Mathematical Foundations for a Compositional Distributional Model of Meaning

We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, for which we rely on the algebra of Pregroups, introduced by Lambek. This mathematical framework enables us to compute the meaning of a well-typed sentence from the meanings of its constituents. Concretely, the type reductions of Pregroups are `lifted' to morphisms in a category, a procedure that transforms meanings of constituents into a meaning of the (well-typed) whole. Importantly, meanings of whole sentences live in a single space, independent of the grammatical structure of the sentence. Hence the inner-product can be used to compare meanings of arbitrary sentences, as it is for comparing the meanings of words in the distributional model. The mathematical structure we employ admits a purely diagrammatic calculus which exposes how the information flows between the words in a sentence in order to make up the meaning of the whole sentence. A variation of our `categorical model' which involves constraining the scalars of the vector spaces to the semiring of Booleans results in a Montague-style Boolean-valued semantics.Comment: to appea

Derived classes as a basis for views in UML/OCL data models

UML is the de facto standard language for analysis and design in object-oriented frameworks. Information systems, and in particular information systems based on databases and their applications, rely heavily on sound principles of analysis and design. Many present-day database applications employ object-oriented principles in the phases of analysis and design due to the advantages of expressiveness and clarity of such languages as UML. Database specifications often involve specifications of constraints, and the Object Constraint Language (OCL) - as part of UML - can aid in the unambiguous modelling of database constraints. One of the central notions in database modelling and in constraint specifications is the notion of a database view. A database view closely corresponds to the notion of derived class in UML. This paper will show how the notion of a derived class in UML can be given a precise semantics in terms of OCL. We will then demonstrate that the notion of a relational database view can be correctly expressed as a derived class in UML/OCL. A central part of our investigation concerns the generality of our manner of representing relational views in OCL. An important problem that we address in this respect is the representation of product spaces and relational joins. Joins are often essential in view definitions, and we shall demonstrate how we can express Cartesian products and joins within the current framework of UML/OCL language by employing the notions of derived class. As a consequence, OCL will be shown to be equipped with the full expressive power of the relational algebra, offering support for the claim that OCL can be useful as a general query language within the framework of the UML/OCL data model.

An Improved Point-Line Incidence Bound Over Arbitrary Fields

We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field $\mathbb{F}$, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that $m$ points and $n$ lines in $\mathbb{F}^2$, with $m^{7/8}<n<m^{8/7}$, determine at most $O(m^{11/15}n^{11/15})$ incidences (where, if $\mathbb{F}$ has positive characteristic $p$, we assume $m^{-2}n^{13}\ll p^{15}$). This improves on the previous best known bound, due to Jones. To obtain our bound, we first prove an optimal point-line incidence bound on Cartesian products, using a reduction to a point-plane incidence bound of Rudnev. We then cover most of the point set with Cartesian products, and we bound the incidences on each product separately, using the bound just mentioned. We give several applications, to sum-product-type problems, an expander problem of Bourgain, the distinct distance problem and Beck's theorem.Comment: 18 pages. To appear in the Bulletin of the London Mathematical Societ