2,377 research outputs found
Computation with Advice
Computation with advice is suggested as generalization of both computation
with discrete advice and Type-2 Nondeterminism. Several embodiments of the
generic concept are discussed, and the close connection to Weihrauch
reducibility is pointed out. As a novel concept, computability with random
advice is studied; which corresponds to correct solutions being guessable with
positive probability. In the framework of computation with advice, it is
possible to define computational complexity for certain concepts of
hypercomputation. Finally, some examples are given which illuminate the
interplay of uniform and non-uniform techniques in order to investigate both
computability with advice and the Weihrauch lattice
Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals
We describe the basic theory of infinite time Turing machines and some recent
developments, including the infinite time degree theory, infinite time
complexity theory, and infinite time computable model theory. We focus
particularly on the application of infinite time Turing machines to the
analysis of the hierarchy of equivalence relations on the reals, in analogy
with the theory arising from Borel reducibility. We define a notion of infinite
time reducibility, which lifts much of the Borel theory into the class
in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference,
200
A constructive version of Birkhoff's ergodic theorem for Martin-L\"of random points
A theorem of Ku\v{c}era states that given a Martin-L\"of random infinite
binary sequence {\omega} and an effectively open set A of measure less than 1,
some tail of {\omega} is not in A. We first prove several results in the same
spirit and generalize them via an effective version of a weak form of
Birkhoff's ergodic theorem. We then use this result to get a stronger form of
it, namely a very general effective version of Birkhoff's ergodic theorem,
which improves all the results previously obtained in this direction, in
particular those of V'Yugin, Nandakumar and Hoyrup, Rojas.Comment: Improved version of the CiE'10 paper, with the strong form of
Birkhoff's ergodic theorem for random point
Computability of probability measures and Martin-Lof randomness over metric spaces
In this paper we investigate algorithmic randomness on more general spaces
than the Cantor space, namely computable metric spaces. To do this, we first
develop a unified framework allowing computations with probability measures. We
show that any computable metric space with a computable probability measure is
isomorphic to the Cantor space in a computable and measure-theoretic sense. We
show that any computable metric space admits a universal uniform randomness
test (without further assumption).Comment: 29 page
Revising Type-2 Computation and Degrees of Discontinuity
By the sometimes so-called MAIN THEOREM of Recursive Analysis, every
computable real function is necessarily continuous. Weihrauch and Zheng
(TCS'2000), Brattka (MLQ'2005), and Ziegler (ToCS'2006) have considered
different relaxed notions of computability to cover also discontinuous
functions. The present work compares and unifies these approaches. This is
based on the concept of the JUMP of a representation: both a TTE-counterpart to
the well known recursion-theoretic jump on Kleene's Arithmetical Hierarchy of
hypercomputation: and a formalization of revising computation in the sense of
Shoenfield.
We also consider Markov and Banach/Mazur oracle-computation of discontinuous
fu nctions and characterize the computational power of Type-2 nondeterminism to
coincide with the first level of the Analytical Hierarchy.Comment: to appear in Proc. CCA'0
The descriptive theory of represented spaces
This is a survey on the ongoing development of a descriptive theory of
represented spaces, which is intended as an extension of both classical and
effective descriptive set theory to deal with both sets and functions between
represented spaces. Most material is from work-in-progress, and thus there may
be a stronger focus on projects involving the author than an objective survey
would merit.Comment: survey of work-in-progres
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