Steven Vickers

QUANTALES, OBSERVATIONAL LOGIC AND PROCESS SEMANTICS

Various notions of observing and testing processes are placed in a uniform algebraic framework in which observations are taken as constituting a quantale. General completeness criteria are stated, and proved in our applications.

Information systems for continuous posets

Abstract The method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive. This develops the results of Raney on completely distributive lattices and of Hoofman on continuous Scott domains, and also generalizes Smyths “R-structures”. Various constructions on continuous posets have neat descriptions in terms of these continuous information systems; here we describe Hoffmann-Lawson duality (which could not be done easily with R-structures) and Vietoris power locales. We also use the method to give a partial answer to a question of Johnstones: in the context of continuous posets. Vietoris algebras are the same as localic semilattices.

A universal characterization of the double powerlocale

The double powerlocale P(X) (found by composing, in either order, the upper and lower powerlocale constructions PU and PL) is shown to be isomolphic in [Locop, Set] to the double exponential SSX where S is the Sierpinski locale. Further PU(X) and PL(X) are shown to be the subobjects of P(X) comprising, respectively, the meet semilattice and join semilattice homomorphisms. A key lemma shows that, for any locales X and Y, natural transformations from SX (the presheaf Loc(_ × X, S)) to SY (i.e. Loc(_ × Y, S)) are equivalent to dcpo morphisms (Scott continuous maps) from the flame ΩX to ΩY. It is also shown that SX has a localic reflection in [Locop, Set] whose frame is the Scott topology on ΩX.The reasoning is constructive in the sense of topos validity.

Constructive points of powerlocales

Results of Bunge and Funk and of Johnstone, providing constructively sound descriptions of the global points of the lower and upper powerlocales, are extended here to describe the generalized points and proved in a way that displays in a symmetric fashion two complementary treatments of frames: as suplattices and as preframes. We then also describe the points of the Vietoris powerlocale. In each of two special cases, an exponential

Topical categories of domains

D (

Partial Horn logic and cartesian categories

being the Sierpinsky locale) is shown to be homeomorphic to a powerlocale: to the lower powerlocale when D is discrete, and to the upper powerlocale when D is compact regular.

Compactness in locales and in formal topology.

This paper shows how it is possible to express many techniques of categorical domain theory in the general context of topical categories (where ‘topical’ means internal in the category Top of Grothendieck toposes with geometric morphisms). The underlying topos machinery is hidden by using a geometric form of constructive mathematics, which enables toposes as ‘generalized topological spaces’ to be treated in a transparently spatial way, and also shows the constructivity of the arguments. The theory of strongly algebraic (SFP) domains is given as a case study in which the topical category is Cartesian closed.

Localic sup-lattices and tropological systems

Abstract A logic is developed in which function symbols are allowed to represent partial functions. It has the usual rules of logic (in the form of a sequent calculus) except that the substitution rule has to be modified. It is developed here in its minimal form, with equality and conjunction, as “partial Horn logic”. Various kinds of logical theory are equivalent: partial Horn theories, “quasi-equational” theories (partial Horn theories without predicate symbols), cartesian theories and essentially algebraic theories. The logic is sound and complete with respect to models in Set , and sound with respect to models in any cartesian (finite limit) category. The simplicity of the quasi-equational form allows an easy predicative constructive proof of the free partial model theorem for cartesian theories: that if a theory morphism is given from one cartesian theory to another, then the forgetful (reduct) functor from one model category to the other has a left adjoint. Various examples of quasi-equational theory are studied, including those of cartesian categories and of other classes of categories. For each quasi-equational theory T another, Cart ϖ T , is constructed, whose models are cartesian categories equipped with models of T . Its initial model, the “classifying category” for T , has properties similar to those of the syntactic category, but more precise with respect to strict cartesian functors.

Geometric logic in computer science

Abstract If a locale is presented by a “flat site”, it is shown how its frame can be presented by generators and relations as a dcpo. A necessary and sufficient condition is derived for compactness of the locale (and also for its openness). Although its derivation uses impredicative constructions, it is also shown predicatively using the inductive generation of formal topologies. A predicative proof of the binary Tychonoff theorem is given, including a characterization of the finite covers of the product by basic opens. The discussion is then related to the double powerlocale.

Strongly algebraic = SFP (topically)

The approach to process semantics using quantales and modules is topologized by considering tropological systems whose sets of states are replaced by locales and which satisfy a suitable stability axiom. A corresponding notion of localic sup-lattice (algebra for the lower powerlocale monad) is described, and it is shown that there are contravariant functors from sup-lattices to localic sup-lattices and, for each quantale Q, from left Q-modules to localic right Q-modules. A proof technique for third completeness due to Abramsky and Vickers is reset constructively, and an example of application to failures semantics is given.