Kleene under a Demonic Star

Desharnais, Jules; Möller, Bernhard; Tchier, Fairouz

doi:10.1007/3-540-45499-3_26

Cited by 14 publications

(14 citation statements)

References 18 publications

(15 reference statements)

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“…Other researchers [6] advocated a different proof style based on monotypes and residuals. We have shown in the present note that all the results can be generalized by using the monotypes and the residuals introduced by [16].…”

Section: Discussionmentioning

confidence: 78%

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Demonic semantics: using monotypes and residuals

Tchier

2004

International Journal of Mathematics and Mathematical Sciences

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Relations and relational operators can be used to define the semantics of programming languages. The operations ∨ and • serve to give angelic semantics by defining a program to go right when there is a possibility to go right. On the other hand, the demonic operations and do the opposite: if there is a possibility to go wrong, a program whose semantics is given by these operators will go wrong; it is the demonic semantics. This type of semantics is known at least since Dijkstra's introduction of the language of guarded commands. Recently, there has been a growing interest in demonic relational semantics of sequential programs. Usually, a construct is given an ad hoc semantic definition based on an intuitive understanding of its behavior. In this note, we show how the notion of relational flow diagram (essentially a matrix whose entries are relations on the set of states of the program), introduced by Schmidt, can be used to give a single demonic definition for a wide range of programming constructs. This research had originally been carried out by J. Desharnais and F. Tchier (1996) in the same framework of the binary homogeneous relations. We show that all the results can be generalized by using the monotypes and the residuals introduced by Desharnais et al. (2000).2000 Mathematics Subject Classification: 18C10, 18C50, 68Q55, 68Q65, 03B70, 06B35.1. Introduction. The approaches to semantics are categorized as operational, axiomatic, or denotational. We will be concerned with the operational and the denotational semantics of nondeterministic programs. The operational semantics is described by the relation between the initial and final states. In our case, we consider the worst execution of the program; that is, we suppose that the program behaves as badly as possible according to the demonic relational semantics . Usually this last one is given an ad hoc semantics definition based on an intuitive understanding of the behavior of the program. Denotational semantics has been introduced by Scott and Strachey. To give the denotational semantics, we associate to a program a mathematical object. In our case, this object is a flow diagram which is a graph whose arrows are weighted by the different steps of the program. The operations are "the demonic choice" and "demonic composition." In this note, we show how the notion of the flow diagram can be exploited to give single demonic operational semantics (with only demonic operators) for a wide range of programming constructs.

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Section: Discussionmentioning

confidence: 78%

“…This research had originally been carried out by J. Desharnais and F. Tchier (1996) in the same framework of the binary homogeneous relations. We show that all the results can be generalized by using the monotypes and the residuals introduced by Desharnais et al (2000).…”

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confidence: 91%

Demonic semantics: using monotypes and residuals

Tchier

2004

International Journal of Mathematics and Mathematical Sciences

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“…Associativity follows from the fact that relational composition is associative. (3,3), (4, 4)} From the composition table and the definition of domain, one obtains…”

Section: Non-closure Of Sc1 Under Compositionmentioning

confidence: 99%

“…is called type implication, an operation which is useful for dealing with assertions in demonic semantics and which enjoys many useful properties, see [3].…”

Section: A Modal Characterizationmentioning

confidence: 99%

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Characterizing determinacy in Kleene algebras

Desharnais

Möller

2001

Information Sciences

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Elements of Kleene algebras can be used, among others, as abstractions of the inputoutput semantics of nondeterministic programs or as models for the association of pointers with their target objects. In the first case, one seeks to distinguish the subclass of elements that correspond to deterministic programs. In the second case one is only interested in functional correspondences, since it does not make sense for a pointer to point to two different objects.We discuss several candidate notions of determinacy and clarify their relationship. Some characterizations that are equivalent in the case where the underlying Kleene algebra is an (abstract) relation algebra are not equivalent for general Kleene algebras.

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Demonic Algebra with Domain

Carufel

Desharnais

2006

Relations and Kleene Algebra in Computer Science

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Kleene under a Demonic Star

Cited by 14 publications

References 18 publications

Demonic semantics: using monotypes and residuals

Demonic semantics: using monotypes and residuals

Characterizing determinacy in Kleene algebras

Demonic Algebra with Domain

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