Results 1 
5 of
5
Kolmogorov complexity and computably enumerable sets
 Ann. Pure Appl. Logic
"... Abstract. We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent development ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
Abstract. We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the computably enumerable sets with trivial initial segment prefixfree complexity. A computably enumerable set A is Ktrivial if and only if the family of sets with complexity bounded by the complexity of A is uniformly computable from the halting problem.
EXACT PAIRS FOR THE IDEAL OF THE KTRIVIAL SEQUENCES IN THE TURING DEGREES
, 2012
"... The Ktrivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable (c.e.) members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [MN06, Question 4.2] and later i ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
The Ktrivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable (c.e.) members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [MN06, Question 4.2] and later in [Nie09, Problem 5.5.8]. We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists a Ktrivial degree d such that for all degrees a, b which are not Ktrivial and a> d, b> d there exists a degree v which is not Ktrivial and a> v, b> v. This work sheds light to the question of the definability of the Ktrivial degrees in the c.e. degrees.
Algorithmic randomness and measures of complexity
 The Bulletin of Symbolic Logic
, 2013
"... Abstract. We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.
ANALOGUES OF CHAITIN’S OMEGA IN THE COMPUTABLY ENUMERABLE SETS
, 2012
"... We show that there are computably enumerable (c.e.) sets with maximum initial segment Kolmogorov complexity amongst all c.e. sets (with respect to both the plain and the prefixfree version of Kolmogorov complexity). These c.e. sets belong to the weak truth table degree of the halting problem, but ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We show that there are computably enumerable (c.e.) sets with maximum initial segment Kolmogorov complexity amongst all c.e. sets (with respect to both the plain and the prefixfree version of Kolmogorov complexity). These c.e. sets belong to the weak truth table degree of the halting problem, but not every weak truth table complete set has maximum initial segment Kolmogorov complexity. Moreover, every c.e. set with maximum initial segment prefixfree complexity is the disjoint union of two c.e. sets with the same property; and is also the disjoint union of two c.e. sets of lesser initial segment complexity.