291 research outputs found
Complexity of Grammar Induction for Quantum Types
Most categorical models of meaning use a functor from the syntactic category
to the semantic category. When semantic information is available, the problem
of grammar induction can therefore be defined as finding preimages of the
semantic types under this forgetful functor, lifting the information flow from
the semantic level to a valid reduction at the syntactic level. We study the
complexity of grammar induction, and show that for a variety of type systems,
including pivotal and compact closed categories, the grammar induction problem
is NP-complete. Our approach could be extended to linguistic type systems such
as autonomous or bi-closed categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
Lambek vs. Lambek: Functorial Vector Space Semantics and String Diagrams for Lambek Calculus
The Distributional Compositional Categorical (DisCoCat) model is a
mathematical framework that provides compositional semantics for meanings of
natural language sentences. It consists of a computational procedure for
constructing meanings of sentences, given their grammatical structure in terms
of compositional type-logic, and given the empirically derived meanings of
their words. For the particular case that the meaning of words is modelled
within a distributional vector space model, its experimental predictions,
derived from real large scale data, have outperformed other empirically
validated methods that could build vectors for a full sentence. This success
can be attributed to a conceptually motivated mathematical underpinning, by
integrating qualitative compositional type-logic and quantitative modelling of
meaning within a category-theoretic mathematical framework.
The type-logic used in the DisCoCat model is Lambek's pregroup grammar.
Pregroup types form a posetal compact closed category, which can be passed, in
a functorial manner, on to the compact closed structure of vector spaces,
linear maps and tensor product. The diagrammatic versions of the equational
reasoning in compact closed categories can be interpreted as the flow of word
meanings within sentences. Pregroups simplify Lambek's previous type-logic, the
Lambek calculus, which has been extensively used to formalise and reason about
various linguistic phenomena. The apparent reliance of the DisCoCat on
pregroups has been seen as a shortcoming. This paper addresses this concern, by
pointing out that one may as well realise a functorial passage from the
original type-logic of Lambek, a monoidal bi-closed category, to vector spaces,
or to any other model of meaning organised within a monoidal bi-closed
category. The corresponding string diagram calculus, due to Baez and Stay, now
depicts the flow of word meanings.Comment: 29 pages, pending publication in Annals of Pure and Applied Logi
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
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
A Generalised Quantifier Theory of Natural Language in Categorical Compositional Distributional Semantics with Bialgebras
Categorical compositional distributional semantics is a model of natural
language; it combines the statistical vector space models of words with the
compositional models of grammar. We formalise in this model the generalised
quantifier theory of natural language, due to Barwise and Cooper. The
underlying setting is a compact closed category with bialgebras. We start from
a generative grammar formalisation and develop an abstract categorical
compositional semantics for it, then instantiate the abstract setting to sets
and relations and to finite dimensional vector spaces and linear maps. We prove
the equivalence of the relational instantiation to the truth theoretic
semantics of generalised quantifiers. The vector space instantiation formalises
the statistical usages of words and enables us to, for the first time, reason
about quantified phrases and sentences compositionally in distributional
semantics
Command injection attacks, continuations, and the Lambek calculus
This paper shows connections between command injection attacks,
continuations, and the Lambek calculus: certain command injections, such as the
tautology attack on SQL, are shown to be a form of control effect that can be
typed using the Lambek calculus, generalizing the double-negation typing of
continuations. Lambek's syntactic calculus is a logic with two implicational
connectives taking their arguments from the left and right, respectively. These
connectives describe how strings interact with their left and right contexts
when building up syntactic structures. The calculus is a form of propositional
logic without structural rules, and so a forerunner of substructural logics
like Linear Logic and Separation Logic.Comment: In Proceedings WoC 2015, arXiv:1606.0583
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