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Operational Interpretations of Linear Logic
, 1998
"... Two different operational interpretations of intuitionistic linear logic have been proposed in the literature. The simplest interpretation recomputes nonlinear values every time they are required. It has good memorymanagement properties, but is often dismissed as being too inefficient. Alternative ..."
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Cited by 27 (0 self)
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Two different operational interpretations of intuitionistic linear logic have been proposed in the literature. The simplest interpretation recomputes nonlinear values every time they are required. It has good memorymanagement properties, but is often dismissed as being too inefficient. Alternatively, one can memoize the results of evaluating nonlinear values. This avoids any recomputation, but has weaker memorymanagement properties. Using a novel combination of typetheoretic and operational techniques we give a concise formal comparison of the two interpretations. Moreover, we show that there is a subset of linear logic where the two operational interpretations coincide. In this subset, which is sufficiently expressive to encode callbyvalue lambdacalculus, we can have the best of both worlds: a simple and efficient implementation, and good memorymanagement properties. Keywords: linear logic, operational semantics, callbyvalue lambda calculus, memory management. 1 Introductio...
Local Realizability Toposes and a Modal Logic for Computability (Extended Abstracts)
 Presented at Tutorial Workshop on Realizability Semantics, FLoC'99
, 1999
"... ) Steven Awodey 1 Lars Birkedal 2y Dana S. Scott 2z 1 Department of Philosophy, Carnegie Mellon University 2 School of Computer Science, Carnegie Mellon University April 15, 1999 Abstract This work is a step toward developing a logic for types and computation that includes both the usual ..."
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Cited by 24 (8 self)
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) Steven Awodey 1 Lars Birkedal 2y Dana S. Scott 2z 1 Department of Philosophy, Carnegie Mellon University 2 School of Computer Science, Carnegie Mellon University April 15, 1999 Abstract This work is a step toward developing a logic for types and computation that includes both the usual spaces of mathematics and constructions and spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes, which we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we study first axiomatically, and then by deriving a modal calculus as its internal logic. The resulting framework is intended as a setting for the logical and categorical study of relative computability. 1 Introduction We report here on the current status of research on the Logic of Types and Computation at Carnegie Mellon University [SAB + ]. The general goal of this research program is to develop a logical fra...
A Relevant Analysis of Natural Deduction
 Journal of Logic and Computation
, 1999
"... Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and ..."
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Cited by 23 (7 self)
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Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and Plotkin's LF, with a representation mechanism: the language of RLF is the lLcalculus; the representation mechanism is judgementsastypes, developed for relevant logics. The lLcalculus type theory is a firstorder dependent type theory with two kinds of dependent function spaces: a linear one and an intuitionistic one. We study a natural deduction presentation of the type theory and establish the required prooftheoretic metatheory. The RLF framework is a conservative extension of LF. We show that RLF uniformly encodes (fragments of) intuitionistic linear logic, Curry's l I calculus and ML with references. We describe the CurryHowardde Bruijn correspondence of the lLcalculus with a s...
Logical Modalities and MultiStage Programming
, 1999
"... . Multistage programming is a method for improving the performance of programs through the introduction of controlled program specialization. This paper makes a case for multistage programming with open code and closed values. We argue that a simple language exploiting interactions between two log ..."
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Cited by 22 (13 self)
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. Multistage programming is a method for improving the performance of programs through the introduction of controlled program specialization. This paper makes a case for multistage programming with open code and closed values. We argue that a simple language exploiting interactions between two logical modalities is well suited for multistage programming, and report the results from our study of categorical models for multistage languages. Keywords: Multistage programming, categorical models, semantics, type systems (multilevel typed calculi) , combination of logics (modal and temporal). 1 Introduction Multistage programming is a method for improving the performance of programs through the introduction of controlled program specialization [15, 13]. MetaML was the first language designed specifically to support this method. It provides a type constructor for "code" and staging annotations for building, combining, and executing code, thus allowing the programmer to have finer cont...
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 19 (4 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...
From Action Calculi to Linear Logic
, 1998
"... . Milner introduced action calculi as a framework for investigating models of interactive behaviour. We present a typetheoretic account of action calculi using the propositionsastypes paradigm; the type theory has a sound and complete interpretation in Power's categorical models. We go on to give ..."
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Cited by 19 (7 self)
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. Milner introduced action calculi as a framework for investigating models of interactive behaviour. We present a typetheoretic account of action calculi using the propositionsastypes paradigm; the type theory has a sound and complete interpretation in Power's categorical models. We go on to give a sound translation of our type theory in the (type theory of) intuitionistic linear logic, corresponding to the relation between Benton's models of linear logic and models of action calculi. The conservativity of the syntactic translation is proved by a modelembedding construction using the Yoneda lemma. Finally, we briefly discuss how these techniques can also be used to give conservative translations between various extensions of action calculi. 1 Introduction Action calculi arose directly from the ßcalculus [MPW92]. They were introduced by Milner [Mil96], to provide a uniform notation for capturing many calculi of interaction such as the ßcalculus, the calculus, models of distribut...
Complete Cuboidal Sets in Axiomatic Domain Theory (Extended Abstract)
 In Proceedings of 12th Annual Symposium on Logic in Computer Science
, 1997
"... ) Marcelo Fiore !mf@dcs.ed.ac.uk? Gordon Plotkin y !gdp@dcs.ed.ac.uk? John Power !ajp@dcs.ed.ac.uk? Department of Computer Science Laboratory for Foundations of Computer Science University of Edinburgh, The King's Buildings Edinburgh EH9 3JZ, Scotland Abstract We study the enrichment of ..."
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Cited by 16 (4 self)
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) Marcelo Fiore !mf@dcs.ed.ac.uk? Gordon Plotkin y !gdp@dcs.ed.ac.uk? John Power !ajp@dcs.ed.ac.uk? Department of Computer Science Laboratory for Foundations of Computer Science University of Edinburgh, The King's Buildings Edinburgh EH9 3JZ, Scotland Abstract We study the enrichment of models of axiomatic domain theory. To this end, we introduce a new and broader notion of domain, viz. that of complete cuboidal set, that complies with the axiomatic requirements. We show that the category of complete cuboidal sets provides a general notion of enrichment for a wide class of axiomatic domaintheoretic structures. Introduction The aim of Axiomatic Domain Theory (ADT) is to provide a conceptual understanding of why domains are adequate as mathematical models of computation. (For a discussion see [12, x Axiomatic Domain Theory ].) The approach taken is to axiomatise the structure needed on a category so that its objects can be considered as domains, and its maps as continuous...
A mixed modal/linear lambda calculus with applications to BellantoniCook safe recursion
, 1998
"... . This paper introduces a simplytyped lambda calculus with both modal and linear function types. Through the use of subtyping extra term formers associated with modality and linearity are avoided. We study the basic metatheory of this system including existence and inference of principal types. The ..."
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Cited by 16 (7 self)
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. This paper introduces a simplytyped lambda calculus with both modal and linear function types. Through the use of subtyping extra term formers associated with modality and linearity are avoided. We study the basic metatheory of this system including existence and inference of principal types. The system serves as a platform for certain higherorder generalisations of BellantoniCook's function algebra capturing polynomial time using a separation of the variables into "safe" and "normal" ones. The distinction between and the syntactic restrictions involved with the safe and normal variables in the BellantoniCook framework are captured by the modal function space and the associated typing rules. The linear function spaces on the other hand are used to enable a certain form of primitive recursion with functional result type which is conservative over polynomial time. The proofs associated with these applications are based on an interpretation of the lambda calculus in a categorytheor...
On traced monoidal closed categories
, 2008
"... ... focus on a simple observation that a traced monoidal category C is closed if and only if the canonical inclusion from C into Int C has a right adjoint. Thus, every traced monoidal closed category arises as a monoidal coreflexive full subcategory of a tortile monoidal category. From this, we der ..."
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Cited by 12 (2 self)
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... focus on a simple observation that a traced monoidal category C is closed if and only if the canonical inclusion from C into Int C has a right adjoint. Thus, every traced monoidal closed category arises as a monoidal coreflexive full subcategory of a tortile monoidal category. From this, we derive a series of facts for traced models of linear logic, and some for models of fixedpoint computation. To make the paper more selfcontained, we also include various background results for traced monoidal categories.