Demonic fixed points

We deal with a relational model for the demonic semantics of programs. The demonic semantics of a while loop is given as a fixed point of a function involving the demonic operators. This motivates us to investigate the fixed points of these functions. We give the expression of the greatest fixed point with respect to the demonic ordering (demonic inclusion) of the semantic function. We prove that this greatest fixed coincides with the least fixed point with respect to the usual ordering (angelic inclusion) of the same function. This is followed by an example of application

Kleene algebra with domain

We propose Kleene algebra with domain (KAD), an extension of Kleene algebra with two equational axioms for a domain and a codomain operation, respectively. KAD considerably augments the expressiveness of Kleene algebra, in particular for the specification and analysis of state transition systems. We develop the basic calculus, discuss some related theories and present the most important models of KAD. We demonstrate applicability by two examples: First, an algebraic reconstruction of Noethericity and well-foundedness; second, an algebraic reconstruction of propositional Hoare logic.Comment: 40 page

Demonic Kleene Algebra

Nous rappelons d’abord le concept d’algèbre de Kleene avec domaine (AKD). Puis, nous expliquons comment utiliser les opérateurs des AKD pour définir un ordre partiel appelé raffinement démoniaque ainsi que d’autres opérateurs démoniaques (plusieurs de ces définitions proviennent de la littérature). Nous cherchons à comprendre comment se comportent les AKD munies des opérateurs démoniaques quand on exclut les opérateurs angéliques usuels. C’est ainsi que les propriétés de ces opérateurs démoniaques nous servent de base pour axiomatiser une algèbre que nous appelons Algèbre démoniaque avec domaine et opérateur t-conditionnel (ADD-[opérateur t-conditionnel]). Les lois des ADD-[opérateur t-conditionnel] qui ne concernent pas l’opérateur de domaine correspondent à celles présentées dans l’article Laws of programming par Hoare et al. publié dans la revue Communications of the ACM en 1987. Ensuite, nous étudions les liens entre les ADD-[opérateur t-conditionnel] et les AKD munies des opérateurs démoniaques. La question est de savoir si ces structures sont isomorphes. Nous démontrons que ce n’est pas le cas en général et nous caractérisons celles qui le sont. En effet, nous montrons qu’une AKD peut être transformée en une ADD-[opérateur t-conditionnel] qui peut être transformée à son tour en l’AKD de départ. Puis, nous présentons les conditions nécessaires et suffisantes pour qu’une ADD-[opérateur t-conditionnel] puisse être transformée en une AKD qui peut être transformée à nouveau en l’ADD-[opérateur t-conditionnel] de départ. Les conditions nécessaires et suffisantes mentionnées précédemment font intervenir un nouveau concept, celui de décomposition. Dans un contexte démoniaque, il est difficile de distinguer des transitions qui, à partir d’un même état, mènent à des états différents. Le concept de décomposition permet d’y arriver simplement. Nous présentons sa définition ainsi que plusieurs de ses propriétés.We first recall the concept of Kleene algebra with domain (KAD). Then we explain how to use the operators of KAD to define a demonic refinement ordering and demonic operators (many of these definitions come from the literature). We want to know how do KADs with the demonic operators but without the usual angelic ones behave. Then, taking the properties of the KAD-based demonic operators as a guideline, we axiomatise an algebra that we call Demonic algebra with domain and t-conditional (DAD-[opérateur t-conditionnel]). The laws of DAD-[opérateur t-conditionnel] not concerning the domain operator agree with those given in the 1987 Communications of the ACM paper Laws of programming by Hoare et al. Then, we investigate the relationship between DAD-[opérateur t-conditionnel] and KAD-based demonic algebras. The question is whether every DAD-[opérateur t-conditionnel] is isomorphic to a KAD-based demonic algebra. We show that it is not the case in general. However, we characterise those that are. Indeed, we demonstrate that a KAD can be transformed into a DAD-[opérateur t-conditionnel] which can be transformed back into the initial KAD. We also establish necessary and sufficient conditions for which a DAD-[opérateur t-conditionnel] can be transformed into a KAD which can be transformed back into the initial DAD-[opérateur t-conditionnel]. Finally, we define the concept of decomposition. This notion is involved in the necessary and sufficient conditions previously mentioned. In a demonic context, it is difficult to distinguish between transitions that, from a given state, go to different states. The concept of decomposition enables to do it easily. We present its definition together with some of its properties

Characterizing determinacy in Kleene algebras

Elements of Kleene algebras can be used, among others, as abstractions of the input-output 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

Characterizing Determinacy in Kleene Algebras

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 di#erent 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

Characterizing determinacy in Kleene algebras (revised version)

Elements of Kleene algebras can be used, among others, as abstractions of the input-output 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