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Focusing the inverse method for linear logic
 Proceedings of CSL 2005
, 2005
"... 1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10 ..."
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1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10
Mechanizing structural induction
 Theor. Comput. Sci
, 1979
"... This thesis has been submitted in fulfilment of the requirements for a postgraduate degree (e.g. PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following terms and conditions of use: • This work is protected by copyright and other intellectual property rights, which are re ..."
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Cited by 41 (0 self)
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This thesis has been submitted in fulfilment of the requirements for a postgraduate degree (e.g. PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following terms and conditions of use: • This work is protected by copyright and other intellectual property rights, which are retained by the thesis author, unless otherwise stated. • A copy can be downloaded for personal noncommercial research or study, without prior permission or charge. • This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the author. • The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the author. • When referring to this work, full bibliographic details including the author, title,
Labelled Propositional Modal Logics: Theory and Practice
, 1996
"... We show how labelled deductive systems can be combined with a logical framework to provide a natural deduction implementation of a large and wellknown class of propositional modal logics (including K, D, T , B, S4, S4:2, KD45, S5). Our approach is modular and based on a separation between a base lo ..."
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Cited by 40 (8 self)
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We show how labelled deductive systems can be combined with a logical framework to provide a natural deduction implementation of a large and wellknown class of propositional modal logics (including K, D, T , B, S4, S4:2, KD45, S5). Our approach is modular and based on a separation between a base logic and a labelling algebra, which interact through a fixed interface. While the base logic stays fixed, different modal logics are generated by plugging in appropriate algebras. This leads to a hierarchical structuring of modal logics with inheritance of theorems. Moreover, it allows modular correctness proofs, both with respect to soundness and completeness for semantics, and faithfulness and adequacy of the implementation. We also investigate the tradeoffs in possible labelled presentations: We show that a narrow interface between the base logic and the labelling algebra supports modularity and provides an attractive prooftheory (in comparision to, e.g., semantic embedding) but limits th...
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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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 Linear Logic
, 1993
"... The paper deals with two versions of the fragment with unit, tensor, linear implication and storage operator (the exponential !) of intuitionistic linear logic. The first version, ILL, appears in a paper by Benton, Bierman, Hyland and de Paiva; the second one, ILL + , is described in this paper. I ..."
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The paper deals with two versions of the fragment with unit, tensor, linear implication and storage operator (the exponential !) of intuitionistic linear logic. The first version, ILL, appears in a paper by Benton, Bierman, Hyland and de Paiva; the second one, ILL + , is described in this paper. ILL has a contraction rule and an introduction rule !I for the exponential; in ILL + , instead of a contraction rule, multiple occurrences of labels for assumptions are permitted under certain conditions; moreover, there is a different introduction rule for the exponential, !I + , which is closer in spirit to the necessitation rule for the normalizable version of S4 discussed by Prawitz in his monograph "Natural Deduction". It is relatively easy to adapt Prawitz's treatment of natural deduction for intuitionistic logic to ILL + ; in particular one can formulate a notion of strong validity (as in Prawitz's "Ideas and Results in Proof Theory") permitting a proof of strong normalization. T...
On an Intuitionistic Modal Logic
 Studia Logica
, 2001
"... . In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4our formulation has several important metatheoretic properties. In addition, we study models ..."
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Cited by 27 (6 self)
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. In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4our formulation has several important metatheoretic properties. In addition, we study models of IS4, not in the framework of Kripke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also a means of modelling proofs as well as provability. 1. Introduction Modal logics are traditionally extensions of classical logic with new operators, or modalities, whose operation is intensional. Modal logics are most commonly justified by the provision of an intuitive semantics based upon `possible worlds', an idea originally due to Kripke. Kripke also provided a possible worlds semantics for intuitionistic logic, and so it is natural to consider intuitionistic logic extended with intensional modalities...
Confluence Properties of Extensional and NonExtensional lambdaCalculi with Explicit Substitutions (Extended Abstract)
 in Proceedings of the Seventh International Conference on Rewriting Techniques and Applications
, 1996
"... ) Delia Kesner CNRS and LRI, B at 490, Universit e ParisSud  91405 Orsay Cedex, France. email:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and nonextensional #calculi with explicit substitutions, where extensionality is interpreted by #expansion. For ..."
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Cited by 24 (5 self)
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) Delia Kesner CNRS and LRI, B at 490, Universit e ParisSud  91405 Orsay Cedex, France. email:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and nonextensional #calculi with explicit substitutions, where extensionality is interpreted by #expansion. For that, we propose a general scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our general scheme makes it possible to treat at the same time many wellknown calculi such as ## , ## # and ## , or some other new calculi that we propose in this paper. We also show for those calculi not fitting in the general scheme that can be translated to another one fitting the scheme, such as #s , how to reason about confluence properties of their extensional and nonextensional versions. 1 Introduction The #calculus is a convenient framework to study functional programming, where the evaluation process is modeled by #reduction. The...
A short proof of the Strong Normalization of Classical Natural Deduction with Disjunction
 Journal of symbolic Logic
, 2003
"... We give a direct, purely arithmetical and elementary proof of the strong normalization of the cutelimination procedure for full (i.e. in presence of all the usual connectives) classical natural deduction. 1 ..."
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Cited by 23 (14 self)
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We give a direct, purely arithmetical and elementary proof of the strong normalization of the cutelimination procedure for full (i.e. in presence of all the usual connectives) classical natural deduction. 1
Some Lambda Calculi With Categorical Sums and Products
, 1993
"... . We consider the simply typed calculus with primitive recursion operators and types corresponding to categorical products and coproducts.. The standard equations corresponding to extensionality and to surjectivity of pairing and its dual are oriented as expansion rules. Strong normalization an ..."
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Cited by 22 (1 self)
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. We consider the simply typed calculus with primitive recursion operators and types corresponding to categorical products and coproducts.. The standard equations corresponding to extensionality and to surjectivity of pairing and its dual are oriented as expansion rules. Strong normalization and ground (basetype) confluence is proved for the full calculus; full confluence is proved for the calculus omitting the rule for strong sums. In the latter case, fixedpoint constructors may be added while retaining confluence. 1 Introduction The systems investigated here are simply typed caluli whose types include pairs, unit, sums, an empty type, and a type of natural numbers supporting constructions by primitive recursion. In the core system the types behave as categorical product and coproducts, so the subject at hand is equivalently ([LS86]) the equational theory of the free bicartesian closed category (generated by objects for the base types) with weak natural numbers object. Su...
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re