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29
Possible Worlds and Resources: The Semantics of BI
 THEORETICAL COMPUTER SCIENCE
, 2003
"... The logic of bunched implications, BI, is a substructural system which freely combines an additive (intuitionistic) and a multiplicative (linear) implication via bunches (contexts with two combining operations, one which admits Weakening and Contraction and one which does not). BI may be seen to a ..."
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Cited by 46 (17 self)
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The logic of bunched implications, BI, is a substructural system which freely combines an additive (intuitionistic) and a multiplicative (linear) implication via bunches (contexts with two combining operations, one which admits Weakening and Contraction and one which does not). BI may be seen to arise from two main perspectives. On the one hand, from prooftheoretic or categorical concerns and, on the other, from a possibleworlds semantics based on preordered (commutative) monoids. This semantics may be motivated from a basic model of the notion of resource. We explain BI's prooftheoretic, categorical and semantic origins. We discuss in detail the question of completeness, explaining the essential distinction between BI with and without ? (the unit of _). We give an extensive discussion of BI as a semantically based logic of resources, giving concrete models based on Petri nets, ambients, computer memory, logic programming, and money.
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 32 (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
Resourcedistribution via Boolean constraints
 Proceedings of CADE14
, 1997
"... We consider the problem of searching for proofs in sequential presentations of logics with multiplicative (or intensional) connectives. Specifically, we start with the multiplicative fragment of linear logic and extend, on the one hand, to linear logic with its additives and, on the other, to the ..."
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Cited by 28 (8 self)
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We consider the problem of searching for proofs in sequential presentations of logics with multiplicative (or intensional) connectives. Specifically, we start with the multiplicative fragment of linear logic and extend, on the one hand, to linear logic with its additives and, on the other, to the additives of the logic of bunched implications, BI. We give an algebraic method for calculating the distribution of the sideformul in multiplicative rules which allows the occurrence or nonoccurrence of a formula on a branch of a proof to be determined once sufficient information is available. Each formula in the conclusion of such a rule is assigned a Boolean expression. As a search proceeds, a set of Boolean constraint equations is generated. We show that a solution to such a set of equations determines a proof corresponding to the given search. We explain a range of strategies, from the lazy to the eager, for solving sets of constraint equations. We indicate how to apply our methods systematically to large family of relevant systems. 1
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.
The Semantics of BI and Resource Tableaux
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2005
"... The logic of bunched implications, BI, provides a logical analysis of a basic notion of resource rich enough, for example, to form the logical basis for “pointer logic ” and “separation logic” semantics for programs which manipulate mutable data structures. We develop a theory of semantic tableaux f ..."
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Cited by 13 (1 self)
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The logic of bunched implications, BI, provides a logical analysis of a basic notion of resource rich enough, for example, to form the logical basis for “pointer logic ” and “separation logic” semantics for programs which manipulate mutable data structures. We develop a theory of semantic tableaux for BI, so providing an elegant basis for efficient theorem proving tools for BI. It is based on the use of an algebra of labels for BI’s tableaux to solve the resourcedistribution problem, the labels being the elements of resource models. For BI with inconsistency, , the challenge consists in dealing with BI’s Grothendieck topological models within such a proofsearch method, based on labels. We prove soundness and completeness theorems for a resource tableaux method TBI with respect to this semantics and provide a way to build countermodels from socalled dependency graphs. Then, from these results, we can define a new resource semantics of BI, based on partially defined monoids, and prove that this semantics is complete. Such a semantics, based on partiality, is closely related to the semantics of BI’s (intuitionistic) pointer and separation logics. Returning to the tableaux calculus, we propose a new version with liberalized rules for which the countermodels are closely related to the topological Kripke semantics of BI. As consequences of the relationships between semantics of BI and resource tableaux, we prove two strong new results for propositional BI: its decidability and the finite model property with respect to topological semantics.
A Logical and Computational Theory of Located Resource
, 2008
"... Experience of practical systems modelling suggests that the key conceptual components of a model of a system are processes, resources, locations, and environment. In recent work, we have given a processtheoretic account of this view in which resources as well as processes are firstclass citizens. ..."
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Cited by 13 (9 self)
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Experience of practical systems modelling suggests that the key conceptual components of a model of a system are processes, resources, locations, and environment. In recent work, we have given a processtheoretic account of this view in which resources as well as processes are firstclass citizens. This process calculus, SCRP, captures the structural aspects of the semantics of the Demos2k modelling tool. Demos2k represents environment stochastically using a wide range of probability distributions and queuelike data structures. Associated with SCRP is a (bunched) modal logic, MBI, which combines the usual additive connectives of HennessyMilner logic with their multiplicative counterparts. In this paper, we complete our conceptual framework by adding to SCRP and MBI an account of a notion of location that is simple, yet sufficiently expressive to capture naturally a wide range of forms of location, both spatial and logical. We also provide a description of an extension of the Demos2k tool to incorporate this notion of location. 1
Plans, Affordances, and Combinatory Grammar
, 2002
"... The idea that natural language grammar and planned action are related systems has been implicit in psychological theory for more than a century. However, formal theories in the two domains have have tended to look very different. This article argues that both faculties share the formal character ..."
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Cited by 11 (1 self)
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The idea that natural language grammar and planned action are related systems has been implicit in psychological theory for more than a century. However, formal theories in the two domains have have tended to look very different. This article argues that both faculties share the formal character of applicative systems based on operations corresponding to the same two combinatory operations, namely functional composition and typeraising. Viewing them in this way suggests simpler and more cognitively plausible accounts of both systems, and suggests that the language faculty evolved in the species and develops in children by a rather direct adaptation of a more primitive apparatus for planning purposive action in the world by composing affordances of objects or tools. The knowledge representation that underlies such planning is also reflected in the natural language semantics of tense, mood, and aspect, which the paper begins by arguing provides the key to understanding both systems.
Elimination of Negation in a Logical Framework
, 2000
"... Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae (HHF) [15] cannot express directly negative information, although negation is a useful specification tool. Since negationasfailure does not fit well in a logical framework, especially one endowed with ..."
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Cited by 10 (3 self)
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Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae (HHF) [15] cannot express directly negative information, although negation is a useful specification tool. Since negationasfailure does not fit well in a logical framework, especially one endowed with hypothetical and parametric judgements, we adapt the idea of elimination of negation introduced in [21] for Horn logic to a fragment of higherorder HHF. This entails finding a middle ground between the Closed World Assumption usually associated with negation and the Open World Assumption typical of logical frameworks; the main technical idea is to isolate a set of programs where static and dynamic clauses do not overlap.
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.
Kripke Resource Models of a DependentlyTyped, Bunched lambdaCalculus (Extended Abstract)
, 1999
"... The lLcalculus is a dependent type theory with both linear and intuitionistic dependent function spaces. It can be seen to arise in two ways. Firstly, in logical frameworks, where it is the language of the RLF logical framework and can uniformly represent linear and other relevant logics. Second ..."
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Cited by 8 (6 self)
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The lLcalculus is a dependent type theory with both linear and intuitionistic dependent function spaces. It can be seen to arise in two ways. Firstly, in logical frameworks, where it is the language of the RLF logical framework and can uniformly represent linear and other relevant logics. Secondly, it is a presentation of the proofobjects of BI, the logic of bunched implications. BI is a logic which directly combines linear and intuitionistic implication and, in its predicate version, has both linear and intuitionistic quantifiers. The lLcalculus is the dependent type theory which generalizes both implications and quantifiers. In this paper, we describe the categorical semantics of the lLcalculus. This is given by Kripke resource models, which are monoidindexed sets of functorial Kripke models, the monoid giving an account of resource consumption. We describe a class of concrete, settheoretic models. The models are given by the category of families of sets, parametrized over a small monoidal category, in which the intuitionistic dependent function space is described in the established way, but the linear dependent function space is described using Day's tensor product.