Sub-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-L\"of tests, (2) Kolmogorov complexity, and (3) martingales. We show these notions can be equivalently defined with prefix-free Kolmogorov complexity. We prove that one direction of van Lambalgen's theorem holds for relative computability, but the other direction fails. We discuss statistical properties of these notions of randomness

Effective Capacity and Randomness of Closed Sets

We investigate the connection between measure and capacity for the space of nonempty closed subsets of {0,1}*. For any computable measure, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets which have nonempty intersection with Q. We prove an effective version of Choquet's capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions that characterize when the capacity of a random closed set equals zero or is >0. We construct for certain measures an effectively closed set with positive capacity and with Lebesgue measure zero

Index sets in computable analysis

AbstractΠ01 classes in a space X where X equals {0, 1}ω, ωω, [0, 1], or the real line real are given an effective enumeration Pe,X and the computably continuous functions are given an effective enumeration Fe,X. The notion of index sets associated with Π01 classes and with computably continuous functions is developed. The complexity of various problems of analysis is determined by the complexity of the associated index set

Effectively closed sets and graphs of computable real functions

AbstractIn this paper, we compare the computability and complexity of a continuous real function F with the computability and complexity of the graph G of the function F. A similar analysis will be carried out for functions on subspaces of the real line such as the Cantor space, the Baire space and the unit interval. In particular, we define four basic types of effectively closed sets C depending on whether (i) the set of closed intervals which with nonempty intersection with C is recursively enumerable (r.e.), (ii) the set of closed intervals with empty intersection with C is r.e., (iii) the set of open intervals which with nonempty intersection with C is r.e., and (iv) the set of open intervals with empty intersection with C is r.e. We study the relationships between these four types of effectively closed sets in general and the relationships between these four types of effectively closed sets for closed sets which are graphs of continuous functions

Effective Symbolic Dynamics

AbstractWe investigate computable subshifts and the connection with effective symbolic dynamics. It is shown that a decidable Π10 class P is a subshift if and only if there is a computable function F mapping 2N to 2N such that P is the set of itineraries of elements of 2N. A Π10 subshift is constructed which has no computable element. We also consider the symbolic dynamics of maps on the unit interval

Algorithmic Randomness and Capacity of Closed Sets

We investigate the connection between measure, capacity and algorithmic randomness for the space of closed sets. For any computable measure m, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We prove an effective version of Choquet's capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions on the measure m that characterize when the capacity of an m-random closed set equals zero. This includes new results in classical probability theory as well as results for algorithmic randomness. For certain computable measures, we construct effectively closed sets with positive capacity and with Lebesgue measure zero. We show that for computable measures, a real q is upper semi-computable if and only if there is an effectively closed set with capacity q