2,331 research outputs found
Characterization theorem for the conditionally computable real functions
The class of uniformly computable real functions with respect to a small
subrecursive class of operators computes the elementary functions of calculus,
restricted to compact subsets of their domains. The class of conditionally
computable real functions with respect to the same class of operators is a
proper extension of the class of uniformly computable real functions and it
computes the elementary functions of calculus on their whole domains. The
definition of both classes relies on certain transformations of infinitistic
names of real numbers. In the present paper, the conditional computability of
real functions is characterized in the spirit of Tent and Ziegler, avoiding the
use of infinitistic names
Computable de Finetti measures
We prove a computable version of de Finetti's theorem on exchangeable
sequences of real random variables. As a consequence, exchangeable stochastic
processes expressed in probabilistic functional programming languages can be
automatically rewritten as procedures that do not modify non-local state. Along
the way, we prove that a distribution on the unit interval is computable if and
only if its moments are uniformly computable.Comment: 32 pages. Final journal version; expanded somewhat, with minor
corrections. To appear in Annals of Pure and Applied Logic. Extended abstract
appeared in Proceedings of CiE '09, LNCS 5635, pp. 218-23
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
An extension of Chaitin's halting probability \Omega to a measurement operator in an infinite dimensional quantum system
This paper proposes an extension of Chaitin's halting probability \Omega to a
measurement operator in an infinite dimensional quantum system. Chaitin's
\Omega is defined as the probability that the universal self-delimiting Turing
machine U halts, and plays a central role in the development of algorithmic
information theory. In the theory, there are two equivalent ways to define the
program-size complexity H(s) of a given finite binary string s. In the standard
way, H(s) is defined as the length of the shortest input string for U to output
s. In the other way, the so-called universal probability m is introduced first,
and then H(s) is defined as -log_2 m(s) without reference to the concept of
program-size.
Mathematically, the statistics of outcomes in a quantum measurement are
described by a positive operator-valued measure (POVM) in the most general
setting. Based on the theory of computability structures on a Banach space
developed by Pour-El and Richards, we extend the universal probability to an
analogue of POVM in an infinite dimensional quantum system, called a universal
semi-POVM. We also give another characterization of Chaitin's \Omega numbers by
universal probabilities. Then, based on this characterization, we propose to
define an extension of \Omega as a sum of the POVM elements of a universal
semi-POVM. The validity of this definition is discussed.
In what follows, we introduce an operator version \hat{H}(s) of H(s) in a
Hilbert space of infinite dimension using a universal semi-POVM, and study its
properties.Comment: 24 pages, LaTeX2e, no figures, accepted for publication in
Mathematical Logic Quarterly: The title was slightly changed and a section on
an operator-valued algorithmic information theory was adde
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