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Towards Semantics of SelfAdaptive Software
, 2000
"... When people perform computations, they routinely monitor their results, and try to adapt and improve their algorithms when a need arises. The idea of selfadaptive software is to implement this common facility of human mind within the framework of the standard logical methods of software engineering ..."
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Cited by 9 (1 self)
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When people perform computations, they routinely monitor their results, and try to adapt and improve their algorithms when a need arises. The idea of selfadaptive software is to implement this common facility of human mind within the framework of the standard logical methods of software engineering. The ubiquitous practice of testing, debugging and improving programs at the design time should be automated, and established as a continuing run time routine. Technically, the task thus requires combining functionalities of automated software development tools and of runtime environments. Such combinations lead not just to challenging engineering problems, but also to novel theoretical questions. Formal methods are needed, and the standard techniques do not suffice. As a first contribution in this direction, we present a basic mathematical framework suitable for describing selfadaptive software at a high level of semantical abstraction. A static view leads to a structure akin...
Logic Programming in Tau Categories
 in Computer Science Logic '94 , LNCS 933
, 1995
"... Many features of current logic programming languages are not captured by conventional semantics. Their fundamentally nonground character, and the uniform way in which such languages have been extended to typed domains, subject to constraints, suggest that a categorical treatment of constraint domai ..."
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Many features of current logic programming languages are not captured by conventional semantics. Their fundamentally nonground character, and the uniform way in which such languages have been extended to typed domains, subject to constraints, suggest that a categorical treatment of constraint domains, of programming syntax and of semantics may be closer in spirit to what declarative programming is really about, than conventional settheoretic semantics. We generalize the notion of a (manysorted) logic program and of a resolution proof, by defining them both over a (not necessarily free) category C , a category with products enriched with a mechanism for canonically manipulating nary relations [8]. Computing over this domain includes computing over the Herbrand Universe, and over equationally presented constraint domains as special cases. We give a categorical treatment of the fixpoint semantics of Kowalski and van Emden, which establishes completeness in a very general setting. 1 In...
Classifying Toposes for First Order Theories
 Annals of Pure and Applied Logic
, 1997
"... By a classifying topos for a firstorder theory T, we mean a topos E such that, for any topos F , models of T in F correspond exactly to open geometric morphisms F ! E . We show that not every (infinitary) firstorder theory has a classifying topos in this sense, but we characterize those which ..."
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Cited by 7 (3 self)
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By a classifying topos for a firstorder theory T, we mean a topos E such that, for any topos F , models of T in F correspond exactly to open geometric morphisms F ! E . We show that not every (infinitary) firstorder theory has a classifying topos in this sense, but we characterize those which do by an appropriate `smallness condition', and we show that every Grothendieck topos arises as the classifying topos of such a theory. We also show that every firstorder theory has a conservative extension to one which possesses a classifying topos, and we obtain a Heytingvalued completeness theorem for infinitary firstorder logic.
Integrals and valuations
, 2008
"... We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and mostly consist of explicit manipul ..."
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Cited by 6 (2 self)
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We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and mostly consist of explicit manipulations on a distributive lattice associated to a given Riesz space.
ELEMENTARY QUOTIENT COMPLETION
"... Abstract. We extend the notion of exact completion on a category with weak finite limits to Lawvere’s elementary doctrines. We show how any such doctrine admits an elementary quotient completion, which is the universal solution to adding certain quotients. We note that the elementary quotient compl ..."
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Cited by 5 (1 self)
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Abstract. We extend the notion of exact completion on a category with weak finite limits to Lawvere’s elementary doctrines. We show how any such doctrine admits an elementary quotient completion, which is the universal solution to adding certain quotients. We note that the elementary quotient completion can be obtained as the composite of two other universal constructions: one adds effective quotients, the other forces extensionality of morphisms. We also prove that each construction preserves comprehension.
An institutionindependent generalization of Tarski’s Elementary Chain Theorem
 Journal of Logic and Computation
, 2006
"... Abstract. We prove an institutional version of Tarski’s Elementary Chain Theorem applicable to a whole plethora of “firstorderaccessible” logics, which are, roughly speaking, logics whose sentences can be constructed from atomic formulae by means of classical firstorder connectives and quantifi ..."
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Abstract. We prove an institutional version of Tarski’s Elementary Chain Theorem applicable to a whole plethora of “firstorderaccessible” logics, which are, roughly speaking, logics whose sentences can be constructed from atomic formulae by means of classical firstorder connectives and quantifiers. These include the unconditional equational, positive, (Π ∪ Σ)0n, and full firstorder logics, as well as less conventional logics, used in computer science, such as hidden or rewriting logic.