22,896 research outputs found
Renormalization and Computation II: Time Cut-off and the Halting Problem
This is the second installment to the project initiated in [Ma3]. In the
first Part, I argued that both philosophy and technique of the perturbative
renormalization in quantum field theory could be meaningfully transplanted to
the theory of computation, and sketched several contexts supporting this view.
In this second part, I address some of the issues raised in [Ma3] and provide
their development in three contexts: a categorification of the algorithmic
computations; time cut--off and Anytime Algorithms; and finally, a Hopf algebra
renormalization of the Halting Problem.Comment: 28 page
Grilliot's trick in Nonstandard Analysis
The technique known as Grilliot's trick constitutes a template for explicitly
defining the Turing jump functional in terms of a given
effectively discontinuous type two functional. In this paper, we discuss the
standard extensionality trick: a technique similar to Grilliot's trick in
Nonstandard Analysis. This nonstandard trick proceeds by deriving from the
existence of certain nonstandard discontinuous functionals, the Transfer
principle from Nonstandard analysis limited to -formulas; from this
(generally ineffective) implication, we obtain an effective implication
expressing the Turing jump functional in terms of a discontinuous functional
(and no longer involving Nonstandard Analysis). The advantage of our
nonstandard approach is that one obtains effective content without paying
attention to effective content. We also discuss a new class of functionals
which all seem to fall outside the established categories. These functionals
directly derive from the Standard Part axiom of Nonstandard Analysis.Comment: 21 page
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Recognising the small Ree groups in their natural representations
We present Las Vegas algorithms for constructive recognition and constructive
membership testing of the Ree groups 2G_2(q) = Ree(q), where q = 3^{2m + 1} for
some m > 0, in their natural representations of degree 7. The input is a
generating set X.
The constructive recognition algorithm is polynomial time given a discrete
logarithm oracle. The constructive membership testing consists of a
pre-processing step, that only needs to be executed once for a given X, and a
main step. The latter is polynomial time, and the former is polynomial time
given a discrete logarithm oracle.
Implementations of the algorithms are available for the computer algebra
system MAGMA
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