107 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
Total Representations
Almost all representations considered in computable analysis are partial. We
provide arguments in favor of total representations (by elements of the Baire
space). Total representations make the well known analogy between numberings
and representations closer, unify some terminology, simplify some technical
details, suggest interesting open questions and new invariants of topological
spaces relevant to computable analysis.Comment: 30 page
Computability in partial combinatory algebras
We prove a number of elementary facts about computability in partial
combinatory algebras (pca's). We disprove a suggestion made by Kreisel about
using Friedberg numberings to construct extensional pca's. We then discuss
separability and elements without total extensions. We relate this to Ershov's
notion of precompleteness, and we show that precomplete numberings are not 1-1
in general
Computing Solution Operators of Boundary-value Problems for Some Linear Hyperbolic Systems of PDEs
We discuss possibilities of application of Numerical Analysis methods to
proving computability, in the sense of the TTE approach, of solution operators
of boundary-value problems for systems of PDEs. We prove computability of the
solution operator for a symmetric hyperbolic system with computable real
coefficients and dissipative boundary conditions, and of the Cauchy problem for
the same system (we also prove computable dependence on the coefficients) in a
cube . Such systems describe a wide variety of physical
processes (e.g. elasticity, acoustics, Maxwell equations). Moreover, many
boundary-value problems for the wave equation also can be reduced to this case,
thus we partially answer a question raised in Weihrauch and Zhong (2002).
Compared with most of other existing methods of proving computability for PDEs,
this method does not require existence of explicit solution formulas and is
thus applicable to a broader class of (systems of) equations.Comment: 31 page
On approximate decidability of minimal programs
An index in a numbering of partial-recursive functions is called minimal
if every lesser index computes a different function from . Since the 1960's
it has been known that, in any reasonable programming language, no effective
procedure determines whether or not a given index is minimal. We investigate
whether the task of determining minimal indices can be solved in an approximate
sense. Our first question, regarding the set of minimal indices, is whether
there exists an algorithm which can correctly label 1 out of indices as
either minimal or non-minimal. Our second question, regarding the function
which computes minimal indices, is whether one can compute a short list of
candidate indices which includes a minimal index for a given program. We give
some negative results and leave the possibility of positive results as open
questions
- β¦