2,699 research outputs found
How powerful are integer-valued martingales?
In the theory of algorithmic randomness, one of the central notions is that
of computable randomness. An infinite binary sequence X is computably random if
no recursive martingale (strategy) can win an infinite amount of money by
betting on the values of the bits of X. In the classical model, the martingales
considered are real-valued, that is, the bets made by the martingale can be
arbitrary real numbers. In this paper, we investigate a more restricted model,
where only integer-valued martingales are considered, and we study the class of
random sequences induced by this model.Comment: Long version of the CiE 2010 paper
Sub-computable Boundedness Randomness
This paper defines a new notion of bounded computable randomness for certain
classes of sub-computable functions which lack a universal machine. In
particular, we define such versions of randomness for primitive recursive
functions and for PSPACE functions. These new notions are robust in that there
are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov
complexity, and (3) martingales. We show these notions can be equivalently
defined with prefix-free Kolmogorov complexity. We prove that one direction of
van Lambalgen's theorem holds for relative computability, but the other
direction fails. We discuss statistical properties of these notions of
randomness
Uniform distribution and algorithmic randomness
A seminal theorem due to Weyl states that if (a_n) is any sequence of
distinct integers, then, for almost every real number x, the sequence (a_n x)
is uniformly distributed modulo one. In particular, for almost every x in the
unit interval, the sequence (a_n x) is uniformly distributed modulo one for
every computable sequence (a_n) of distinct integers. Call such an x "UD
random". Here it is shown that every Schnorr random real is UD random, but
there are Kurtz random reals that are not UD random. On the other hand, Weyl's
theorem still holds relative to a particular effectively closed null set, so
there are UD random reals that are not Kurtz random
Arithmetic complexity via effective names for random sequences
We investigate enumerability properties for classes of sets which permit
recursive, lexicographically increasing approximations, or left-r.e. sets. In
addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably,
Schnorr, and Kurtz random sets, weakly 1-generics and their complementary
classes, we find that there exist characterizations of the third and fourth
levels of the arithmetic hierarchy purely in terms of these notions.
More generally, there exists an equivalence between arithmetic complexity and
existence of numberings for classes of left-r.e. sets with shift-persistent
elements. While some classes (such as Martin-L\"{o}f randoms and Kurtz
non-randoms) have left-r.e. numberings, there is no canonical, or acceptable,
left-r.e. numbering for any class of left-r.e. randoms.
Finally, we note some fundamental differences between left-r.e. numberings
for sets and reals
Calibrating the complexity of Delta 2 sets via their changes
The computational complexity of a Delta 2 set will be calibrated by the
amount of changes needed for any of its computable approximations. Firstly, we
study Martin-Loef random sets, where we quantify the changes of initial
segments. Secondly, we look at c.e. sets, where we quantify the overall amount
of changes by obedience to cost functions. Finally, we combine the two
settings. The discussions lead to three basic principles on how complexity and
changes relate
Randomness extraction and asymptotic Hamming distance
We obtain a non-implication result in the Medvedev degrees by studying
sequences that are close to Martin-L\"of random in asymptotic Hamming distance.
Our result is that the class of stochastically bi-immune sets is not Medvedev
reducible to the class of sets having complex packing dimension 1
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