336 research outputs found
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
An Introduction to Different Approaches to Initial Semantics
Characterizing programming languages with variable binding as initial
objects, was first achieved by Fiore, Plotkin, and Turi in their seminal paper
published at LICS'99. To do so, in particular to prove initiality theorems,
they developed a framework based on monoidal categories, functors with
strengths, and -monoids. An alternative approach using modules over
monads was later introduced by Hirschowitz and Maggesi, for endofunctor
categories, that is, for particular monoidal categories. This approach has the
advantage of providing a more general and abstract definition of signatures and
models; however, no general initiality result is known for this notion of
signature. Furthermore, Matthes and Uustalu provided a categorical formalism
for constructing (initial) monads via Mendler-style recursion, that can also be
used for initial semantics. The different approaches have been developed
further in several articles. However, in practice, the literature is difficult
to access, and links between the different strands of work remain
underexplored.
In the present work, we give an introduction to initial semantics that
encompasses the three different strands. We develop a suitable "pushout" of
Hirschowitz and Maggesi's framework with Fiore's, and rely on Matthes and
Uustalu's formalism to provide modular proofs. For this purpose, we generalize
both Hirschowitz and Maggesi's framework, and Matthes and Uustalu's formalism
to the general setting of monoidal categories studied by Fiore and
collaborators. Moreover, we provide fully worked out presentation of some basic
instances of the literature, and an extensive discussion of related work
explaining the links between the different approaches
Unification and Logarithmic Space
We present an algebraic characterization of the complexity classes Logspace
and NLogspace, using an algebra with a composition law based on unification.
This new bridge between unification and complexity classes is inspired from
proof theory and more specifically linear logic and Geometry of Interaction.
We show how unification can be used to build a model of computation by means
of specific subalgebras associated to finite permutations groups. We then prove
that whether an observation (the algebraic counterpart of a program) accepts a
word can be decided within logarithmic space. We also show that the
construction can naturally represent pointer machines, an intuitive way of
understanding logarithmic space computing
Effective lambda-models vs recursively enumerable lambda-theories
A longstanding open problem is whether there exists a non syntactical model
of the untyped lambda-calculus whose theory is exactly the least lambda-theory
(l-beta). In this paper we investigate the more general question of whether the
equational/order theory of a model of the (untyped) lambda-calculus can be
recursively enumerable (r.e. for brevity). We introduce a notion of effective
model of lambda-calculus calculus, which covers in particular all the models
individually introduced in the literature. We prove that the order theory of an
effective model is never r.e.; from this it follows that its equational theory
cannot be l-beta or l-beta-eta. We then show that no effective model living in
the stable or strongly stable semantics has an r.e. equational theory.
Concerning Scott's semantics, we investigate the class of graph models and
prove that no order theory of a graph model can be r.e., and that there exists
an effective graph model whose equational/order theory is minimum among all
theories of graph models. Finally, we show that the class of graph models
enjoys a kind of downwards Lowenheim-Skolem theorem.Comment: 34
The Lazy Lambda Calculus : an investigation into the foundations of functional programming
Imperial Users onl
A computable expression of closure to efficient causation
International audienceIn this paper, we propose a mathematical expression of closure to efficient causation in terms of lambda-calculus; we argue that this opens up the perspective of developing principled computer simulations of systems closed to efficient causation in an appropriate programming language. An important implication of our formulation is that, by exhibiting an expression in lambda-calculus, which is a paradigmatic formalism for computability and programming, we show that there are no conceptual or principled problems in realizing a computer simulation or model of closure to efficient causation. We conclude with a brief discussion of the question whether closure to efficient causation captures all relevant properties of living systems. We suggest that it might not be the case, and that more complex definitions could indeed create crucial some obstacles to computability
Virtual classes and virtual motives of Quot schemes on threefolds
For a simple, rigid vector bundle on a Calabi-Yau -fold , we
construct a symmetric obstruction theory on the Quot scheme
, and we solve the associated enumerative theory. We
discuss the case of other -folds. Exploiting the critical structure on
, we construct a virtual motive
(in the sense of Behrend-Bryan-Szendr\H{o}i) for for an
arbitrary vector bundle on a smooth -fold . We compute the associated
motivic partition function. We obtain new examples of higher rank (motivic)
Donaldson-Thomas invariants.Comment: 22 pages. Removed appendix. To appear in Adv. Mat
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