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Quantum logic in dagger kernel categories
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"... This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial inject ..."
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This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/ordertheoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and andthen connectives are obtained, as adjoints, via the existentialpullback adjunction between fibres. 1
A New Framework for Declarative Programming
, 2001
"... We propose a new indexedcategory syntax and semantics of Weak Hereditarily Harrop logic programming with constraints, based on resolution over taucategories:finite product categories with canonical structure. ..."
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We propose a new indexedcategory syntax and semantics of Weak Hereditarily Harrop logic programming with constraints, based on resolution over taucategories:finite product categories with canonical structure.
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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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.
Proving Semantical Equivalence of Data Specifications
 J. Pure and Applied Algebra
, 2005
"... More than two decades ago, Peter Freyd introduced essentially algebraic specifications, a wellbehaved generalization of algebraic specifications, allowing for equational partiality. These essentially algebraic specifications turn out to have a number of very interesting applications in computer ..."
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More than two decades ago, Peter Freyd introduced essentially algebraic specifications, a wellbehaved generalization of algebraic specifications, allowing for equational partiality. These essentially algebraic specifications turn out to have a number of very interesting applications in computer science.
Formalized proof, computation, and the construction problem in algebraic geometry
, 2004
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B.: Integrals and valuations
 2009) for Algebraic Quantum Theory
"... Abstract. 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 ..."
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Abstract. 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. 1.
Encapsulating data in Logic Programming via Categorical Constraints
 Meinke (Eds.), Principles ofDeclarative Programming, Lecture Notes in Computer Sciences
, 1998
"... We define a framework for writing executable declarative specifications which incorporate categorical constraints on data, Horn Clauses and datatype specification over finiteproduct categories. We construct a generic extension of a base syntactic category of constraints in which arrows correspond t ..."
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We define a framework for writing executable declarative specifications which incorporate categorical constraints on data, Horn Clauses and datatype specification over finiteproduct categories. We construct a generic extension of a base syntactic category of constraints in which arrows correspond to resolution proofs subject to the specified data constraints. 1 Introduction Much of the research in logic programming is aimed at expanding the expressive power and efficiency of declarative languages without compromising the logical transparency commitment: programs should (almost) read like specifications. One approach is to place more expressive power and more of the control components into the logic itself, possibly by expanding the scope of the underlying mathematical formalism. This has been the goal of constraint logic programming (CLP, Set constraints, Prolog III), and extensions to higherorder and linear logic, to name a few such efforts. This paper is a step in this direction. ...