19 research outputs found
Random semicomputable reals revisited
The aim of this expository paper is to present a nice series of results,
obtained in the papers of Chaitin (1976), Solovay (1975), Calude et al. (1998),
Kucera and Slaman (2001). This joint effort led to a full characterization of
lower semicomputable random reals, both as those that can be expressed as a
"Chaitin Omega" and those that are maximal for the Solovay reducibility. The
original proofs were somewhat involved; in this paper, we present these results
in an elementary way, in particular requiring only basic knowledge of
algorithmic randomness. We add also several simple observations relating lower
semicomputable random reals and busy beaver functions.Comment: 15 page
Things that can be made into themselves
One says that a property of sets of natural numbers can be made into
itself iff there is a numbering of all left-r.e.
sets such that the index set satisfies has the property
as well. For example, the property of being Martin-L\"of random can be made
into itself. Herein we characterize those singleton properties which can be
made into themselves. A second direction of the present work is the
investigation of the structure of left-r.e. sets under inclusion modulo a
finite set. In contrast to the corresponding structure for r.e. sets, which has
only maximal but no minimal members, both minimal and maximal left-r.e. sets
exist. Moreover, our construction of minimal and maximal left-r.e. sets greatly
differs from Friedberg's classical construction of maximal r.e. sets. Finally,
we investigate whether the properties of minimal and maximal left-r.e. sets can
be made into themselves
Optimal asymptotic bounds on the oracle use in computations from Chaitin’s Omega
Chaitin’s number is the halting probability of a universal prefix-free machine, and although it depends on the underlying enumeration of prefix-free machines, it is always Turing-complete. It can be observed, in fact, that for every computably enumerable (c.e.) real �, there exists a Turing functional via which computes �, and such that the number of bits of that are needed for the computation of the first n bits of � (i.e. the use on argument n) is bounded above by a computable function h(n) = n + o (n). We characterise the asymptotic upper bounds on the use of Chaitin’s in oracle computations of halting probabilities (i.e. c.e. reals). We show that the following two conditions are equivalent for any computable function h such that h(n)
Kolmogorov complexity and recursive events
Please read the abstract in the section 00front of this documentThesis (PhD (Mathematics))--University of Pretoria, 2000.Mathematics and Applied Mathematicsunrestricte
Algorithmic information and incompressibility of families of multidimensional networks
This article presents a theoretical investigation of string-based generalized
representations of families of finite networks in a multidimensional space.
First, we study the recursive labeling of networks with (finite) arbitrary node
dimensions (or aspects), such as time instants or layers. In particular, we
study these networks that are formalized in the form of multiaspect graphs. We
show that, unlike classical graphs, the algorithmic information of a
multidimensional network is not in general dominated by the algorithmic
information of the binary sequence that determines the presence or absence of
edges. This universal algorithmic approach sets limitations and conditions for
irreducible information content analysis in comparing networks with a large
number of dimensions, such as multilayer networks. Nevertheless, we show that
there are particular cases of infinite nesting families of finite
multidimensional networks with a unified recursive labeling such that each
member of these families is incompressible. From these results, we study
network topological properties and equivalences in irreducible information
content of multidimensional networks in comparison to their isomorphic
classical graph.Comment: Extended preprint version of the pape