10,332 research outputs found
Unifying Functional Interpretations: Past and Future
This article surveys work done in the last six years on the unification of
various functional interpretations including G\"odel's dialectica
interpretation, its Diller-Nahm variant, Kreisel modified realizability,
Stein's family of functional interpretations, functional interpretations "with
truth", and bounded functional interpretations. Our goal in the present paper
is twofold: (1) to look back and single out the main lessons learnt so far, and
(2) to look forward and list several open questions and possible directions for
further research.Comment: 18 page
The Challenge of Unifying Semantic and Syntactic Inference Restrictions
While syntactic inference restrictions don't play an important role for SAT, they are an essential reasoning technique for more expressive logics, such as first-order logic, or fragments thereof. In particular, they can result in short proofs or model representations. On the other hand, semantically guided inference systems enjoy important properties, such as the generation of solely non-redundant clauses. I discuss to what extend the two paradigms may be unifiable
The Structure of Matter in Spacetime from the Substructure of Time
The nature of the change in perspective that accompanies the proposal of a
unified physical theory deriving from the single dimension of time is
elaborated. On expressing a temporal interval in a multi-dimensional form, via
a direct arithmetic decomposition, both the geometric structure of
4-dimensional spacetime and the physical structure of matter in spacetime can
be derived from the substructure of time. While reviewing this construction,
here we emphasise how the new conceptual picture differs from the more typical
viewpoint in theoretical physics of accounting for the properties of matter by
first postulating entities on top of a given spacetime background or by
geometrically augmenting 4-dimensional spacetime itself. With reference to
historical and philosophical sources we argue that the proposed perspective,
centred on the possible arithmetic forms of time, provides an account for how
the mathematical structures of the theory can relate directly to the physical
structures of the empirical world.Comment: 32 pages, 2 figure
Generalised Proper Time as a Unifying Basis for Models with Two Right-Handed Neutrinos
Models with two right-handed neutrinos are able to accommodate solar and
atmospheric neutrino oscillation observations as well as a mechanism for the
baryon asymmetry of the universe. While economical in terms of the required new
states beyond the Standard Model, given that there are three generations of the
other leptons and quarks this raises the question concerning why only two
right-handed neutrino states should exist. Here we develop from first
principles a fundamental unification scheme based upon a direct generalisation
and analysis of a simple proper time interval with a structure beyond that of
local 4-dimensional spacetime and further augmenting that of models with extra
spatial dimensions. This theory leads to properties of matter fields that
resemble the Standard Model, with an intrinsic left-right asymmetry which is
particularly marked for the neutrino sector. It will be shown how the theory
can provide a foundation for the natural incorporation of two right-handed
neutrinos and may in principle underlie firm predictions both in the neutrino
sector and for other new physics beyond the Standard Model. While connecting
with contemporary and future experiments the origins of the theory are
motivated in a similar spirit as for the earliest unified field theories.Comment: 68 pages, 2 figure
Optimization and NP_R-Completeness of Certain Fewnomials
We give a high precision polynomial-time approximation scheme for the
supremum of any honest n-variate (n+2)-nomial with a constant term, allowing
real exponents as well as real coefficients. Our complexity bounds count field
operations and inequality checks, and are polynomial in n and the logarithm of
a certain condition number. For the special case of polynomials (i.e., integer
exponents), the log of our condition number is quadratic in the sparse
encoding. The best previous complexity bounds were exponential in the sparse
encoding, even for n fixed. Along the way, we extend the theory of
A-discriminants to real exponents and certain exponential sums, and find new
and natural NP_R-complete problems.Comment: 9 pages, 7 figures (3 of them tiny). This is close to the final
conference proceedings versio
Computing Stable Models of Normal Logic Programs Without Grounding
We present a method for computing stable models of normal logic programs,
i.e., logic programs extended with negation, in the presence of predicates with
arbitrary terms. Such programs need not have a finite grounding, so traditional
methods do not apply. Our method relies on the use of a non-Herbrand universe,
as well as coinduction, constructive negation and a number of other novel
techniques. Using our method, a normal logic program with predicates can be
executed directly under the stable model semantics without requiring it to be
grounded either before or during execution and without requiring that its
variables range over a finite domain. As a result, our method is quite general
and supports the use of terms as arguments, including lists and complex data
structures. A prototype implementation and non-trivial applications have been
developed to demonstrate the feasibility of our method
Heisenberg and the Levels of Reality
We first analyze the transdisciplinary model of Reality and its key-concept
of "Levels of Reality". We then compare this model with the one elaborated by
Werner Heisenberg in 1942.Comment: 12 pages, Reference added to the journal in which the paer is
publishe
Approximation and Estimation for High-Dimensional Deep Learning Networks
It has been experimentally observed in recent years that multi-layer
artificial neural networks have a surprising ability to generalize, even when
trained with far more parameters than observations. Is there a theoretical
basis for this? The best available bounds on their metric entropy and
associated complexity measures are essentially linear in the number of
parameters, which is inadequate to explain this phenomenon. Here we examine the
statistical risk (mean squared predictive error) of multi-layer networks with
-type controls on their parameters and with ramp activation functions
(also called lower-rectified linear units). In this setting, the risk is shown
to be upper bounded by , where is the input
dimension to each layer, is the number of layers, and is the sample
size. In this way, the input dimension can be much larger than the sample size
and the estimator can still be accurate, provided the target function has such
controls and that the sample size is at least moderately large
compared to . The heart of the analysis is the development of a
sampling strategy that demonstrates the accuracy of a sparse covering of deep
ramp networks. Lower bounds show that the identified risk is close to being
optimal
Automatic Differentiation using Constraint Handling Rules in Prolog
Automatic differentiation is a technique which allows a programmer to define
a numerical computation via compositions of a broad range of numeric and
computational primitives and have the underlying system support the computation
of partial derivatives of the result with respect to any of its inputs, without
making any finite difference approximations, and without manipulating large
symbolic expressions representing the computation. This note describes a novel
approach to reverse mode automatic differentiation using constraint logic
programmming, specifically, the constraint handling rules (CHR) library of SWI
Prolog, resulting in a very small (50 lines of code) implementation. When
applied to a differentiation-based implementation of the inside-outside
algorithm for parameter learning in probabilistic grammars, the CHR based
implementations outperformed two well-known frameworks for optimising
differentiable functions, Theano and TensorFlow, by a large margin
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