Browsing by Author "Brink, Chris"

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    A topological framework for program semantics
    (1995) Rewitzky, Ingrid Moira; Brink, Chris
    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.
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    Algebraic terminological representation
    (1991) Schmidt, Renate Anneliese; Brink, Chris
    This thesis investigates terminological representation languages, as used in KL-ONE-type knowledge representation systems, from an algebraic point of view. Terminological representation languages are based on two primitive syntactic types, called concepts and roles, which are usually interpreted model-theoretically as sets and relations, respectively. I propose an algebraic rather than a model-theoretic approach. I show that terminological representations can be naturally accommodated in equational algebras of sets interacting with relations, and I use equational logic as a vehicle for reasoning about concepts interacting with roles.
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    Formulas of first-order logic in distributive normal form
    (1997) Nelte, Karen; Brink, Chris; Kieseppä, Ilkka
    It was shown by Jaakko Hintikka that every formula of first-order logic can be written as a disjunction of formulas called constituents. Such a disjunction is called a distributive normal form of the formula. It is a generalization of the disjunctive normal form for propositional logic. However, there are some significant differences between these two normal forms, caused chiefly by the impossibility of defining the constituents in such a way that they are all consistent. Distributive normal forms and some of their properties are studied. For example, the size of distributive normal forms is examined, and although we can't determine exactly how many constituents (of each form) are consistent, it is shown that the vast majority are inconsistent. Hintikka's definition of trivial inconsistency is studied, and a new definition of trivial inconsistency is given in terms of a necessary condition for the consistency of a constituent which is stronger than the condition which Hintikka used in his definition of trivial inconsistency. An error in Hintikka's attempted proof of the completeness theorem of the theory of distributive normal forms is pointed out, and a similar completeness theorem is proved using the new definition of trivial inconsistency.
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    Logical presentations of domains
    (1993) Hulley, Hardy; Brink, Chris
    This thesis combines a fairly general overview of domain theory with a detailed examination of recent work which establishes a connection between domain theory and logic. To start with, the theory of domains is developed with such issues as the semantics of recursion and iteration; the solution of recursive domain equations; and non-determinism in mind. In this way, a reasonably comprehensive account of domains, as ordered sets, is given. The topological dimension of domain theory is then revealed, and the logical insights gained by regarding domains as topological spaces are emphasised. These logical insights are further reinforced by an examination of pointless topology and Stone duality. A few of the more prominent categories of domains are surveyed, and Stone-type dualities for the objects of some of these categories are presented. The above dualities are then applied to the task of presenting domains as logical theories. Two types of logical theory are considered, namely axiomatic systems, and Gentzen-style deductive systems. The way in which these theories describe domains is by capturing the relationships between the open subsets of domains.
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    Modelling the algebra of weakest preconditions
    (1991) Rewitzky, Ingrid Moira; Brink, Chris
    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.
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    Power constructs and propositional systems
    (1999) Britz, Katarina; Brink, Chris
    Propositional systems are deductively closed sets of sentences phrased in the language of some propositional logic. The set of systems of a given logic is turned into an algebra by endowing it with a number of operations, and into a relational structure by endowing it with a number of relations. Certain operations and relations on systems arise from some corresponding base operation or relation, either on sentences in the logic or on propositional valuations. These operations and relations on systems are called power constructs. The aim of this thesis is to investigate the use of power constructs in propositional systems. Some operations and relations on systems that arise as power constructs include the Tarskian addition and product operations, the contraction and revision operations of theory change, certain multiple- conclusion consequence relations, and certain relations of verisimilitude and simulation. The logical framework for this investigation is provided by the definition and comparison of a number of multiple-conclusion logics, including a paraconsistent three-valued logic of partial knowledge.