The Kleene hierarchy classification of recursively random sequences

The Kleene Hierarchy Classification of Recursively Random Sequenceso)

Introduction. In [1], Church suggests a precise formulation for the notion of random sequence as conceived by von Mises. This paper introduces a modified formulation to avoid some objections inherent in the "classical" case (see Loveland [8] or [9]), and defines corresponding sets which are then located in the Kleene hierarchy of recursive unsolvability. More precisely, we locate a collection o...

A Finite Hierarchy of the Recursively Enumerable Real Numbers

For any set A of natural numbers, denote by xA the corresponding real number such that A is just the set of “1” positions in its binary expansion. In this paper we characterize the number xA for some classes of recursively enumerable sets A. Applying finite injury priority methods we show that there is a d-r.e. set A such that xA is not a semicomputable real number (which corresponds to the lim...

Recursive classification of pseudo-random sequences

Rates of Convergence of Recursively Defined Sequences

This paper gives a generalization of a result by Matiyasevich which gives explicit rates of convergence for monotone recursively defined sequences. The generalization is motivated by recent developments in fixed point theory and the search for applications of proof mining to the field. It relaxes the requirement for monotonicity to the form xn+1 ≤ (1 + an)xn + bn where the parameter sequences h...

Programming and automating mathematics in the Tarski-Kleene hierarchy

We present examples from a reference implementation of variants of Kleene algebras and Tarski’s relation algebras in the theorem proving environment Isabelle/HOL. For Kleene algebras we show how models can be programmed, including sets of traces and paths, languages, binary relations, max-plus and min-plus algebra, matrices, formal power series. For relation algebras we discuss primarily proof ...