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Concurrent Kleene Algebra
"... A concurrent Kleene algebra offers, next to choice and iteration, operators for sequential and concurrent composition, related by an inequational form of the exchange law. We show applicability of the algebra to a partiallyordered trace model of program execution semantics and demonstrate its usefu ..."
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A concurrent Kleene algebra offers, next to choice and iteration, operators for sequential and concurrent composition, related by an inequational form of the exchange law. We show applicability of the algebra to a partiallyordered trace model of program execution semantics and demonstrate its usefulness by validating familiar proof rules for sequential programs (Hoare triples) and for concurrent ones (Jones’s rely/guarantee calculus). This involves an algebraic notion of invariants; for these the exchange inequation strengthens to an equational distributivity law. Most of our reasoning has been checked by computer.
On the Compositionality and Analysis of Algebraic HighLevel Nets
 RESEARCH REPORT A16, DIGITAL SYSTEMS LABORATORY
, 1991
"... This work discusses three aspects of net theory: compositionality of nets, analysis of nets and highlevel nets. Net theory has often been criticised for the difficulty of giving a compositional semantics to a net. In this work we discuss this problem form a category theoretic point of view. In cate ..."
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Cited by 3 (1 self)
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This work discusses three aspects of net theory: compositionality of nets, analysis of nets and highlevel nets. Net theory has often been criticised for the difficulty of giving a compositional semantics to a net. In this work we discuss this problem form a category theoretic point of view. In category theory compositionality is represented by colimits. We show how a highlevel net can be mapped into a lowlevel net that represents its behaviour. This construction is functorial and preserves colimits, giving a compositional semantics for these highlevel nets. Using this semantics we propose some proof rules for compositional reasoning with highlevel nets. Linear logic is a recently discovered logic that has many interesting properties. From a net theoretic point of view its interest lies in the fact that it is able to express resources in an analogous manner to nets. Several interpretations of Petri nets in terms of linear logic exist. In this work we study a model theoretic inter...
Foundations of Concurrent Kleene Algebra
"... A Concurrent Kleene Algebra offers two composition operators, one that stands for sequential execution and the other for concurrent execution [10]. In this paper we investigate the abstract background of this law in terms of independence relations on which a concrete trace model of the algebra is b ..."
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A Concurrent Kleene Algebra offers two composition operators, one that stands for sequential execution and the other for concurrent execution [10]. In this paper we investigate the abstract background of this law in terms of independence relations on which a concrete trace model of the algebra is based. Moreover, we show the interdependence of the basic properties of such relations and two further laws that are essential in the application of the algebra to a Jones style rely/guarantee calculus. Finally we reconstruct the trace model in a more abstract setting based on the notion of atoms from lattice theory.
Universität Augsburg Concurrent Kleene Algebra
, 2009
"... Abstract. A concurrent Kleene algebra offers, next to choice and iteration, two composition operators, one that stands for sequential execution and the other for concurrent execution. They are related by an inequational form of the exchange law. We show the applicability of the algebra to a partiall ..."
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Abstract. A concurrent Kleene algebra offers, next to choice and iteration, two composition operators, one that stands for sequential execution and the other for concurrent execution. They are related by an inequational form of the exchange law. We show the applicability of the algebra to a partiallyordered trace model of program execution semantics and demonstrate its usefulness by validating familiar proof rules for sequential programs (Hoare triples) and for concurrent programming (Jones’s rely/guarantee calculus). The latter involves an algebraic notion of invariants; for these the exchange inequation strengthens to an equational distributivity law. Most of our reasoning has been checked by computer. 1
Foundations of Concurrent Kleene Algebra
, 2009
"... Abstract. A Concurrent Kleene Algebra offers two composition operators, one that stands for sequential execution and the other for concurrent execution [9]. In this paper we investigate the abstract background of this law in terms of independence relations on which a concrete trace model of the alge ..."
Abstract
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Abstract. A Concurrent Kleene Algebra offers two composition operators, one that stands for sequential execution and the other for concurrent execution [9]. In this paper we investigate the abstract background of this law in terms of independence relations on which a concrete trace model of the algebra is based. Moreover, we show the interdependence of the basic properties of such relations and two further laws that are essential in the application of the algebra to a Jones style rely/guarantee calculus. Finally we reconstruct the trace model in a more abstract setting based on the notion of atoms from lattice theory. 1
Convergence in formal topology: a unifying notion
, 2013
"... Several variations on the definition of a Formal Topology exist in the literature. They differ on how they express convergence, the formal property corresponding to the fact that open subsets are closed under finite intersections. We introduce a general notion of convergence of which any previous ..."
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Several variations on the definition of a Formal Topology exist in the literature. They differ on how they express convergence, the formal property corresponding to the fact that open subsets are closed under finite intersections. We introduce a general notion of convergence of which any previous definition is a special case. This leads to a predicative presentation and inductive generation of locales (formal covers), commutative quantales (convergent covers) and suplattices (basic covers) in a uniform way. Thanks to our abstract treatment of convergence, we are able to specify categorically the precise sense according to which our inductively generated structures are free, thus refining Johnstone’s coverage theorem. We also obtain a natural and predicative version of a fundamental result by Joyal and Tierney: convergent covers (commutative quantales) correspond to commutative