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:

: In certain areas of artificial intelligence there is need to represent continuous change and to make statements that are interpreted with respect to time intervals rather than time points. To this end we develop a modal temporal logic based on time intervals, a logic which can be viewed as a generalization of pointbased modal temporal logic. We discuss related logics, give an intuitive presentation of the new logic, and define its formal syntax and semantics. We make no assumption about the underlying nature of time, allowing it to be discrete (such as the natural numbers) or continuous (such as the rationals or the reals), linear or branching, complete (such as the reals) or not (such as the rationals). We show, however, that there are formulas in the logic that allow us to distinguish all these situations. We also give a translation of our logic into first-order logic, which allows us to apply some results on first-order logic to our modal one. Finally, we consider the difficulty o...

It has been argued that knowledge is a useful tool for designing and analyzing complex systems. The notion of knowledge that seems most relevant in this context is an external, information-based notion that can be shown to satisfy all the axioms of the modal logic S5. We carefully examine the properties of this notion of knowledge and show that they depend crucially, and in subtle ways, on assumptions we make about the system and about the language used for describing knowledge. We present a formal model in which we can capture various assumptions frequently made about systems, such as whether they are deterministic or nondeterministic, whether knowledge is cumulative (which means that processes never "forget"), and whether or not the "environment" affects the state transitions of the processes. We then show that under some assumptions about the system and the language, certain states of knowledge are not attainable and the axioms of S5 do not completely characterize the pr...

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 }

A simple extension of the propositional temporal logic of linear time is proposed. The extension consists of strengthening the until operator by indexing it with the regular programs of propositional dynamic logic (PDL). It is shown that DLTL, the resulting logic, is expressively equivalent to S1S, the monadic second-order theory of !-sequences. In fact a sublogic of DLTL which corresponds to propositional dynamic logic with a linear time semantics is already as expressive as S1S. We pin down in an obvious manner the sublogic of DLTL which correponds to the first order fragment of S1S. We show that DLTL has an exponential time decision procedure. We also obtain an axiomatization of DLTL. Finally, we point to some natural extensions of the approach presented here for bringing together propositional dynamic and temporal logics in a linear time setting.

Abstract. We propose a simple compositional program logic for an imperative extension of call-by-value PCF, built on Hoare logic and our preceding work on program logics for pure higher-order functions. A systematic use of names and operations on them allows precise and general description of complex higher-order imperative behaviour. The proof rules of the logic exactly follow the syntax of the language and can cleanly embed, justify and extend the standard proof rules for total correctness of Hoare logic. The logic offers a foundation for general treatment of aliasing and local state on its basis, with minimal extensions. After establishing soundness, we prove that valid assertions for programs completely characterise their behaviour up to observational congruence, which is proved using a variant of finite canonical forms. The use of the logic is illustrated through reasoning examples which are hard to assert and infer using existing program logics.

Boolean logic treats disjunction and conjunction symmetrically and algebraically. The corresponding operations for computation are respectively nondeterminism (choice) and concurrency. Petri nets treat these symmetrically but not algebraically, while event structures treat them algebraically but not symmetrically. Here we achieve both via the notion of an event space as a poset with all nonempty joins representing concurrence and a top representing the unreachable event. The symmetry is with the dual notion of state space, a poset with all nonempty meets representing choice and a bottom representing the start state. The algebra is that of a parallel programming language expanded to the language of full linear logic, Girard's axiomatization of which is satisfied by the event space interpretation of this language. Event spaces resemble finite dimensional vector spaces in distinguishing tensor product from direct product and in being isomorphic to their double dual, but differ from them i...

DATA TYPES A. Comparing Type Objects There has been as much confusion over type identity as there has been over object identity, although the type identity problem is usually referred to as the type equivalence problem [Aho86,s.6.3] [Wegbreit74] [Welsh77]. The type identity problem is to determine when two types are equal, so that type checking can be done in a programming language. 22 Algol-68 takes the point of view of "structural" equivalence, in which nonrecursive types that are built up from primitive types using the same type constructors in the same order should compare equal, while Ada takes the point of view of "name" equivalence, in which types are equivalent if and only if they have the same name. We will ignore the software engineering issues of which kind of type equivalence makes for better-engineered programs, and focus on the basic issue of type equivalence itself. We note that if a type system offers the type TYPE---i.e., it offers first-class representations of typ...

Abstract. People often need to reason about policy changes before they are adopted. For example, suppose a website manager knows that users want to enter her site without going through the welcome page. To decide whether or not to permit this, the wise manager will consider the consequences of modifying the policies (e.g., would this allow users to bypass advertisements and legal notices?). Similiarly, people often need to compare policy sets. For example, consider a person who wants to buy health insurance. Before choosing a provider, the customer will want to compare the different policies. In other words, the customer wants to reason about the effect of choosing one policy set over another. We introduce a logic, based on propositional dynamic logic, in which these tasks can be done. We give a sound and complete axiomatization for our logic, and also show that it is decidable. More precisely, the satisfiability problem is decidable in nondeterministic exponential time. 1

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...