15 research outputs found
Semantic Unification A sheaf theoretic approach to natural language
Language is contextual and sheaf theory provides a high level mathematical
framework to model contextuality. We show how sheaf theory can model the
contextual nature of natural language and how gluing can be used to provide a
global semantics for a discourse by putting together the local logical
semantics of each sentence within the discourse. We introduce a presheaf
structure corresponding to a basic form of Discourse Representation Structures.
Within this setting, we formulate a notion of semantic unification --- gluing
meanings of parts of a discourse into a coherent whole --- as a form of
sheaf-theoretic gluing. We illustrate this idea with a number of examples where
it can used to represent resolutions of anaphoric references. We also discuss
multivalued gluing, described using a distributions functor, which can be used
to represent situations where multiple gluings are possible, and where we may
need to rank them using quantitative measures.
Dedicated to Jim Lambek on the occasion of his 90th birthday.Comment: 12 page
Translating and Evolving: Towards a Model of Language Change in DisCoCat
The categorical compositional distributional (DisCoCat) model of meaning
developed by Coecke et al. (2010) has been successful in modeling various
aspects of meaning. However, it fails to model the fact that language can
change. We give an approach to DisCoCat that allows us to represent language
models and translations between them, enabling us to describe translations from
one language to another, or changes within the same language. We unify the
product space representation given in (Coecke et al., 2010) and the functorial
description in (Kartsaklis et al., 2013), in a way that allows us to view a
language as a catalogue of meanings. We formalize the notion of a lexicon in
DisCoCat, and define a dictionary of meanings between two lexicons. All this is
done within the framework of monoidal categories. We give examples of how to
apply our methods, and give a concrete suggestion for compositional translation
in corpora.Comment: In Proceedings CAPNS 2018, arXiv:1811.0270
Type grammar meets Japanese particles
PACLIC 20 / Wuhan, China / 1-3 November, 200
On Pregroups, Freedom, and (Virtual) Conceptual Necessity
Pregroups were introduced in (Lambek, 1999), and provide a founda-tion for a particularly simple syntactic calculus. Buszkowski (2001) showed that free pregroup grammars generate exactly the -free context-free lan-guages. Here we characterize the class of languages generable by all pre-groups, which will be shown to be the entire class of recursively enumerable languages. To show this result, we rely on the well-known representation of recursively enumerable languages as the homomorphic image of the inter-section of two context-free languages (Ginsburg et al., 1967). We define an operation of cross-product over grammars (so-called because of its behaviour on the types), and show that the cross-product of any two free-pregroup grammars generates exactly the intersection of their respective languages. The representation theorem applies once we show that allowing ‘empty cat-egories ’ (i.e. lexical items without overt phonological content) allows us to mimic the effects of any string homomorphism.
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
Towards Functorial Language-Games
In categorical compositional semantics of natural language one studies
functors from a category of grammatical derivations (such as a Lambek pregroup)
to a semantic category (such as real vector spaces). We compositionally build
game-theoretic semantics of sentences by taking the semantic category to be the
category whose morphisms are open games. This requires some modifications to
the grammar category to compensate for the failure of open games to form a
compact closed category. We illustrate the theory using simple examples of
Wittgenstein's language-games.Comment: In Proceedings CAPNS 2018, arXiv:1811.0270
Extended pregroup grammars applied to natural languages
Pregroups and pregroup grammars were introduced by Lambek in 1999 [14] as an algebraic tool for the syntactic analysis of natural lan-guages. The main focus in that paper was on certain extended pregroup grammars such as pregroups with modalities, product pregroup grammars and tupled pregroup grammars. Their applications to different syntactic structures of natural languages, mainly Polish, are explored/shown here