Domain Theory in Logical Form

Abstract Abramsky, S., Domain theory in logical form, Annals of Pure and Applied Logic 51 (1991) 1–77. • Domain theory, the mathematical theory of computation introduced by Scott as a foundation for detonational semantics • The theory of concurrency and systems behaviour developed by Milner, Hennesy based on operational semantics. • Logics of programs Stone duality provides a junction between semantics (spaces of points=detonations 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. (1) A metalanguage is introduced, comprising • types = universes of discourse for various computational situations; • terms = programs = syntactic intensions for models or points. (2) A standard denotational interpretation of the metalanguage is given, assigning domains to types and domain elements to terms. (3) The metalanguage is also given a logical interpretation, in which types are interpreted as propositional theories and terms are interpreted via a program logic, which axiomatises the properties they satisfy. (4) The two interpretations are related by showing that they are Stone duals of each other. Hence, semantics and logic are guaranteed to be in harmony with each other, and in fact each determines the other up to isomorphism. (5) This opens the way to a whole range of applications. Given a denotational description of a computational situation in our metalanguage, we can turn the handle to obtain a logic for that situation.