Ingrid Rewitzky

Algebras for Galois-style connections and their discrete duality

Bounded distributive lattices with pairs of operators forming Galois, dual Galois, residuation and dual residuation connections, respectively, are considered. A discrete duality for these classes of algebras and the corresponding classes of relational systems is established. These results are extended to the fuzzy logic MTL.

Modelling angelic and demonic nondeterminism with multirelations

This paper presents an introduction to a calculus of binary multirelations, which can model both angelic and demonic kinds of non-determinism. The isomorphism between up-closed multirelations and monotonic predicate transformers allows a different view of program transformation, and program transformation calculations using multirelations are easier to perform in some circumstances. Multirelations are illustrated by modelling both kinds of nondeterministic behaviour in games and resource-sharing protocols.

Discrete Duality and Its Applications to Reasoning with Incomplete Information

We present general principles of establishing a duality between a class of algebras and a class of relational systems such that topology is not involved. We show how such a discrete duality contributes to proving completeness of logical systems and to correspondence theory. Next, we outline applications of discrete dualities to analysis of data in information systems with incomplete information in the rough set-style, and in contexts of formal concept analysis.

Monotone predicate transformers as up-closed multirelations

In the study of semantic models for computations two independent views predominate: relational models and predicate transformer semantics. Recently the traditional relational view of computations as binary relations between states has been generalised to multirelations between states and properties allowing the simultaneous treatment of angelic and demonic nondeterminism. In this paper the two-level nature of multirelations is exploited to provide a factorisation of up-closed multirelations which clarifies exactly how multirelations model nondeterminism. Moreover, monotone predicate transformers are, in the precise sense of duality, up-closed multirelations. As such they are shown to provide a notion of effectivity of a specification for achieving a given postcondition.

Three Dual Ontologies

In this paper we give an example of intertranslatability between an ontology of individuals (nominalism), an ontology of properties (realism), and an ontology of facts (factualism). We demonstrate that these three ontologies are dual to each other, meaning that each ontology can be translated into, and recaptured from, each of the others. The aim of the enterprise is to raise the possibility that, at least in some settings, there may be no need for considerations of ontological primacy. Whether the world is made up of things, or properties, or facts, may be no more than a matter of how we look at it.

Context Algebras, Context Frames, and Their Discrete Duality

The data structures dealt with in formal concept analysis are referred to as contexts. In this paper we study contexts within the framework of discrete duality. We show that contexts can be adequately represented by a class of sufficiency algebras called context algebras. On the logical side we define a class of context frames which are the semantic structures for context logic, a lattice-based logic associated with the class of context algebras. We prove a discrete duality between context algebras and context frames, and we develop a Hilbert style axiomatization of context logic and prove its completeness with respect to context frames. Then we prove a duality via truth theorem showing that both context algebras and context frames provide the adequate semantic structures for context logic. We discuss applications of context algebras and context logic to the specification and verification of various problems concerning contexts such as implications (attribute dependencies) in contexts, and derivation of implications from finite sets of implications.

Towards reasoning about Hoare relations

A logical framework is presented for defining semantics of programs that satisfy Hoare postulates. The two families of logical systems are given: modal systems and relational systems. In the modal systems semantics of Hoare-style programming languages is provided in terms of relations and sets, and in relational systems in terms of relations only. Proof theory for the given logics is presented.

Discrete Duality for Relation Algebras and Cylindric Algebras

Following the representation theorems for relation algebras and cylindric algebras presented in [5] and [7] we develop discrete duality for relation algebras and relation frames, and for cylindric algebras and cylindric frames.

Predicate transformers as power operations

In predicate transformer semantics, a program is represented as a mapping from predicates to predicates. In relational semantics, a program is represented as an (input-output) binary relation over some state space. We show how each of these approaches can be obtained from the other by using thepower construction.

Structures with Multirelations, their Discrete Dualities and Applications

In this paper we show that the problem of discrete duality can be extended beyond the clasical setting of duality between a class of algebras and a class of relational structures. Namely, for some classes of algebras, the relevant dual structures are the structures with multirelations. Several applications of multirelations will be described.