144 research outputs found
Computability Theory
Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science
Kolmogorov complexity and computably enumerable sets
We study the computably enumerable sets in terms of the: (a) Kolmogorov
complexity of their initial segments; (b) Kolmogorov complexity of finite
programs when they are used as oracles. We present an extended discussion of
the existing research on this topic, along with recent developments and open
problems. Besides this survey, our main original result is the following
characterization of the computably enumerable sets with trivial initial segment
prefix-free complexity. A computably enumerable set is -trivial if and
only if the family of sets with complexity bounded by the complexity of is
uniformly computable from the halting problem
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
Characterizing the strongly jump-traceable sets via randomness
We show that if a set is computable from every superlow 1-random set,
then is strongly jump-traceable. This theorem shows that the computably
enumerable (c.e.) strongly jump-traceable sets are exactly the c.e.\ sets
computable from every superlow 1-random set.
We also prove the analogous result for superhighness: a c.e.\ set is strongly
jump-traceable if and only if it is computable from every superhigh 1-random
set.
Finally, we show that for each cost function with the limit condition
there is a 1-random set such that every c.e.\ set
obeys . To do so, we connect cost function strength and the strength of
randomness notions. This result gives a full correspondence between obedience
of cost functions and being computable from 1-random sets.Comment: 41 page
The hierarchy of equivalence relations on the natural numbers under computable reducibility
The notion of computable reducibility between equivalence relations on the
natural numbers provides a natural computable analogue of Borel reducibility.
We investigate the computable reducibility hierarchy, comparing and contrasting
it with the Borel reducibility hierarchy from descriptive set theory.
Meanwhile, the notion of computable reducibility appears well suited for an
analysis of equivalence relations on the c.e.\ sets, and more specifically, on
various classes of c.e.\ structures. This is a rich context with many natural
examples, such as the isomorphism relation on c.e.\ graphs or on computably
presented groups. Here, our exposition extends earlier work in the literature
concerning the classification of computable structures. An abundance of open
questions remains.Comment: To appear in Computabilit
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