52 research outputs found
Curry-Howard-Lambek Correspondence for Intuitionistic Belief
This paper introduces a natural deduction calculus for intuitionistic logic
of belief which is easily turned into a modal
-calculus giving a computational semantics for deductions in
. By using that interpretation, it is also proved that
has good proof-theoretic properties. The correspondence
between deductions and typed terms is then extended to a categorical semantics
for identity of proofs in showing the general structure of
such a modality for belief in an intuitionistic framework.Comment: Submitted to Studia Logica on January 31st, 202
Continuity as a computational effect
The original purpose of component-based development was to provide techniques to master complex software, through composition, reuse and parametrisation. However, such systems are rapidly moving towards a level in which software becomes prevalently intertwined with (continuous) physical processes. A possible way to accommodate the latter in component calculi relies on a suitable encoding of continuous behaviour as (yet another) computational effect.
This paper introduces such an encoding through a monad which, in the compositional development of hybrid systems, may play a role similar to the one played by 1+, powerset, and distribution monads in the characterisation of partial, nondeterministic and probabilistic components, respectively. This monad and its Kleisli category provide a universe in which the effects of continuity over (different forms of) composition can be suitably studied.This work is financed by the ERDF - European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT - Fundacao para a Ciencia e a Tecnologia within project POCI-01-0145-FEDER-016692.The first author is also sponsored by FCT grant SFRH/BD/52234/2013, and the second one by FCT grant SFRH/BSAB/113890/2015. Moreover, D. Hofmann and M. Martins are supported by the EU FP7 Marie Curie PIRSES-GA-2012-318986 project GeTFun: Generalizing Truth-Functionality and FCT project UID/MAT/04106/2013 through CIDMA
Physics, Topology, Logic and Computation: A Rosetta Stone
In physics, Feynman diagrams are used to reason about quantum processes. In
the 1980s, it became clear that underlying these diagrams is a powerful analogy
between quantum physics and topology: namely, a linear operator behaves very
much like a "cobordism". Similar diagrams can be used to reason about logic,
where they represent proofs, and computation, where they represent programs.
With the rise of interest in quantum cryptography and quantum computation, it
became clear that there is extensive network of analogies between physics,
topology, logic and computation. In this expository paper, we make some of
these analogies precise using the concept of "closed symmetric monoidal
category". We assume no prior knowledge of category theory, proof theory or
computer science.Comment: 73 pages, 8 encapsulated postscript figure
Which simple types have a unique inhabitant?
International audienceWe study the question of whether a given type has a unique inhabitant modulo program equivalence. In the setting of simply-typed lambda-calculus with sums, equipped with the strong βη-equivalence, we show that uniqueness is decidable. We present a saturating focused logic that introduces irreducible cuts on positive types "as soon as possible". Backward search in this logic gives an effective algorithm that returns either zero, one or two distinct inhabitants for any given type. Preliminary application studies show that such a feature can be useful in strongly-typed programs, inferring the code of highly-polymorphic library functions, or "glue code" inside more complex terms
Category-Theoretic Quantitative Compositional Distributional Models of Natural Language Semantics
This thesis is about the problem of compositionality in distributional
semantics. Distributional semantics presupposes that the meanings of words are
a function of their occurrences in textual contexts. It models words as
distributions over these contexts and represents them as vectors in high
dimensional spaces. The problem of compositionality for such models concerns
itself with how to produce representations for larger units of text by
composing the representations of smaller units of text.
This thesis focuses on a particular approach to this compositionality
problem, namely using the categorical framework developed by Coecke, Sadrzadeh,
and Clark, which combines syntactic analysis formalisms with distributional
semantic representations of meaning to produce syntactically motivated
composition operations. This thesis shows how this approach can be
theoretically extended and practically implemented to produce concrete
compositional distributional models of natural language semantics. It
furthermore demonstrates that such models can perform on par with, or better
than, other competing approaches in the field of natural language processing.
There are three principal contributions to computational linguistics in this
thesis. The first is to extend the DisCoCat framework on the syntactic front
and semantic front, incorporating a number of syntactic analysis formalisms and
providing learning procedures allowing for the generation of concrete
compositional distributional models. The second contribution is to evaluate the
models developed from the procedures presented here, showing that they
outperform other compositional distributional models present in the literature.
The third contribution is to show how using category theory to solve linguistic
problems forms a sound basis for research, illustrated by examples of work on
this topic, that also suggest directions for future research.Comment: DPhil Thesis, University of Oxford, Submitted and accepted in 201
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