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29
On Bunched Typing
, 2002
"... We study a typing scheme derived from a semantic situation where a single category possesses several closed structures, corresponding to dierent varieties of function type. In this scheme typing contexts are trees built from two (or more) binary combining operations, or in short, bunches. Bunched ..."
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Cited by 33 (2 self)
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We study a typing scheme derived from a semantic situation where a single category possesses several closed structures, corresponding to dierent varieties of function type. In this scheme typing contexts are trees built from two (or more) binary combining operations, or in short, bunches. Bunched typing and its logical counterpart, bunched implications, have arisen in joint work of the author and David Pym. The present paper gives a basic account of the type system, and then focusses on concrete models that illustrate how it may be understood in terms of resource access and sharing. The most
Topological and Limitspace subcategories of Countablybased Equilogical Spaces
, 2001
"... this paper we show that the two approaches are equivalent for a ..."
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Cited by 22 (4 self)
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this paper we show that the two approaches are equivalent for a
Safe recursion with higher types and BCKalgebra
 Annals of Pure and Applied Logic
, 2000
"... In previous work the author has introduced a lambda calculus SLR with modal and linear types which serves as an extension of BellantoniCook's function algebra BC to higher types. It is a step towards a functional programming language in which all programs run in polynomial time. In this paper we de ..."
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Cited by 21 (3 self)
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In previous work the author has introduced a lambda calculus SLR with modal and linear types which serves as an extension of BellantoniCook's function algebra BC to higher types. It is a step towards a functional programming language in which all programs run in polynomial time. In this paper we develop a semantics of SLR using BCKalgebras consisting of certain polynomialtime algorithms. It will follow from this semantics that safe recursion with arbitrary result type built up from N and ( as well as recursion over trees and other data structures remains within polynomial time. In its original formulation SLR supported only natural numbers and recursion on notation with first order functional result type. 1 Introduction In [10] and [11] we have introduced a lambda calculus SLR which generalises the BellantoniCook characterisation of PTIME [2] to higherorder functions. The separation between normal and safe variables which is crucial to the BellantoniCook system has been achieved...
A Cellular Nerve for Higher Categories
, 2002
"... ... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there ..."
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Cited by 21 (2 self)
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... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there exists a dense subcategory YA of the category of Aalgebras for each ooperad A in Batanin’s sense. Whenever A is contractible, the resulting homotopy category of Aalgebras (i.e. weak ocategories) is
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...
Algebra and Logic for Resourcebased Systems Modelling
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2009
"... ... often, models are required to be executable, as a simulation, on a computer. In this paper, we present some contributions to the processtheoretic and logical foundations of discreteevent modelling with resources and processes. We present a process calculus with an explicit representation of re ..."
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Cited by 17 (10 self)
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... often, models are required to be executable, as a simulation, on a computer. In this paper, we present some contributions to the processtheoretic and logical foundations of discreteevent modelling with resources and processes. We present a process calculus with an explicit representation of resources in which processes and resources coevolve. The calculus is closely connected to a logic that may be used as a specification language for properties of models. The logic is strong enough to allow requirements that a system has certain structure; for example, that it is a parallel composite of subsystems. This work consolidates, extends, and improves upon aspects of earlier work of ours in this area. An extended example, consisting of a semantics for a simple parallel programming language, indicates a connection with separating logics for concurrency.
Exact Completions and Toposes
 University of Edinburgh
, 2000
"... Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and ..."
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Cited by 13 (4 self)
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Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to
Systems Modelling via Resources and Processes: Philosophy, Calculus, Semantics, and Logic
 GDP FESTSCHRIFT ENTCS, TO APPEAR
"... We describe a programme of research in resource semantics, concurrency theory, bunched logic, and stochastic processes, as applied to mathematical systems modelling. Motivated by a desire for structurally and semantically rigorous discrete event modelling tools, applicable to enterprisescale as wel ..."
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Cited by 9 (6 self)
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We describe a programme of research in resource semantics, concurrency theory, bunched logic, and stochastic processes, as applied to mathematical systems modelling. Motivated by a desire for structurally and semantically rigorous discrete event modelling tools, applicable to enterprisescale as well as componentscale systems, we introduce a new approach to compositional reasoning based on a development of SCCS with an explicit model of resource. Our calculus models the coevolution of resources and processes with synchronization constrained by the availability of resources. We provide a simple denotational semantics as a parametrization of Abramsky’s synchronization trees semantics for SCCS. We also provide a logical characterization, analogous to HennessyMilner logic’s characterization of bisimulation in CCS, of bisimulation between resource processes which is compositional in the concurrent and local structure of systems. We discuss applications to ideas such as location and access control.