Chris Brink

Relational methods in computer science

No wonder you activities are, reading will be always needed. It is not only to fulfil the duties that you need to finish in deadline time. Reading will encourage your mind and thoughts. Of course, reading will greatly develop your experiences about everything. Reading relational methods in computer science is also a way as one of the collective books that gives many advantages. The advantages are not only for you, but for the other peoples with those meaningful benefits.

A Verisimilar Ordering of Theories Phrased in a Propositional Language

Popper introduced the concept of verisimilitude in the early sixties; Miller and Tichy deflated his definition in the early seventies. These facts, and the ensuing debate on verisimilitude, have been well chronicled in the pages of this Journal, as a look at, for example, Oddie [1981] and Urbach [19831 will show. Various attempts have been made to construct an acceptable definition of verisimilitude; these have mostly centred around the idea of distance from the truth. But this shared approach has not led to any substantial agreement. Moreover, of late there has been growing pessimism concerning the very possibility of success. This is due mostly to an argument first raised in Miller [1974], which says that our intuitions concerning verisimilitude are dependent on the language in which theories are expressed. In this paper we do not tread any of the well-worn paths: this is not another de novo investigation of the issues involved in verisimilitude. It is rather the serendipitous application to this concept of a certain construction applied to algebraic structures. Namely, to any structure there corresponds its power structure, essentially built up by taking the power relation of each relation in that structure. And for any relation R between elements of a set A, its power relation R+ relates subsets of A in a way dependent on R. It turns out that there is a natural power relation to be found between formulae of a propositional language. We offer this relation for consideration as a verisimilar ordering of theories phrased in a proposition language. Our approach, then, if we are to be charged with one, is to think of verisimilitude as an ordering relation between theories, and to present a model of this relation. We agree that an acceptable definition of verisimilitude should lead to an ordering of theories phrased in a first-order language. Nevertheless, for a start, to illustrate the ideas involved, and because the subject

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.

Subsumption computed algebraically

Abstract This paper deals with terminological representation languages for KL-ONE -type knowledge representation systems. Such languages are based on the two primitive syntactic types called concepts and roles, which are usually represented model-theoretically as sets and binary relations, respectively. Rather than following the model-theoretic route, we show that the semantics can be naturally accomodated in the context of an equational algebra of relations interacting with sets. Equational logic is then a natural vehicle for computing subsumptions, both of concepts and of roles. We thus propose the algebraic rather than model-theoretic computation of subsumption.

Verisimilitude: views and reviews

This paper is both a survey and a review of the current state of the debate concerning verisimilitude. As a survey it is intended for the interested outsider who wants both easy access to and some comparison between the respective approaches. As a review it covers the first three books on the topic: those of Oddie. Niiniluoto and Kuipers.

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.

SECOND-ORDER BOOLEAN ALGEBRAS

Opgedra aan Prof. Hennie Schutte by geleentheid van sy sestigste verjaarsdag. Abstract A Boolean algebra is the algebraic version of a field of sets. The complex algebra C(B) of a Boolean algebra B is defined over the power set of B; it is a field of sets with extra operations. The notion of a second-order Boolean algebra is intended to be the algebraic version of the complex algebra of a Boolean algebra. To this end a representation theorem is proved.

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.

POWER STRUCTURES AND LOGIC

ABSTRACT This paper generalizes the concept of a power alge bra to that of a power structure, and gives three application of power structures to logic.

R⌝-algebras and R⌝-model structures as power constructs

In relevance logic it has become commonplace to associate with each logic both an algebraic counterpart and a relational counterpart. The former comes from the Lindenbaum construction; the latter, called a model structure, is designed for semantical purposes. Knowing that they are related through the logic, we may enquire after the algebraic relationship between the algebra and the model structure. This paper offers a complete solution for the relevance logic R⌝. Namely, R⌝-algebras and R⌝-model structures can be obtained from each other, and represented in terms of each other, by application of power constructions.