25,135 research outputs found
Kolmogorov complexity and the Recursion Theorem
Several classes of DNR functions are characterized in terms of Kolmogorov
complexity. In particular, a set of natural numbers A can wtt-compute a DNR
function iff there is a nontrivial recursive lower bound on the Kolmogorov
complexity of the initial segments of A. Furthermore, A can Turing compute a
DNR function iff there is a nontrivial A-recursive lower bound on the
Kolmogorov complexity of the initial segements of A. A is PA-complete, that is,
A can compute a {0,1}-valued DNR function, iff A can compute a function F such
that F(n) is a string of length n and maximal C-complexity among the strings of
length n. A solves the halting problem iff A can compute a function F such that
F(n) is a string of length n and maximal H-complexity among the strings of
length n. Further characterizations for these classes are given. The existence
of a DNR function in a Turing degree is equivalent to the failure of the
Recursion Theorem for this degree; thus the provided results characterize those
Turing degrees in terms of Kolmogorov complexity which do no longer permit the
usage of the Recursion Theorem.Comment: Full version of paper presented at STACS 2006, Lecture Notes in
Computer Science 3884 (2006), 149--16
Convergence and Convergence Rate of Stochastic Gradient Search in the Case of Multiple and Non-Isolated Extrema
The asymptotic behavior of stochastic gradient algorithms is studied. Relying
on results from differential geometry (Lojasiewicz gradient inequality), the
single limit-point convergence of the algorithm iterates is demonstrated and
relatively tight bounds on the convergence rate are derived. In sharp contrast
to the existing asymptotic results, the new results presented here allow the
objective function to have multiple and non-isolated minima. The new results
also offer new insights into the asymptotic properties of several classes of
recursive algorithms which are routinely used in engineering, statistics,
machine learning and operations research
Randomness on computable probability spaces - A dynamical point of view
We extend the notion of randomness (in the version introduced by Schnorr) to computable probability spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is typical for some dynamic, if it follows the statistical behavior of the system (Birkhoff’s pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every mixing computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications
Uniform Diagonalization Theorem for Complexity Classes of Promise Problems including Randomized and Quantum Classes
Diagonalization in the spirit of Cantor's diagonal arguments is a widely used
tool in theoretical computer sciences to obtain structural results about
computational problems and complexity classes by indirect proofs. The Uniform
Diagonalization Theorem allows the construction of problems outside complexity
classes while still being reducible to a specific decision problem. This paper
provides a generalization of the Uniform Diagonalization Theorem by extending
it to promise problems and the complexity classes they form, e.g. randomized
and quantum complexity classes. The theorem requires from the underlying
computing model not only the decidability of its acceptance and rejection
behaviour but also of its promise-contradicting indifferent behaviour - a
property that we will introduce as "total decidability" of promise problems.
Implications of the Uniform Diagonalization Theorem are mainly of two kinds:
1. Existence of intermediate problems (e.g. between BQP and QMA) - also known
as Ladner's Theorem - and 2. Undecidability if a problem of a complexity class
is contained in a subclass (e.g. membership of a QMA-problem in BQP). Like the
original Uniform Diagonalization Theorem the extension applies besides BQP and
QMA to a large variety of complexity class pairs, including combinations from
deterministic, randomized and quantum classes.Comment: 15 page
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