A topological framework for program semantics

Program semantics can be viewed relationally as in relational semantics, algebraically as in predicate transformer semantics, logically as in information systems and order-theoretically as in denotational semantics. This can be compared to a common situation in non-classical logics. Namely, a logic can often be presented as a formal deductive system, as an algebra and as a relational structure, with each of the presentations derivable from each of the other two. The central hypothesis of this thesis is that this situation can serve as a paradigm for unifying the various versions of program semantics. Starting with a relational semantics based on certain ordered topological spaces, called Priestley spaces, and invoking the techniques of Priestley duality, an algebraic. a logical and an order-theoretic presentation of program semantics are derived. Each of these four presentations are also derivable from each of the other three. The topological model of program semantics based on Priestley spaces thus serves as a unifying framework for other versions of program semantics, essentially as in the logic-algebra-semantics paradigm

Modelling the algebra of weakest preconditions

In expounding the notions of pre- and postconditions, of termination and nontermination, of correctness and of predicate transformers I found that the same trivalent distinction played a major role in all contexts. Namely: Initialisation properties: An execution of a program always, sometimes or never starts from an initial state. Termination/nontermination properties: If it starts, the execution always, sometimes or never terminates. Clean-/messy termination properties: A terminating execution always, sometimes or never terminates cleanly. Final state properties: All, some or no final states of α from s have a given property

Monotonic Distributive Semilattices

In the study of algebras related to non-classical logics, (distributive) semilattices are always present in the background. For example, the algebraic semantic of the {→, ∧, ⊤}-fragment of intuitionistic logic is the variety of implicative meet-semilattices (Chellas 1980; Hansen 2003). In this paper we introduce and study the class of distributive meet-semilattices endowed with a monotonic modal operator m. We study the representation theory of these algebras using the theory of canonical extensions and we give a topological duality for them. Also, we show how our new duality extends to some particular subclasses.Fil: Celani, Sergio Arturo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Departamento de Matemática; ArgentinaFil: Menchón, María Paula. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Approximating Simulations

: Simulation rules provide a simple, sound and complete technique for proving that one data type refines another. This paper expresses the dependence of the soundness and completeness on the programming language by showing how sound and complete simulation rules for a given language are derived from those for its sublanguages. The dependence of the completeness on the semantic model is expressed using a J'onsson/Tarski duality translation between the relational and predicate transformer semantics of data types and simulation rules. Both these dependencies are considered in the context of a simple language for terminating programs. Keywords: Data refinement, data structure, formal semantics, J'onsson/Tarski duality, simulation rule. 1 Introduction Simulation rules have been introduced [10, 12, 8] as a technique for developing or verifying an implementation against its more abstract specification. Their importance for establishing refinement of abstract by concrete data structures resu..

Towards Automated Testing

This paper investigates an approach for automated testing. A system for generating questions and tests is described, and a case study is given where such questions and tests were used for continuous assessment in a first-year mathematics course. The very positive feedback from the students and staff indicates that this approach to continuous assessment is an excellent way of helping students master the course material and keep up to date, while simultaneously minimizing the time spent by lecturers in setting and marking tests