The theory of Kolmogorov complexity is based on the discovery, by Alan Turing in 1936, of the universal Turing machine. After proposing the Turing machine as an explanation of the notion of a computing machine, Turing found that there exists one Turing machine which can simulate any other Turing machine. Complexity, according to Kolmogorov, can be measured by the length of the shortest program ...

We study in which way Kolmogorov complexity and instance complexity affect properties of r.e. sets. We show that the well-known 2 log n upper bound on the Kolmogorov complexity of initial segments of r.e. sets is optimal and characterize the T-degrees of r.e. sets which attain this bound. The main part of the paper is concerned with instance complexity of r.e. sets. We construct a nonrecursive ...

This is a short introduction to Kolmogorov complexity and information theory. The interested reader is referred to the literature, especially the textbooks [CT91] and [LV97] which cover the fields of information theory and Kolmogorov complexity in depth and with all the necessary rigor. They are well to read and require only a minimum of prior knowledge. Kolmogorov complexity. Also known as alg...

The equivalence of universal measure, a priori probability and the negative exponential of Kolmogorov complexity is a well known result with signiicance both to the theory of Kolmogorov complexity and many of its applications. In this paper we consider deenitions for the time-bounded setting where similar equivalences hold. Unlike the unbounded case, time-bounded universal distributions give ri...

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 prefix-free 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. Moreove...

This invited Descriptional Complexity of Formal Systems lecture provides an opportunity to raise awareness of the tight connection that exists between Kolmogorov Complexity and Circuit Complexity, and to argue that this connection is useful and interesting. Kolmogorov complexity has been used to provide examples of complete sets for classes such as EXP and PSPACE that are fundamentally differen...