The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and systems behaviour developed by Milner, Hennessy et al. based on operational semantics. • Logics of programs. Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of properties of processes). Moreover, the underlying logic is geometric, which can be computationally interpreted as the logic of observable properties—i.e. properties which can be determined to hold of a process on the basis of a finite amount of information about its execution. These ideas lead to the following programme:

this paper have natural algebraic and topological interpretations: Let L be the Boolean algebra of formulas of PL modulo the PL axioms of Section 4, and let rim= {nXlXe Z}, fL=/fXlXe m }

The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect to representable relation algebras, when expressed in their DA form are complete with respect to representable dynamic algebras. Moreover, whereas the theory of RA is undecidable, that of DA is decidable in exponential time. These results follow from representability of the free intensional dynamic algebras. Dept. of Computer Science, Stanford, CA 94305. This paper is based on a talk given at the conference Algebra and Computer Science, Ames, Iowa, June 2-4, 1988. It will appear in the proceedings of that conference, to be published by SpringerVerlag in the Lecture Notes in Computer Science series. This work was supported by the National Science Foundation under grant number CCR-8814921 ...

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 expressibility 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.

We present a two-sorted 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 set-forming 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 so-called terminological logics arising in knowledge representation have evolved a semantics best described as a calculus of relations interacting with sets.

Dynamic algebras combine the classes of Boolean (B 0 0) and regular (R [ ; ) algebras into a single finitely axiomatized variety (B R 3) resembling an R-module with "scalar" multiplication 3. The basic result is that is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main result is that free dynamic algebras are residually finite (i.e. factor as a subdirect product of finite dynamic algebras), important because finite separable dynamic algebras are isomorphic to Kripke structures. Applications include a new completeness proof for the Segerberg axiomatization of propositional dynamic logic, and yet another notion of regular algebra. Key words: Dynamic algebra, logic, program verification, regular algebra. This paper or...

Abstract not found

To appear in E. Orlowska, editor, Logic at Work , Kluwer Academic Publishers.

We re-examine the modal µ-calculus in the light of some classical theory of Boolean algebras and recent results on duality theory for a modal logic with fixed points. We propose interpreting formulas into a field of subsets of states instead of the full power set lattice used by Kozen. Under this interpretation we relate image compact modal frames with Scott continuity of the box modality, m-saturated transition systems and descriptive modal frames. Also, it is shown that the class of image compact modal frames satisfies the Hennessy-Milner property. We conclude by showing that for descriptive modal µ-frames the standard interpretation coincides with the one we proposed.