4,842 research outputs found
Types and forgetfulness in categorical linguistics and quantum mechanics
The role of types in categorical models of meaning is investigated. A general
scheme for how typed models of meaning may be used to compare sentences,
regardless of their grammatical structure is described, and a toy example is
used as an illustration. Taking as a starting point the question of whether the
evaluation of such a type system 'loses information', we consider the
parametrized typing associated with connectives from this viewpoint.
The answer to this question implies that, within full categorical models of
meaning, the objects associated with types must exhibit a simple but subtle
categorical property known as self-similarity. We investigate the category
theory behind this, with explicit reference to typed systems, and their
monoidal closed structure. We then demonstrate close connections between such
self-similar structures and dagger Frobenius algebras. In particular, we
demonstrate that the categorical structures implied by the polymorphically
typed connectives give rise to a (lax unitless) form of the special forms of
Frobenius algebras known as classical structures, used heavily in abstract
categorical approaches to quantum mechanics.Comment: 37 pages, 4 figure
Relational Graph Models at Work
We study the relational graph models that constitute a natural subclass of
relational models of lambda-calculus. We prove that among the lambda-theories
induced by such models there exists a minimal one, and that the corresponding
relational graph model is very natural and easy to construct. We then study
relational graph models that are fully abstract, in the sense that they capture
some observational equivalence between lambda-terms. We focus on the two main
observational equivalences in the lambda-calculus, the theory H+ generated by
taking as observables the beta-normal forms, and H* generated by considering as
observables the head normal forms. On the one hand we introduce a notion of
lambda-K\"onig model and prove that a relational graph model is fully abstract
for H+ if and only if it is extensional and lambda-K\"onig. On the other hand
we show that the dual notion of hyperimmune model, together with
extensionality, captures the full abstraction for H*
Introducing a Calculus of Effects and Handlers for Natural Language Semantics
In compositional model-theoretic semantics, researchers assemble
truth-conditions or other kinds of denotations using the lambda calculus. It
was previously observed that the lambda terms and/or the denotations studied
tend to follow the same pattern: they are instances of a monad. In this paper,
we present an extension of the simply-typed lambda calculus that exploits this
uniformity using the recently discovered technique of effect handlers. We prove
that our calculus exhibits some of the key formal properties of the lambda
calculus and we use it to construct a modular semantics for a small fragment
that involves multiple distinct semantic phenomena
The dagger lambda calculus
We present a novel lambda calculus that casts the categorical approach to the
study of quantum protocols into the rich and well established tradition of type
theory. Our construction extends the linear typed lambda calculus with a linear
negation of "trivialised" De Morgan duality. Reduction is realised through
explicit substitution, based on a symmetric notion of binding of global scope,
with rules acting on the entire typing judgement instead of on a specific
subterm. Proofs of subject reduction, confluence, strong normalisation and
consistency are provided, and the language is shown to be an internal language
for dagger compact categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
Fourier spectral methods for fractional-in-space reaction-diffusion equations
Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is computationally demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reactiondiffusion equations. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is show-cased by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models,together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator
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