Kolmogorov Complexity and Solovay Functions
نویسندگان
چکیده
Solovay [19] proved that there exists a computable upper bound f of the prefix-free Kolmogorov complexity function K such that f(x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) ≤ f(x)+O(1) for all x and f(x) ≤ K(x) + O(1) for infinitely many x, which we call Solovay functions. We show that Solovay functions present interesting connections with randomness notions such as Martin-Löf randomness and K-triviality.
منابع مشابه
Solovay functions and their applications in algorithmic randomness
Classical versions of Kolmogorov complexity are incomputable. Nevertheless, in 1975 Solovay showed that there are computable functions f ≥ K + O(1) such that for infinitely many strings σ, f(σ) = K(σ) + O(1), where K denotes prefix-free Kolmogorov complexity (while C denotes plain Kolmogorov complexity). Such an f is now called a Solovay function. We prove that many classical results about K ca...
متن کاملSolovay functions and K-triviality
As part of his groundbreaking work on algorithmic randomness, Solovay demonstrated in the 1970s the remarkable fact that there are computable upper bounds of prefix-free Kolmogorov complexity K that are tight on infinitely many values (up to an additive constant). Such computable upper bounds are called Solovay functions. Recent work of Bienvenu and Downey [STACS 2009, LIPIcs 3, pp 147-158] ind...
متن کاملAn Oracle Strongly Separating Deterministic Time from Nondeterministic Time, via Kolmogorov Complexity
Hartmanis used Kolmogorov complexity to provide an alternate proof of the classical result of Baker, Gill, and Solovay that there is an oracle relative to which P is not NP. We refine the technique to strengthen the result, constructing an oracle relative to which a conjecture of Lipton is false.
متن کاملEvery 2-random real is Kolmogorov random
We study reals with infinitely many incompressible prefixes. Call A ∈ 2 Kolmogorov random if (∃∞n) C(A n) > n − O(1), where C denotes plain Kolmogorov complexity. This property was suggested by Loveland and studied by Martin-Löf, Schnorr and Solovay. We prove that 2-random reals are Kolmogorov random.1 Together with the converse—proved by Nies, Stephan and Terwijn [11]—this provides a natural c...
متن کاملRandomness and Reducibility
How random is a real? Given two reals, which is more random? If we partition reals into equivalence classes of reals of the “same degrees of randomness”, what does the resulting structure look like? The goal of this paper is to look at questions like these, specifically by studying the properties of reducibilities that act as measures of relative randomness, as embodied in the concept of initia...
متن کامل