Results 1  10
of
19
On Presentations of Algebraic Structures
 in Complexity, Logic and Recursion Theory
, 1995
"... This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicat ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicated to the memory of my friend and teacher Chris Ash who contributed so much to effective structure theory and who left us far too young early in 1995
The KolmogorovLoveland stochastic sequences are not closed under selecting subsequences
 Journal of Symbolic Logic
, 2002
"... It is shown that the class of KolmogorovLoveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the notorious open problem whether the KolmogorovLoveland stochastic sequences are closed unde ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
It is shown that the class of KolmogorovLoveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the notorious open problem whether the KolmogorovLoveland stochastic sequences are closed under selecting subsequences by KolmogorovLoveland selection rules, i.e., by not necessarily monotonic partially computable selection rules. As a corollary, we obtain an easy proof for the previously known result that the KolmogorovLoveland stochastic sequences form a proper subclass of the MisesWaldChurch stochastic sequences.
The Complexity of Stochastic Sequences
 In Conference on Computational Complexity 2003
, 2003
"... We observe that known results on the Kolmogorov complexity of pre xes of eectively stochastic sequences extend to corresponding random sequences. First, there are recursively random random sequences such that for any nondecreasing and unbounded computable function f and for almost all n, the unifor ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
We observe that known results on the Kolmogorov complexity of pre xes of eectively stochastic sequences extend to corresponding random sequences. First, there are recursively random random sequences such that for any nondecreasing and unbounded computable function f and for almost all n, the uniform complexity of the length n pre x of the sequence is bounded by f(n). Second, a similar result with bounds of the form f(n) log n holds for partialrecursively random sequences.
On Selection Functions that Do Not Preserve Normality
 of Lecture Notes in Computer Science
, 2006
"... The sequence selected from a sequence R(0)R(1)... by a language L is the subsequence of all bits R(n + 1) such that the prefix R(0)... R(n) is in L. By a result of Agafonoff [1], a sequence is normal if and only if any subsequence selected by a regular language is again normal. Kamae and Weiss [11] ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
The sequence selected from a sequence R(0)R(1)... by a language L is the subsequence of all bits R(n + 1) such that the prefix R(0)... R(n) is in L. By a result of Agafonoff [1], a sequence is normal if and only if any subsequence selected by a regular language is again normal. Kamae and Weiss [11] and others have raised the question of how complex a language must be such that selecting according to the language does not preserve normality. We show that there are such languages that are only slightly more complicated than regular ones, namely, normality is neither preserved by linear languages nor by deterministic onecounter languages. In fact, for both types of languages it is possible to select a constant sequence from a normal one.
A Limiting First Order Realizability Interpretation
"... Constructive Mathematics might be regarded as a fragment of classical mathematics in which any proof of an existence theorem is equipped with a computable function giving the solution of the theorem. Limit Computable Mathematics (LCM) considered in this note is a fragment of classical mathematics ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Constructive Mathematics might be regarded as a fragment of classical mathematics in which any proof of an existence theorem is equipped with a computable function giving the solution of the theorem. Limit Computable Mathematics (LCM) considered in this note is a fragment of classical mathematics in which any proof of an existence theorem is equipped with a function computing the solution of the theorem in the limit.
Questions in Computable Algebra and Combinatorics
, 1999
"... this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25
THE COMPLEXITY OF THE INDEX SETS OF ℵ0CATEGORICAL THEORIES AND OF EHRENFEUCHT THEORIES
, 2006
"... Abstract. We classify the computabilitytheoretic complexity of two index sets of classes of firstorder theories: We show that the property of being an ℵ0categorical theory is Π0 3complete; and the property of being an Ehrenfeucht theory Π1 1complete. We also show that the property of having con ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. We classify the computabilitytheoretic complexity of two index sets of classes of firstorder theories: We show that the property of being an ℵ0categorical theory is Π0 3complete; and the property of being an Ehrenfeucht theory Π1 1complete. We also show that the property of having continuum many models is Σ1 1 hard. Finally, as a corollary, we note that the properties of having only decidable models, and of having only computable models, are both Π1 1complete. 1. The Main Theorem Measuring the complexity of mathematical notions is one of the main tasks of mathematical logic. Two of the main tools to classify complexity are provided by Kleene’s arithmetical and analytical hierarchy. These two hierarchies provide convenient ways to determine the exact complexity of properties by various notions of completeness, and to give lower bounds on the complexity by various notions of hardness. (See, e.g., Kleene [1], Soare [10] or Odifreddi [4, 5] for the definitions.) This paper will investigate the complexity of properties of a firstorder theory, more precisely, the complexity of a countable firstorder theory having a certain number of models. Recall that a theory is called ℵ0categorical if it has only one countable model up to isomorphism, and an Ehrenfeucht theory if it has more than one but only finitely many countable models up to isomorphism. In order to measure the complexity of these notions, we will use decidable firstorder theories,
Computability, Definability and Algebraic Structures
, 1999
"... In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set
Computably enumerable reals and uniformly presentable ideals
 Mathematical Logic Quarterly
"... We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefixfree set of strings such that α = ∑ σ∈A 2−σ . Note that ∑ σ∈A 2−σ  is precisely the measure of the set of reals that have a ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefixfree set of strings such that α = ∑ σ∈A 2−σ . Note that ∑ σ∈A 2−σ  is precisely the measure of the set of reals that have a string in A as an initial segment. So we will simply abbreviate ∑ σ∈A 2−σ  by µ(A). It is known that whenever A so presents α then A ≤wtt α, where ≤wtt denotes weak truth table reducibility, and that the wtt degrees of presentations form an ideal I(α) in the computably enumerable wtt degrees. We prove that any such ideal is Σ 0 3, and conversely that if I is any Σ 0 3 ideal in the computably enumerable wtt degrees then there is a computable enumerable real α such that I = I(α). 1
Lattice Embeddings for Abstract Bounded Reducibilities
, 1997
"... We give an abstract account of resourcebounded reducibilities as exemplified by the polynomially time or logarithmically spacebounded reducibilities of Turing, truthtable, and manyone type. We introduce a small set of axioms that are satisfied for most of the specific resourcebounded reducibil ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We give an abstract account of resourcebounded reducibilities as exemplified by the polynomially time or logarithmically spacebounded reducibilities of Turing, truthtable, and manyone type. We introduce a small set of axioms that are satisfied for most of the specific resourcebounded reducibilities appearing in the literature. Some of the axioms are of a more algebraic nature, such as the requirement that the reducibility under consideration is a reexive relation, while others are formulated in terms of recursion theory and for example are related to delayed computations of arbitrary recursive sets. The main technical result shown is that for any reducibility that satisfies these axioms, every countable distributive lattice can be embedded into any proper interval of the structure induced on the recursive sets. This extends a corresponding result for polynomially timebounded reducibilities due to AmbosSpies, as well as a result on embeddings of partial orderings for axiomatically described reducibilities due to Mehlhorn.