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A system of interaction and structure
 ACM TRANSACTIONS ON COMPUTATIONAL LOGIC
, 2004
"... This paper introduces a logical system, called BV, which extends multiplicative linear logic by a noncommutative selfdual logical operator. This extension is particularly challenging for the sequent calculus, and so far it is not achieved therein. It becomes very natural in a new formalism, call ..."
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Cited by 87 (15 self)
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This paper introduces a logical system, called BV, which extends multiplicative linear logic by a noncommutative selfdual logical operator. This extension is particularly challenging for the sequent calculus, and so far it is not achieved therein. It becomes very natural in a new formalism, called the calculus of structures, which is the main contribution of this work. Structures are formulae subject to certain equational laws typical of sequents. The calculus of structures is obtained by generalising the sequent calculus in such a way that a new topdown symmetry of derivations is observed, and it employs inference rules that rewrite inside structures at any depth. These properties, in addition to allowing the design of BV, yield a modular proof of cut elimination.
Noncommutativity and MELL in the Calculus of Structures
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2001
"... We introduce the calculus of structures: it is more general than the sequent calculus and it allows for cut elimination and the subformula property. We show a simple extension of multiplicative linear logic, by a selfdual noncommutative operator inspired by CCS, that seems not to be expressible in ..."
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Cited by 58 (22 self)
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We introduce the calculus of structures: it is more general than the sequent calculus and it allows for cut elimination and the subformula property. We show a simple extension of multiplicative linear logic, by a selfdual noncommutative operator inspired by CCS, that seems not to be expressible in the sequent calculus. Then we show that multiplicative exponential linear logic benefits from its presentation in the calculus of structures, especially because we can replace the ordinary, global promotion rule by a local version. These formal systems, for which we prove cut elimination, outline a range of techniques and properties that were not previously available. Contrarily to what happens in the sequent calculus, the cut elimination proof is modular.
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 36 (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"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.
A NonCommutative Extension of MELL
, 2002
"... We extend multiplicative exponential linear logic (MELL) by a noncommutative, selfdual logical operator. The extended system, called NEL, is defined in the formalism of the calculus of structures, which is a generalisation of the sequent calculus and provides a more refined analysis of proofs. We ..."
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Cited by 27 (12 self)
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We extend multiplicative exponential linear logic (MELL) by a noncommutative, selfdual logical operator. The extended system, called NEL, is defined in the formalism of the calculus of structures, which is a generalisation of the sequent calculus and provides a more refined analysis of proofs. We should then be able to extend the range of applications of MELL, by modelling a broad notion of sequentiality and providing new properties of proofs. We show some proof theoretical results: decomposition and cut elimination. The new operator represents a significant challenge: to get our results we use here for the first time some novel techniques, which constitute a uniform and modular approach to cut elimination, contrary to what is possible in the sequent calculus.
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 27 (14 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...
Using Linear Logic to Reason About Sequent Systems
 PROCEEDINGS OF TABLEAUX 2002, LNAI 2381
, 2002
"... Linear logic can be used as a metalogic for the specification of some sequent calculus proof systems. We explore in this paper properties of such linear logic specifications. We show that derivability of one proof system from another has a simple decision procedure that is implemented simply vi ..."
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Cited by 22 (4 self)
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Linear logic can be used as a metalogic for the specification of some sequent calculus proof systems. We explore in this paper properties of such linear logic specifications. We show that derivability of one proof system from another has a simple decision procedure that is implemented simply via bounded logic programming search. We also provide conditions to ensure that an encoded proof system has the cutelimination property and show that this can be decided again by simple, bounded proof search algorithms.
Linear logic as a framework for specifying sequent calculus
 Lecture Notes in Logic 17, Logic Colloquium’99
, 2004
"... Abstract. In recent years, intuitionistic logic and type systems have been used in numerous computational systems as frameworks for the specification of natural deduction proof systems. As we shall illustrate here, linear logic can be used similarly to specify the more general setting of sequent cal ..."
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Cited by 11 (3 self)
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Abstract. In recent years, intuitionistic logic and type systems have been used in numerous computational systems as frameworks for the specification of natural deduction proof systems. As we shall illustrate here, linear logic can be used similarly to specify the more general setting of sequent calculus proof systems. Linear logic’s meta theory can be used also to analyze properties of a specified objectlevel proof system. We shall present several example encodings of sequent calculus proof systems using the Forum presentation of linear logic. Since the objectlevel encodings result in logic programs (in the sense of Forum), various aspects of objectlevel proof systems can be automated. §1. Introduction. Logics and type systems have been exploited in recent years as frameworks for the specification of deduction in a number of logics. Such meta logics or logical frameworks have generally been based on intuitionistic logic in which quantification at (nonpredicate) higherorder types is available. Identifying a framework that allows the specification of a wide range of logics has
A system of interaction and structure IV: The exponentials
 IN THE SECOND ROUND OF REVISION FOR MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2007
"... We study some normalisation properties of the deepinference proof system NEL, which can be seen both as 1) an extension of multiplicative exponential linear logic (MELL) by a certain noncommutative selfdual logical operator; and 2) an extension of system BV by the exponentials of linear logic. T ..."
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Cited by 11 (6 self)
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We study some normalisation properties of the deepinference proof system NEL, which can be seen both as 1) an extension of multiplicative exponential linear logic (MELL) by a certain noncommutative selfdual logical operator; and 2) an extension of system BV by the exponentials of linear logic. The interest of NEL resides in: 1) its being Turing complete, while the same for MELL is not known, and is widely conjectured not to be the case; 2) its inclusion of a selfdual, noncommutative logical operator that, despite its simplicity, cannot be axiomatised in any analytic sequent calculus system; 3) its ability to model the sequential composition of processes. We present several decomposition results for NEL and, as a consequence of those and via a splitting theorem, cut elimination. We use, for the first time, an induction measure based on flow graphs associated to the exponentials, which captures their rather complex behaviour in the normalisation process. The results are presented in the calculus of structures, which is the first, developed formalism in deep inference.
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...