535 research outputs found
Distributional Sentence Entailment Using Density Matrices
Categorical compositional distributional model of Coecke et al. (2010)
suggests a way to combine grammatical composition of the formal, type logical
models with the corpus based, empirical word representations of distributional
semantics. This paper contributes to the project by expanding the model to also
capture entailment relations. This is achieved by extending the representations
of words from points in meaning space to density operators, which are
probability distributions on the subspaces of the space. A symmetric measure of
similarity and an asymmetric measure of entailment is defined, where lexical
entailment is measured using von Neumann entropy, the quantum variant of
Kullback-Leibler divergence. Lexical entailment, combined with the composition
map on word representations, provides a method to obtain entailment relations
on the level of sentences. Truth theoretic and corpus-based examples are
provided.Comment: 11 page
Abstract Tensor Systems as Monoidal Categories
The primary contribution of this paper is to give a formal, categorical
treatment to Penrose's abstract tensor notation, in the context of traced
symmetric monoidal categories. To do so, we introduce a typed, sum-free version
of an abstract tensor system and demonstrate the construction of its associated
category. We then show that the associated category of the free abstract tensor
system is in fact the free traced symmetric monoidal category on a monoidal
signature. A notable consequence of this result is a simple proof for the
soundness and completeness of the diagrammatic language for traced symmetric
monoidal categories.Comment: Dedicated to Joachim Lambek on the occasion of his 90th birthda
Constructing applicative functors
Applicative functors define an interface to computation that is more general, and correspondingly weaker, than that of monads. First used in parser libraries, they are now seeing a wide range of applications. This paper sets out to explore the space of non-monadic applicative functors useful in programming. We work with a generalization, lax monoidal functors, and consider several methods of constructing useful functors of this type, just as transformers are used to construct computational monads. For example, coends, familiar to functional programmers as existential types, yield a range of useful applicative functors, including left Kan extensions. Other constructions are final fixed points, a limited sum construction, and a generalization of the semi-direct product of monoids. Implementations in Haskell are included where possible
Introduction to Categories and Categorical Logic
The aim of these notes is to provide a succinct, accessible introduction to
some of the basic ideas of category theory and categorical logic. The notes are
based on a lecture course given at Oxford over the past few years. They contain
numerous exercises, and hopefully will prove useful for self-study by those
seeking a first introduction to the subject, with fairly minimal prerequisites.
The coverage is by no means comprehensive, but should provide a good basis for
further study; a guide to further reading is included. The main prerequisite is
a basic familiarity with the elements of discrete mathematics: sets, relations
and functions. An Appendix contains a summary of what we will need, and it may
be useful to review this first. In addition, some prior exposure to abstract
algebra - vector spaces and linear maps, or groups and group homomorphisms -
would be helpful.Comment: 96 page
Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics
This paper deals with topos-theoretic truth-value valuations of quantum
propositions. Concretely, a mathematical framework of a specific type of modal
approach is extended to the topos theory, and further, structures of the
obtained truth-value valuations are investigated. What is taken up is the modal
approach based on a determinate lattice \Dcal(e,R), which is a sublattice of
the lattice \Lcal of all quantum propositions and is determined by a quantum
state and a preferred determinate observable . Topos-theoretic extension
is made in the functor category \Sets^{\CcalR} of which base category
\CcalR is determined by . Each true atom, which determines truth values,
true or false, of all propositions in \Dcal(e,R), generates also a
multi-valued valuation function of which domain and range are \Lcal and a
Heyting algebra given by the subobject classifier in \Sets^{\CcalR},
respectively. All true propositions in \Dcal(e,R) are assigned the top
element of the Heyting algebra by the valuation function. False propositions
including the null proposition are, however, assigned values larger than the
bottom element. This defect can be removed by use of a subobject
semi-classifier. Furthermore, in order to treat all possible determinate
observables in a unified framework, another valuations are constructed in the
functor category \Sets^{\Ccal}. Here, the base category \Ccal includes all
\CcalR's as subcategories. Although \Sets^{\Ccal} has a structure
apparently different from \Sets^{\CcalR}, a subobject semi-classifier of
\Sets^{\Ccal} gives valuations completely equivalent to those in
\Sets^{\CcalR}'s.Comment: LaTeX2
A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
This paper is the first in a series whose goal is to develop a fundamentally
new way of constructing theories of physics. The motivation comes from a desire
to address certain deep issues that arise when contemplating quantum theories
of space and time. Our basic contention is that constructing a theory of
physics is equivalent to finding a representation in a topos of a certain
formal language that is attached to the system. Classical physics arises when
the topos is the category of sets. Other types of theory employ a different
topos. In this paper we discuss two different types of language that can be
attached to a system, S. The first is a propositional language, PL(S); the
second is a higher-order, typed language L(S). Both languages provide deductive
systems with an intuitionistic logic. The reason for introducing PL(S) is that,
as shown in paper II of the series, it is the easiest way of understanding, and
expanding on, the earlier work on topos theory and quantum physics. However,
the main thrust of our programme utilises the more powerful language L(S) and
its representation in an appropriate topos.Comment: 36 pages, no figure
The Grail theorem prover: Type theory for syntax and semantics
As the name suggests, type-logical grammars are a grammar formalism based on
logic and type theory. From the prespective of grammar design, type-logical
grammars develop the syntactic and semantic aspects of linguistic phenomena
hand-in-hand, letting the desired semantics of an expression inform the
syntactic type and vice versa. Prototypical examples of the successful
application of type-logical grammars to the syntax-semantics interface include
coordination, quantifier scope and extraction.This chapter describes the Grail
theorem prover, a series of tools for designing and testing grammars in various
modern type-logical grammars which functions as a tool . All tools described in
this chapter are freely available
Variable binding, symmetric monoidal closed theories, and bigraphs
This paper investigates the use of symmetric monoidal closed (SMC) structure
for representing syntax with variable binding, in particular for languages with
linear aspects. In our setting, one first specifies an SMC theory T, which may
express binding operations, in a way reminiscent from higher-order abstract
syntax. This theory generates an SMC category S(T) whose morphisms are, in a
sense, terms in the desired syntax. We apply our approach to Jensen and
Milner's (abstract binding) bigraphs, which are linear w.r.t. processes. This
leads to an alternative category of bigraphs, which we compare to the original.Comment: An introduction to two more technical previous preprints. Accepted at
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