Results 1 
3 of
3
Limits on the Computational Power of Random Strings
, 2010
"... Let C(x) andK(x) denote plain and prefix Kolmogorov complexity, respectively, and let RC and RK denote the sets of strings that are “random ” according to these measures; both RK and RC are undecidable. Earlier work has shown that every set in NEXP is in NP relative to both RK and RC, and that every ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
Let C(x) andK(x) denote plain and prefix Kolmogorov complexity, respectively, and let RC and RK denote the sets of strings that are “random ” according to these measures; both RK and RC are undecidable. Earlier work has shown that every set in NEXP is in NP relative to both RK and RC, and that every set in BPP is polynomialtime truthtable reducible to both RK and RC [ABK06a, BFKL10]. (All of these inclusions hold, no matter which “universal ” Turing machine one uses in the definitions of C(x) andK(x).) Since each machine U gives rise to a slightly different measure CU or KU, these inclusions can be stated as: • BPP ⊆ DEC ∩ ⋂ U
On Process Complexity
"... Process complexity is one of the basic variants of Kolmogorov complexity. Unlike plain Kolmogorov complexity, process complexity provides a simple characterization of randomness for real numbers in terms of initial segment complexity. Process complexity was first developed in (Schnorr 1973). Schnorr ..."
Abstract
 Add to MetaCart
(Show Context)
Process complexity is one of the basic variants of Kolmogorov complexity. Unlike plain Kolmogorov complexity, process complexity provides a simple characterization of randomness for real numbers in terms of initial segment complexity. Process complexity was first developed in (Schnorr 1973). Schnorr’s definition of a process, while simple, can be difficult to work with. In many situations, a preferable definition of a process is that given by Levin in (Levin & Zvonkin 1970). In this paper we define a variant of process complexity based on Levin’s definition of a process. We call this variant strict process complexity. Strict process complexity retains the main desirable properties of process complexity. Particularly, it provides simple characterizations of MartinLöf random real numbers, and of computable real numbers. However, we will prove that strict process complexity does not agree within an additive constant with Schnorr’s original process complexity. One of the basic properties of prefixfree complexity is that it is subadditive. Subadditive means that there is some constant d such that for all strings σ, τ the complexity of στ (σ and τ concatenated) is less than or equal to the sum of the complexities of σ and τ plus d. A fundamental question about any complexity measure is whether or not it is subadditive. In this paper we resolve this question for process complexity by proving that neither of these process complexities is subadditive. 1
Some Applications of Randomness in Computational Complexity
, 2013
"... In this dissertation we consider two different notions of randomness and their applications to problems in complexity theory. In part one of the dissertation we consider Kolmogorov complexity, a way to formalize a measure of the randomness of a single finite string, something that cannot be done u ..."
Abstract
 Add to MetaCart
In this dissertation we consider two different notions of randomness and their applications to problems in complexity theory. In part one of the dissertation we consider Kolmogorov complexity, a way to formalize a measure of the randomness of a single finite string, something that cannot be done using the usual distributional definitions. We let R be the set of random strings under this measure and study what resourcebounded machines can compute using R as an oracle. We show the surprising result that under proper definitions we can in fact define wellformed complexity classes using this approach, and that perhaps it is possible to exactly characterize standard classes such as BPP and NEXP in this way. In part two of the dissertation we switch gears and consider the use of randomness as a tool in propositional proof complexity, a subarea of complexity theory that addresses the NP vs. coNP problem. Here we consider the ability of various proof systems to efficiently refute randomly generated unsatisfiable 3CNF and 3XOR formulas. In particular, we show that certain restricted proof systems based on Ordered Binary Decision Diagrams requires exponentialsize refutations of these formulas. We also