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29
A powerdomain construction
 SIAM J. of Computing
, 1976
"... Abstract. We develop a powerdomain construction, [.], which is analogous to the powerset construction and also fits in with the usual sum, product and exponentiation constructions on domains. The desire for such a construction arises when considering programming languages with nondeterministic featu ..."
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Cited by 225 (20 self)
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Abstract. We develop a powerdomain construction, [.], which is analogous to the powerset construction and also fits in with the usual sum, product and exponentiation constructions on domains. The desire for such a construction arises when considering programming languages with nondeterministic features or parallel features treated in a nondeterministic way. We hope to achieve a natural, fully abstract semantics in which such equivalences as (pparq)=(qparp) hold. The domain (D Truthvalues) is not the right one, and instead we take the (finitely) generable subsets of D. When D is discrete they are ordered in an elementwise fashion. In the general case they are given the coarsest ordering consistent, in an appropriate sense, with the ordering given in the discrete case. We then find a restricted class of algebraic inductive partial orders which is closed under [. as well as the sum, product and exponentiation constructions. This class permits the solution of recursive domain equations, and we give some illustrative semantics using 5[.]. It remains to be seen if our powerdomain construction does give rise to fully abstract semantics, although such natural equivalences as the above do hold. The major deficiency is the lack of a convincing treatment of the fair parallel construct. 1. Introduction. When one follows the ScottStrachey approach to the
Impulse differential inclusions: A viability approach to hybrid systems
 IEEE Transactions on Automatic Control
, 2002
"... Abstract. Impulse differential inclusions are introduced as a framework for modelling hybrid phenomena. Connections to standard problems in area of hybrid systems are discussed. Conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an i ..."
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Cited by 41 (5 self)
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Abstract. Impulse differential inclusions are introduced as a framework for modelling hybrid phenomena. Connections to standard problems in area of hybrid systems are discussed. Conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion. For sets that violate these conditions, methods are developed for approximating their viability and invariance kernels, that is the largest subset that is viable or invariant under the action of the impulse differential inclusion. The results are demonstrated on examples. 1.
Chaincomplete posets and directed sets with applications. Algebra univers
, 1976
"... Let a poset P be called chaincomplete when every chain, including the empty chain, has a sup in P. Many authors have investigated properties of posets atisfying some sort of chaincompleteness condition (see [,11, [31, [6], I71, [17], [,181, ['191, [,211, [,221), and used them in a variety o ..."
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Cited by 33 (0 self)
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Let a poset P be called chaincomplete when every chain, including the empty chain, has a sup in P. Many authors have investigated properties of posets atisfying some sort of chaincompleteness condition (see [,11, [31, [6], I71, [17], [,181, ['191, [,211, [,221), and used them in a variety of applications. In this paper we study the
Hoare Logics for Recursive Procedures and Unbounded Nondeterminism
 COMPUTER SCIENCE LOGIC (CSL 2002), VOLUME 2471 OF LNCS
, 2002
"... This paper presents sound and complete Hoare logics for partial and total correctness of recursive parameterless procedures in the context of unbounded nondeterminism. For total correctness, the literature so far has either restricted recursive procedures to be deterministic or has studied unbounde ..."
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Cited by 28 (3 self)
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This paper presents sound and complete Hoare logics for partial and total correctness of recursive parameterless procedures in the context of unbounded nondeterminism. For total correctness, the literature so far has either restricted recursive procedures to be deterministic or has studied unbounded nondeterminism only in conjunction with loops rather than procedures. We consider both single procedures and systems of mutually recursive procedures. All proofs have been checked with the theorem prover Isabelle/HOL.
Peirce Algebras
, 1992
"... We present a twosorted algebra, called a Peirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a relationforming o ..."
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Cited by 25 (10 self)
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We present a twosorted algebra, called a Peirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a relationforming operator on sets (the Peirce product of Boolean modules) and a setforming operator on relations (a cylindrification operation). Two applications of Peirce algebras are given. The first points out that Peirce algebras provide a natural algebraic framework for modelling certain programming constructs. The second shows that the socalled terminological logics arising in knowledge representation have evolved a semantics best described as a calculus of relations interacting with sets.
A CSP Approach To Action Systems
, 1992
"... The communicating sequential processes (CSP) formalism, introduced by Hoare [Hoa85], is an eventbased approach to distributed computing. The actionsystem formalism, introduced by Back & KurkiSuonio [BKS83], is a statebased approach to distributed computing. Using weakestprecondition formula ..."
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Cited by 24 (7 self)
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The communicating sequential processes (CSP) formalism, introduced by Hoare [Hoa85], is an eventbased approach to distributed computing. The actionsystem formalism, introduced by Back & KurkiSuonio [BKS83], is a statebased approach to distributed computing. Using weakestprecondition formulae, Morgan [Mor90a] has defined a correspondence between action systems and the failuresdivergences model for CSP. Simulation is a proof technique for showing refinement of action systems. Using the correspondence of [Mor90a], Woodcock & Morgan [WM90] have shown that simulation is sound and complete in the CSP failuresdivergences model. In this thesis, Morgan's correspondence is extended to the CSP infinitetraces model [Ros88] in order to deal more properly with unbounded nondeterminism. It is shown that simulation is sound in the infinitetraces model, though completeness is lost in certain cases. The new correspondence is then extended to include a notion of internal action. This allows the ...
Exploring Summation and Product Operators in the Refinement Calculus
 Mathematics of Program Construction
, 1994
"... Product and summation operators for predicate transformers were introduced by Naumann [21] and by Martin [15] using category theoretic considerations. In this paper, we formalise these operators in the higher order logic approach to the refinement calculus of [4], and examine various algebraic prope ..."
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Cited by 19 (10 self)
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Product and summation operators for predicate transformers were introduced by Naumann [21] and by Martin [15] using category theoretic considerations. In this paper, we formalise these operators in the higher order logic approach to the refinement calculus of [4], and examine various algebraic properties of these operators. There are several motivating factors for this analysis. The product operator provides a model of simultaneous execution of statements, while the summation operator provides a simple model of late binding. We also generalise the product operator slightly to form an operator that corresponds to conjunction of specifications. We examine several applications of the these operators showing, for example, how a combination of the product and summation operators could be used to model inheritance in an objectoriented programming language. 1 Introduction Dijkstra introduced weakestprecondition predicate transformers as a means of verifying total correctness properties of ...
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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Cited by 19 (0 self)
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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
Extremal Solutions of Inequations over Lattices with Applications to Supervisory Control
 Theoretical Computer Science
"... We study the existence and computation of extremal solutions of a system of inequations defined over lattices. Using the KnasterTarski fixed point theorem, we obtain sufficient conditions for the existence of supremal as well as infimal solution of a given system of inequations. Iterative technique ..."
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Cited by 16 (8 self)
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We study the existence and computation of extremal solutions of a system of inequations defined over lattices. Using the KnasterTarski fixed point theorem, we obtain sufficient conditions for the existence of supremal as well as infimal solution of a given system of inequations. Iterative techniques are presented for the computation of the extremal solutions whenever they exist, and conditions under which the termination occurs in a single iteration are provided. These results are then applied for obtaining extremal solutions of various inequations that arise in computation of maximally permissive supervisors in control of logical discrete event systems (DESs) first studied by Ramadge and Wonham. Thus our work presents a unifying approach for computation of supervisors in a variety of situations. Keywords: Fixed points, lattices, inequations, discrete event systems, supervisory control, language theory. 1 Introduction Given a set X and a function f : X ! X, x 2 X is called a fixed p...
A zeroone law for logic with a fixedpoint operator
 Inform. and Control
"... The logic obtained by adding the leastfixedpoint operator to firstorder logic was proposed as a query language by Aho and Ullman (in "Proc. 6th ACM Sympos. on Principles of Programming Languages, " 1979, pp. 110120) and has been studied, particularly in connection with finite models, b ..."
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Cited by 14 (6 self)
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The logic obtained by adding the leastfixedpoint operator to firstorder logic was proposed as a query language by Aho and Ullman (in "Proc. 6th ACM Sympos. on Principles of Programming Languages, " 1979, pp. 110120) and has been studied, particularly in connection with finite models, by numerous authors. We extend to this logic, and to the logic containing the more powerful iterativefixedpoint operator, the zeroone law proved for firstorder logic in (Glebskii, Kogan, Liogonki, and Talanov (1969), Kibernetika 2, 3142; Fagin (1976), J. Symbolic Logic 41, 5058). For any sentence q ~ of the extended logic, the proportion of models of q ~ among all structures with universe {1, 2,..., n} approaches 0 or 1 as n tends to infinity. We also show that the problem of deciding, for any cp, whether this proportion approaches 1 is complete for exponential time, if we consider only q)'s with a fixed finite vocabulary (or vocabularies of bounded arity) and complete for doubleexponential time if ~0 is unrestricted. In addition, we establish some related results. © 1985 Academic Press, Inc.