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**11 - 19**of**19**### A categorical logic for information systems

- Journal of the Interest Group in Pure and Applied Logic
, 1996

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"... Preface to the TAC reprint This is a reprint of the final version, published by the Centre de Recherche Mathématique at the Université de Montréal. We are aware of only one error, which will be corrected below. If any others are reported, we will post corrections at ftp.math.mcgill.ca/barr/pdffiles/ ..."

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Preface to the TAC reprint This is a reprint of the final version, published by the Centre de Recherche Mathématique at the Université de Montréal. We are aware of only one error, which will be corrected below. If any others are reported, we will post corrections at ftp.math.mcgill.ca/barr/pdffiles/ctcserr. pdf and at

### March 1993Query Languages for Bags

"... In this paper we study theoretical foundations for programming with bags. We fully determine the strength of many polynomial bag operators relative to an ambient query language. Then picking the strongest combination of these operators we obtain the yardstick nested bag query language Nf?,C(monus, u ..."

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In this paper we study theoretical foundations for programming with bags. We fully determine the strength of many polynomial bag operators relative to an ambient query language. Then picking the strongest combination of these operators we obtain the yardstick nested bag query language Nf?,C(monus, unique). The relationship between nested relational algebra and various fragments of NaC(rnonus, unique) is investigated. The precise amount of extra power that NE(monus, unique) possesses over the nested relational algebra is determined. An ordering for dealing with partial information in bags is proposed and a technique for lifting a linear order at base types to linear order at all types is presented. This linear order is used to prove the conservative extension property for several bag languages. Using this property, we prove some inexpressibility results for NaC(monus, unique). In particular, it can not test for a property that is simultaneously infinite and co-infinite (for example, parity). Then non-polynomial primitives such as powerbag, structural recursion and bounded loop are studied. Structural recursion on bags is shown to be strictly more powerful than the powerbag primitive and it is equivalent to the bounded loop operator. Finally, we show that the numerical functions expressible in NEC(monus, unique) augmented by structural recursion are precisely the primitive recursive functions. 1

### Geometric Logic in Computer Science

- Ryan (Eds.): Theory and Formal Methods
, 1993

"... We present an introduction to geometric logic and the mathematical structures associated with it, such as categorical logic and toposes. We also describe some of its applications in computer science including its potential as a logic for specification languages. ..."

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We present an introduction to geometric logic and the mathematical structures associated with it, such as categorical logic and toposes. We also describe some of its applications in computer science including its potential as a logic for specification languages.

### Categorical Domain Theory: Scott Topology, Powercategories, Coherent Categories

, 2001

"... In the present article we continue recent work in the direction of domain theory were certain (accessible) categories are used as generalized domains. We discuss the possibility of using certain presheaf toposes as generalizations of the Scott topology at this level. We show that the toposes associa ..."

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In the present article we continue recent work in the direction of domain theory were certain (accessible) categories are used as generalized domains. We discuss the possibility of using certain presheaf toposes as generalizations of the Scott topology at this level. We show that the toposes associated with Scott complete categories are injective with respect to dense inclusions of toposes. We propose analogues of the upper and lower powerdomain in terms of the Scott topology at the level of categories. We show that the class of finitely accessible categories is closed under this generalized upper powerdomain construction (the respective result about the lower powerdomain construction is essentially known). We also treat the notion of "coherent domain" by introducing two possible notions of coherence for a finitely accessible category (qua generalized domain). The one of them imitates the stability of the compact saturated sets under intersection and the other one imitates the so-called "2/3 SFP" property. We show that the two notions are equivalent. This amounts to characterizing the small categories whose free cocompletion under finite colimits has finite limits.

### Semantics of Binary Choice Constructs

"... This paper is a summary of the following six publications: (1) Stable Power Domains [Hec94d] (2) Product Operations in Strong Monads [Hec93b] (3) Power Domains Supporting Recursion and Failure [Hec92] (4) Lower Bag Domains [Hec94a] (5) Probabilistic Domains [Hec94b] (6) Probabilistic Power Domains, ..."

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This paper is a summary of the following six publications: (1) Stable Power Domains [Hec94d] (2) Product Operations in Strong Monads [Hec93b] (3) Power Domains Supporting Recursion and Failure [Hec92] (4) Lower Bag Domains [Hec94a] (5) Probabilistic Domains [Hec94b] (6) Probabilistic Power Domains, Information Systems, and Locales [Hec94c] After a general introduction in Section 0, the main results of these six publications are summarized in Sections 1 through 6.

### Generalized Lattices Express Parallel Distributed Concept Learning

"... Abstract—Concepts have been expressed mathematically as propositions in a distributive lattice. A more comprehensive formulation is that of a generalized lattice, or category, in which the concepts are related in hierarchical fashion by lattice-like links called concept morphisms. A concept morphism ..."

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Abstract—Concepts have been expressed mathematically as propositions in a distributive lattice. A more comprehensive formulation is that of a generalized lattice, or category, in which the concepts are related in hierarchical fashion by lattice-like links called concept morphisms. A concept morphism describes how a more abstract concept is used within a more specialized concept, as the color ”red ” is used in describing ”apples”. Often, an abstract concept can be used in a more specialized concept in more than one way as with ”color”, which can appear in ”apples ” as either ”red”, ”yellow ” or ”green”. Further, ”color” appears in ”apples ” because it appears in ”red”, ”yellow” and ”green”, which in turn appear in ”apples”, expressed via the composition of concept morphisms. Using categorical constructs based upon composition together with structure-preserving mappings that preserve compositional structure, a recently-developed semantic theory shows how abstract and specialized concepts are learned by a neural network. I.

### M. J. Healy—Preprint–to be published by Springer 1 Category Theory as a Mathematics for Formalizing Ontologies

, 2007

"... Category theory is discussed as an appropriate mathematical basis for the formalization and study of ontologies. It is based upon the notion of the structure manifest in systems of compositional relations and through mappings between systems that preserve composition. With one or two important excep ..."

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Category theory is discussed as an appropriate mathematical basis for the formalization and study of ontologies. It is based upon the notion of the structure manifest in systems of compositional relations and through mappings between systems that preserve composition. With one or two important exceptions, the basic concepts of category theory needed for an understanding of the other chapters involving category theory are presented here. The exceptions will be presented where they are used. 1