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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
Ordered Linear Logic and Applications
, 2001
"... This work is dedicated to my parents. Acknowledgments Firstly, and foremost, I would like to thank my principal advisor, Frank Pfenning, for his patience with me, and for teaching me most of what I know about logic and type theory. I would also like to acknowledge some useful discussions with Kevin ..."
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Cited by 39 (0 self)
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This work is dedicated to my parents. Acknowledgments Firstly, and foremost, I would like to thank my principal advisor, Frank Pfenning, for his patience with me, and for teaching me most of what I know about logic and type theory. I would also like to acknowledge some useful discussions with Kevin Watkins which led me to simplify some of this work. Finally, I would like to thank my other advisor, John Reynolds, for all his kindness and support over the last five years. Abstract This thesis introduces a new logical system, ordered linear logic, which combines reasoning with unrestricted, linear, and ordered hypotheses. The logic conservatively extends (intuitionistic) linear logic, which contains both unrestricted and linear hypotheses, with a notion of ordered hypotheses. Ordered hypotheses must be used exactly once, subject to the order in which they were assumed (i.e., their order cannot be changed during the course of a derivation). This ordering constraint allows for logical representations of simple data structures such as stacks and queues. We construct ordered linear logic in the style of MartinL&quot;of from the basic notion of a hypothetical judgement. We then show normalization for the system by constructing a sequent calculus presentation and proving cutelimination of the sequent system.
Natural Deduction for Intuitionistic NonCommutative Linear Logic
 Proceedings of the 4th International Conference on Typed Lambda Calculi and Applications (TLCA'99
, 1999
"... We present a system of natural deduction and associated term calculus for intuitionistic noncommutative linear logic (INCLL) as a conservative extension of intuitionistic linear logic. We prove subject reduction and the existence of canonical forms in the implicational fragment. ..."
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Cited by 36 (16 self)
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We present a system of natural deduction and associated term calculus for intuitionistic noncommutative linear logic (INCLL) as a conservative extension of intuitionistic linear logic. We prove subject reduction and the existence of canonical forms in the implicational fragment.
Relating Natural Deduction and Sequent Calculus for Intuitionistic NonCommutative Linear Logic
, 1999
"... We present a sequent calculus for intuitionistic noncommutative linear logic (INCLL) , show that it satisfies cut elimination, and investigate its relationship to a natural deduction system for the logic. We show how normal natural deductions correspond to cutfree derivations, and arbitrary natura ..."
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Cited by 29 (16 self)
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We present a sequent calculus for intuitionistic noncommutative linear logic (INCLL) , show that it satisfies cut elimination, and investigate its relationship to a natural deduction system for the logic. We show how normal natural deductions correspond to cutfree derivations, and arbitrary natural deductions to sequent derivations with cut. This gives us a syntactic proof of normalization for a rich system of noncommutative natural deduction and its associated calculus. INCLL conservatively extends linear logic with means to express sequencing, which has applications in functional programming, logical frameworks, logic programming, and natural language parsing. 1 Introduction Linear logic [11] has been described as a logic of state because it views linear hypotheses as resources which may be consumed in the course of a deduction. It thereby significantly extends the expressive power of both classical and intuitionistic logics, yet it does not offer means to express sequencing. Th...
Noncommutative logic II: sequent calculus and phase semantics
, 1998
"... INTRODUCTION Noncommutative logic is a unication of :  commutative linear logic (Girard 1987) and  cyclic linear logic (Girard 1989; Yetter 1990), a classical conservative extension of the Lambek calculus (Lambek 1958). In a previous paper with Abrusci (Abrusci and Ruet 1999) we presented the mu ..."
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Cited by 27 (6 self)
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INTRODUCTION Noncommutative logic is a unication of :  commutative linear logic (Girard 1987) and  cyclic linear logic (Girard 1989; Yetter 1990), a classical conservative extension of the Lambek calculus (Lambek 1958). In a previous paper with Abrusci (Abrusci and Ruet 1999) we presented the multiplicative fragment of noncommutative logic, with proof nets and a sequent calculus based on the structure of order varieties, and a sequentialization theorem. Here we consider full propositional noncommutative logic. Noncommutative logic. Let us rst review the basic ideas. Consider the purely noncommutative fragment of linear logic, obtained by removing the exchange rule entirely : ` ; ; ; , ` ; ; ; y This work has been partly carried out at LIENSCNRS, Ecole Normale Superieure (Paris), at McGill University
Linear Logic Programming with an Ordered Context
 2nd International Conference on Principles and Practice of Declarative Programming (PPDP 2000
, 2000
"... We begin with a review of ordered linear logic (OLL), a refinement of intuitionistic linear logic with an inherent notion of order. We then develop a logic programming interpretation for OLL in two steps: (1) we give a system of ordered uniform derivations which is sound and complete with respect to ..."
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Cited by 15 (0 self)
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We begin with a review of ordered linear logic (OLL), a refinement of intuitionistic linear logic with an inherent notion of order. We then develop a logic programming interpretation for OLL in two steps: (1) we give a system of ordered uniform derivations which is sound and complete with respect to OLL, and (2) we present a model of resource consumption which removes nondeterminism from ordered resource allocation during search for uniform derivations. We also illustrate the expressive power of the resulting ordered linear logic programming language with several example programs.
CLASSICAL NONASSOCIATIVE LAMBEK CALCULUS
"... We introduce nonassociative linear logic, which may be seen as the classical version of the nonassociative Lambek calculus. We define its sequent calculus, its theory of proof nets, for which we give a correctness criterion and a sequentialization theorem, and we show proof search in it is polyno ..."
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Cited by 13 (1 self)
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We introduce nonassociative linear logic, which may be seen as the classical version of the nonassociative Lambek calculus. We define its sequent calculus, its theory of proof nets, for which we give a correctness criterion and a sequentialization theorem, and we show proof search in it is polynomial.
Phase Semantics and Verification of Concurrent Constraint Programs
, 1998
"... The class CC of concurrent constraint programming languages and its nonmonotonic extension LCC based on linear constraint systems can be given a logical semantics in Girard's intuitionistic linear logic for a variety of observables. In this paper we settle basic completeness results and we sho ..."
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Cited by 10 (2 self)
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The class CC of concurrent constraint programming languages and its nonmonotonic extension LCC based on linear constraint systems can be given a logical semantics in Girard's intuitionistic linear logic for a variety of observables. In this paper we settle basic completeness results and we show how the phase semantics of linear logic can be used to provide simple and very concise "semantical" proofs of safety properties for CC or LCC programs.
Ordered Linear Logic Programming
, 1998
"... We begin with a review of intuitionistic noncommutative linear logic (INCLL), a refinement of linear logic with an inherent notion of order proposed by the authors in prior work. We then develop a logic programming interpretation for INCLL in two steps: (1) we give a system of ordered uniform deriv ..."
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Cited by 8 (6 self)
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We begin with a review of intuitionistic noncommutative linear logic (INCLL), a refinement of linear logic with an inherent notion of order proposed by the authors in prior work. We then develop a logic programming interpretation for INCLL in two steps: (1) we give a system of ordered uniform derivations which is sound and complete with respect to INCLL, and (2) we present a model of resource consumption which removes nondeterminism from ordered resource allocation during search for uniform derivations. We also illustrate the expressive power of the resulting ordered linear logic programming language through some examples, including programs for merge sort, insertion sort, and natural language parsing. 1 The authors can be reached at jpolakow@cs.cmu.edu and fp@cs.cmu.edu. This work was sponsored NSF Grants CCR9804014 and CCR9619584. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, ei...