Complexity of Randomness Notions
نویسندگان
چکیده
Schnorr famously proved that Martin-Löf-randomness of a sequence A can be characterised via the complexity of A’s initial segments. Nies, Stephan and Terwijn as well as independently Miller showed that Kolmogorov randomness coincides with Martin-Löf randomness relative to the halting problem K; that is, a set A is Martin-Löf random relative to K iff there is no function f such that for all m and all n > f(m) it holds that C(A(0)A(1) . . . A(n)) ≤ n−m. In the present work it is shown that characterisations of this style can also be given for other randomness criteria like strongly random, Kurtz random relative to K, PA-incomplete Martin-Löf random and strongly Kurtz random; here one does not just quantify over all functions f but over functions f of a specific form. For example, A is Martin-Löf random and PA-incomplete iff there is no A-recursive function f such that for all m and all n > f(m) it holds that C(A(0)A(1) . . . A(n)) ≤ n −m. The characterisation for strong randomness relates to functions which are the concatenation of an A-recursive function executed after a K-recursive function; this solves an open problem of Nies. In addition to this, characterisations of a similar style are also given for Demuth randomness and Schnorr randomness relative to K. Although the unrelativised versions of Kurtz randomness and Schnorr randomness do not admit such a characterisation in terms of plain Kolmogorov complexity, Bienvenu and Merkle gave one in terms of Kolmogorov complexity defined by computable machines.
منابع مشابه
Chaos/Complexity Theory and Education
Sciences exist to demonstrate the fundamental order underlying nature. Chaos/complexity theory is a novel and amazing field of scientific inquiry. Notions of our everyday experiences are somehow in connection to the laws of nature through chaos/complexity theory’s concerns with the relationships between simplicity and complexity, between orderliness and randomness (Retrieved from http://www.inc...
متن کاملDescriptive Set Theoretical Complexity of Randomness Notions
We study the descriptive set theoretical complexity of various randomness notions.
متن کاملSub-computable Boundedness Randomness
This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for PSPACE functions. These new notions are robust in that there are equivalent formulations in terms of (1) Martin-Löf tests, (2) Kolmogorov complexity, and (3)...
متن کاملAround Kolmogorov complexity: basic notions and results
Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. There are several textbooks and monographs devoted to this theory [4, 1, 5, 2, 7] where one can find the detailed exposition of many difficult results as well as historical references. However, it seems that a short survey of its basi...
متن کاملOn Elementary Computability-Theoretic Properties of Algorithmic Randomness
In this paper we apply some elementary computability-theoretic notions to algorithmic complexity theory with the aim of understanding the role and extent of computability techniques for algorithmic complexity theory. We study some computability-theoretic properties of two different notions of randomness for finite strings: randomness based on the blank-endmarker complexity measure and Chaitin’s...
متن کاملCdmtcs Research Report Series on Hypersimple Sets and Chaitin Complexity on Hypersimple Sets and Chaitin Complexity
In this paper we study some computability theoretic properties of two notions of randomness for nite strings: randomness based on the blank-endmarker complexity measure and Chaitin randomness based on the self-delimiting complexity measure. For example, we nd the position of RAND and RAND at the same level in the scale of immunity notions by proving that both of them are not hyperimmune sets. A...
متن کامل