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Between Functions and Relations in Calculating Programs
, 1992
"... This thesis is about the calculational approach to programming, in which one derives programs from specifications. One such calculational paradigm is Ruby, the relational calculus developed by Jones and Sheeran for describing and designing circuits. We identify two shortcomings with derivations made ..."
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This thesis is about the calculational approach to programming, in which one derives programs from specifications. One such calculational paradigm is Ruby, the relational calculus developed by Jones and Sheeran for describing and designing circuits. We identify two shortcomings with derivations made using Ruby. The first is that the notion of a program being an implementation of a specification has never been made precise. The second is to do with types. Fundamental to the use of type information in deriving programs is the idea of having types as special kinds of programs. In Ruby, types are partial equivalence relations (pers). Unfortunately, manipulating some formulae involving types has proved difficult within Ruby. In particular, the preconditions of the `induction' laws that are much used within program derivation often work out to be assertions about types; such assertions have typically been verified either by informal arguments or by using predicate calculus, rather than by ap...
Characterizing Determinacy in Kleene Algebras
 INFORMATION SCIENCES
, 2000
"... Elements of Kleene algebras can be used, among others, as abstractions of the inputoutput semantics of nondeterministic programs or as models for the association of pointers with their target objects. In the first case, one seeks to distinguish the subclass of elements that correspond to determinist ..."
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Cited by 12 (5 self)
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Elements of Kleene algebras can be used, among others, as abstractions of the inputoutput semantics of nondeterministic programs or as models for the association of pointers with their target objects. In the first case, one seeks to distinguish the subclass of elements that correspond to deterministic programs. In the second case one is only interested in functional correspondences, since it does not make sense for a pointer to point to two di#erent objects. We discuss several candidate notions of determinacy and clarify their relationship. Some characterizations that are equivalent in the case where the underlying Kleene algebra is an (abstract) relation algebra are not equivalent for general Kleene algebras.
Logic of Predicates Versus Linear Logic
 ICS PAS Reports, Vol 795
, 1995
"... This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one envisaged by Girard. The latter, cf. [11], is prooftheoretic in nature. Firstly, every sequent of classical, resp., intuitionistic, logic is translated into a sequent of commutative ..."
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Cited by 2 (2 self)
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This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one envisaged by Girard. The latter, cf. [11], is prooftheoretic in nature. Firstly, every sequent of classical, resp., intuitionistic, logic is translated into a sequent of commutative linear logic with exponentials. Then one shows that the former can be proved classically, resp., intuitionistically, iff its translation can be proved linearly. Here it is shown that every theory of classical logic of predicates with equality lives in a sufficiently rich theory built over a noncommutiative intuitionistic substructural logic: the logic of predicates with explicit substitution. This perspective does not require to call upon
Towards program development, specification and verification with Isabelle
, 1995
"... The purpose of this paper is to report on our experiments to use Isabelle  a generic theorem prover  as a universal environment within which specification, development and verification of imperative programs can be performed. The use of a theorem prover for the programming tasks is most appropri ..."
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The purpose of this paper is to report on our experiments to use Isabelle  a generic theorem prover  as a universal environment within which specification, development and verification of imperative programs can be performed. The use of a theorem prover for the programming tasks is most appropriate when the processes of program specification, development and verification can be presented as logical activities. In our case this is achieved by adopting pLSD  a novel programming logic.
Equality In Linear Logic
, 1996
"... reference is [Ros]). Quantales were introduced by Mulvey ([Mul]) as an algebraic tool for studying representations of noncommutative C algebras. Informally, a quantale is a complete lattice Q equipped with a product distributive over arbitrary sup's. The importance of quantales for Linear ..."
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reference is [Ros]). Quantales were introduced by Mulvey ([Mul]) as an algebraic tool for studying representations of noncommutative C algebras. Informally, a quantale is a complete lattice Q equipped with a product distributive over arbitrary sup's. The importance of quantales for Linear Logic is revealed in Yetter's work ([Yet]), who proved that semantics of classical linear logic is given by a class of quantales, named Girard quantales, which coincides with Girard's phase semantics. An analogous result is obtained for a sort of noncommutative linear logic, as well as intuitionistic linear logic without negation, which suggest that the utilisation of the theory of quantales (or even weaker structures, such that *autonomous posets) might be fruitful in studying the semantic of several variants of linear logic. As usual, we denote the order in a lattice by , while W and V denote the operatio
Topological Relational Quantales
"... Abstract We introduce and present results about a class of quantales, the topological relational quantales, that can be associated with tuples (X,R) such that X is a topological space and R is a lowersemicontinuous equivalence relation on X (see Definition 1 below). Our motivating case studies and ..."
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Abstract We introduce and present results about a class of quantales, the topological relational quantales, that can be associated with tuples (X,R) such that X is a topological space and R is a lowersemicontinuous equivalence relation on X (see Definition 1 below). Our motivating case studies and examples of topological relational quantales include concrete quantales such as the relational quantales of [11] and the quantale Pen introduced in [12] (see Examples 1–5 below). In this setting, we focus in particular on the question of the representability of quantales into quantales of binary relations on a set, which has already been studied by some authors in the literature: for instance, in [1] it is shown that every quantale Q can be orderembedded into a quantale of relations in such a way that the noncommutative product is represented as relational composition, but this method does not extend to involutive quantales and the join of Q is not in general represented as the union. In [10], it is shown that any involutive quantale is embeddable into a quantale of joinsemilattice endomorphisms, which give back the quantales of relations when the sup lattice is a powerset. We present sufficient conditions for the representability of unital involutive quantales