Current theoretical work offers measures of schema equivalence based on the information capacity of schemas. This work is based on the existence of abstract functions satisfying various restrictions between the sets of all instances of two schemas. In considering schemas that arise in practice, however, it is not clear how to reason about the existence of such abstract functions. Further, these notions of equivalence tend to be too liberal in that schemas are often considered equivalent when a practitioner would consider them to be different. As a result, practical integration methodologies have not utilized this theoretical foundation and most of them have relied on ad-hoc approaches. We present results that seek to bridge this gap. First, we consider the problem of deciding information capacity equivalence and dominance of schemas that occur in practice, i.e., those that can express inheritance and simple integrity constraints. We show that this problem is undecidable. This undecidab...

Transitive extensional well founded relations provide an intuitionistic notion of ordinals which admits transfinite induction. However these ordinals are not directed and their successor operation is poorly behaved, leading to problems of functoriality. We show how to make the successor monotone by introducing plumpness, which strengthens transitivity. This clarifes the traditional development of successors and unions, making it intuitionistic; even the (classical) proof of trichotomy is made simpler. The definition is, however, recursive, and, as their name suggests, the plump ordinals grow very rapidly. Directedness must be defined hereditarily. It is orthogonal to the other four conditions, and the lower powerdomain construction is shown to be the universal way of imposing it. We treat ordinals as order-types, and develop a corresponding set theory similar to Osius’ transitive set objects. This presents Mostowski’s theorem as a reflection of categories, and set-theoretic union is a corollary of the adjoint functor theorem. Mostowski’s theorem and the rank for some of the notions of ordinal are formulated and proved without the axiom of replacement, but this seems to be unavoidable for the plump rank. The comparison between sets and toposes is developed as far as the identification of replacement with completeness and there are some suggestions for further work in this area. Each notion of set or ordinal defines a free algebra for one of the theories discussed by Joyal and Moerdijk, namely joins of a family of arities together with an operation s satisfying conditions such as x ≤ sx, monotonicity or s(x ∨ y) ≤ sx ∨ sy. Finally we discuss the fixed point theorem for a monotone endofunction s of a poset with least element and directed joins. This may be proved under each of a variety of additional hypotheses. We explain why it is unlikely that any notion of ordinal obeying the induction scheme for arbitrary predicates will prove the pure result.

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The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum �(A) in T (A), which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on �, and self-adjoint elements of A define continuous functions (more precisely, locale maps) from � to Scott’s interval domain. Noting that open subsets of �(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T (A). These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.

Restriction to geometric logic can enable one to define topological structures and continuous maps without explicit reference to topologies. This idea is illustrated with some examples and used to explain 1

We will analyze some extensions of Martin-Löf’s constructive type theory by means of extensional set constructors and we will show that often the most natural requirements over them lead to classical logic or even to inconsistency. 1

Many different "and"- and "or"-operations have been proposed for use in fuzzy logic; ; see, e.g., [4], [13]. It is therefore important to select, for each particular application, the operations which are the best for this particular application. Several papers discuss the optimal choice of "and"- and "or"-operations for fuzzy control, when the main criterion is to get the stablest control (or the smoothest or the most robust or the fastest-tocompute) . In reasoning applications, however, it is more appropriate to select operations which are the best in reflecting human reasoning, i.e., operations which are "the most logical". In this paper, we explain how we can use logic motivations to select fuzzy logic operations, and show the consequences of this choice. As one of the unexpected consequences, we get a surprising relation with the entropy techniques, well known in probabilistic approach to uncertainty. Main Idea. One of the main ideas behind fuzzy logic is that often, we do not ha...

ABSTRACT. Some aspects of the theory of Boolean algebras and distributive lattices-- in particular, the Stone Representation Theorems and the properties of filters and ideals-- are analyzed in a constructive setting.

Abstract. An elementary theory of strict ∞-categories with application to concrete duality is given. New examples of first and second order concrete duality are presented. 1. Categories, functors, natural transformations, modifications There are two kinds of weakness happenning to ∞-categories. One is changing all occurences of equality = with a weaker equivalence realtion ∼. The other one is a weak naturality condition. The first one is not proper and implies strict category theory. The second one is proper and gives a weak category theory. Below we use ∼ instead of =. It is not necessary but has an advantage to treat directly the classification problem (up to ∼). Definition 1.1. • ∞-precategory is a (big) set L endowed with (1) grading L = ∐ Ln n≥0 (2) unary operations d, c: ∐ (3) unary operation e: ∐ n≥1 L n → ∐ L n → ∐ L n≥0

The question “What is category theory ” is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of mathematical interest are usually characterized by some universal mapping property so the general thesis is that category theory is about determination through universals. In recent decades, the notion of adjoint functors has moved to center-stage as category theory’s primary tool to characterize what is important and universal in mathematics. Hence our focus here is to present a theory of adjoint functors, a theory which shows that all adjunctions arise from the birepresentations of “chimeras ” or “heteromorphisms ” between the objects of different categories. Since representations provide universal mapping properties, this theory places adjoints within the framework of determination through universals. The conclusion considers some unreasonably effective analogies between these mathematical