectives\Omega (times, conjunction) and & (par, disjunction) , and henceforth the commutativity of all logic. This a priori commutativity is not always desirable  it is ..."
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Cited by 41 (7 self)
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INTRODUCTION Unrestricted exchange rules of Girard's linear logic [8] force the commutativity of the multiplicative connectives\Omega (times, conjunction) and & (par, disjunction) , and henceforth the commutativity of all logic. This a priori commutativity is not always desirable  it is quite problematic in applications like linguistics or computer science , and actually the desire of a noncommutative logic goes back to the very beginning of LL [9]. Previous works on noncommutativity deal essentially with noncommutative fragments of LL, obtained by removing the exchange rule at all. At that point, a simple remark on the status of exchange in the sequent calculus is necessary to be clear: there are two presentations of exchange in commutative LL, either sequents are finite sets of occurrences of formulas and exchange is obviously implicit, or sequents are fini
Pomset Logic: A NonCommutative Extension of Classical Linear Logic
, 1997
"... We extend the multiplicative fragment of linear logic with a noncommutative connective (called before), which, roughly speaking, corresponds to sequential composition. This lead us to a calculus where the conclusion of a proof is a Partially Ordered MultiSET of formulae. We firstly examine coherenc ..."
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Cited by 41 (10 self)
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We extend the multiplicative fragment of linear logic with a noncommutative connective (called before), which, roughly speaking, corresponds to sequential composition. This lead us to a calculus where the conclusion of a proof is a Partially Ordered MultiSET of formulae. We firstly examine coherence semantics, where we introduce the before connective, and ordered products of formulae. Secondly we extend the syntax of multiplicative proof nets to these new operations. We then prove strong normalisation, and confluence. Coming back to the denotational semantics that we started with, we establish in an unusual way the soundness of this calculus with respect to the semantics. The converse, i.e. a kind of completeness result, is simply stated: we refer to a report for its lengthy proof. We conclude by mentioning more results, including a sequent calculus which is interpreted by both the semantics and the proof net syntax, although we are not sure that it takes all proof nets into account...
Partial Proof Trees as Building Blocks for a Categorial Grammar
 Linguistics and Philosophy
, 1997
"... We describe a categorial system (PPTS) based on partial proof trees (PPTs) as the building blocks of the system. The PPTs are obtained by unfolding the arguments of the type that would be associated with a lexical item in a simple categorial grammar. The PPTs are the basic types in the system and a ..."
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Cited by 39 (9 self)
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We describe a categorial system (PPTS) based on partial proof trees (PPTs) as the building blocks of the system. The PPTs are obtained by unfolding the arguments of the type that would be associated with a lexical item in a simple categorial grammar. The PPTs are the basic types in the system and a derivation proceeds by combining PPTs together. We describe the construction of the finite set of basic PPTs and the operations for combining them. PPTS can be viewed as a categorial system incorporating some of the key insights of lexicalized tree adjoining grammar, namely the notion of an extended domain of locality and the consequent factoring of recursion from the domain of dependencies. PPTS therefore inherits the linguistic and computational properties of that system, and so can be viewed as a `middle ground' between a categorial grammar and a phrase structure grammar. We also discuss the relationship between PPTS, natural deduction, and linear logic proofnets, and argue that natural ...
Lambek calculus is npcomplete
 Theoretical Computer Science
, 2003
"... We prove that for both the Lambek calculus L and the Lambek calculus allowing empty premises L ∗ the derivability problem is NPcomplete. It follows that also for the multiplicative fragments of cyclic linear logic and noncommutative linear logic the derivability problem is NPcomplete. ..."
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Cited by 34 (0 self)
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We prove that for both the Lambek calculus L and the Lambek calculus allowing empty premises L ∗ the derivability problem is NPcomplete. It follows that also for the multiplicative fragments of cyclic linear logic and noncommutative linear logic the derivability problem is NPcomplete.