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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 53 (8 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.
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.
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 33 (15 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 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...
Between Functions and Relations in Calculating Programs
, 1992
"... This thesis is about the calculational approach to programming, in which one derives programs from specifications. One such calculational paradigm is Ruby, the relational calculus developed by Jones and Sheeran for describing and designing circuits. We identify two shortcomings with derivations made ..."
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Cited by 15 (4 self)
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This thesis is about the calculational approach to programming, in which one derives programs from specifications. One such calculational paradigm is Ruby, the relational calculus developed by Jones and Sheeran for describing and designing circuits. We identify two shortcomings with derivations made using Ruby. The first is that the notion of a program being an implementation of a specification has never been made precise. The second is to do with types. Fundamental to the use of type information in deriving programs is the idea of having types as special kinds of programs. In Ruby, types are partial equivalence relations (pers). Unfortunately, manipulating some formulae involving types has proved difficult within Ruby. In particular, the preconditions of the `induction' laws that are much used within program derivation often work out to be assertions about types; such assertions have typically been verified either by informal arguments or by using predicate calculus, rather than by ap...
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...
Logic of Predicates With Explicit Substitutions
 Mathematical Foundations of Computer Science 1996, 21st Symposium
, 1996
"... This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one offered by Girard. The latter, cf. [9], translates every sequent of the usual propositional logic (classical, or intuitionistic) into a sequent of commutative linear logic. Then one ..."
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Cited by 3 (3 self)
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This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one offered by Girard. The latter, cf. [9], translates every sequent of the usual propositional logic (classical, or intuitionistic) into a sequent of commutative linear logic. Then one shows that a sequent can be proved classically, resp., intuitionistically, iff its translation can be proved linearly. By contrast, our embedding only works on the level of predicate logic. We show that every theory of classical logic of predicates with equality lives as a theory within a noncommutative intuitionistic substructural logic: the logic of predicates with equality and explicit substitution. Also, our explanation does not require to call upon so called exponentials  the modalities introduced by Girard just to facilitate his embedding. Our construction is also different from other proposals to move substitutions from the level of metatheory to the theory of logic, cf. [16]. They add substitutions as modal constructions. Here, substitutions are considered new atomic formulae.
Logic of Predicates Versus Linear Logic
 ICS PAS Reports, Vol 795
, 1995
"... This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one envisaged by Girard. The latter, cf. [11], is prooftheoretic in nature. Firstly, every sequent of classical, resp., intuitionistic, logic is translated into a sequent of commutative ..."
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Cited by 2 (2 self)
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This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one envisaged by Girard. The latter, cf. [11], is prooftheoretic in nature. Firstly, every sequent of classical, resp., intuitionistic, logic is translated into a sequent of commutative linear logic with exponentials. Then one shows that the former can be proved classically, resp., intuitionistically, iff its translation can be proved linearly. Here it is shown that every theory of classical logic of predicates with equality lives in a sufficiently rich theory built over a noncommutiative intuitionistic substructural logic: the logic of predicates with explicit substitution. This perspective does not require to call upon
Towards program development, specification and verification with Isabelle
, 1995
"... The purpose of this paper is to report on our experiments to use Isabelle  a generic theorem prover  as a universal environment within which specification, development and verification of imperative programs can be performed. The use of a theorem prover for the programming tasks is most appropri ..."
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Cited by 2 (0 self)
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The purpose of this paper is to report on our experiments to use Isabelle  a generic theorem prover  as a universal environment within which specification, development and verification of imperative programs can be performed. The use of a theorem prover for the programming tasks is most appropriate when the processes of program specification, development and verification can be presented as logical activities. In our case this is achieved by adopting pLSD  a novel programming logic.
Design, Analysis and Reasoning about Tools: Abstracts from the Third Workshop
, 1993
"... s from the Third Workshop Flemming Nielson (editor) October 1993 1 Introduction The third DART workshop took place on Thursday August l9th and Friday August 20th at the Department of Computer Science (DIKU) at the University of Copenhagen; it was organized by Mads Rosendahl and others at DIKU, and ..."
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s from the Third Workshop Flemming Nielson (editor) October 1993 1 Introduction The third DART workshop took place on Thursday August l9th and Friday August 20th at the Department of Computer Science (DIKU) at the University of Copenhagen; it was organized by Mads Rosendahl and others at DIKU, and Torben Amtoft and Susanne BrĂ¸nberg helped producing this report. The first day comprised survey presentations whereas the second contained more research oriented talks. The primary aim of the workshop was to increase the awareness of DART participants for each other's work, to stimulate collaboration between the di#erent groups, and to inform Danish industry about the skills possessed by the groups. The DART project started in March 1991 (prematurely terminating a smaller project on Formal Implementation, Transformation and Analysis of Programs) and is funded by the Danish Research Councils as part of the Danish Research Programme on Informatics. To date it has received about 8 million Danis...