A generalization of Levin-Schnorr's theorem

TL;DR

This paper generalizes the concept of partial randomness and complexity in reals, establishing a duality between randomness and complexity through a generalized Levin-Schnorr theorem, and explores arithmetic-based randomness via relativization.

Contribution

It introduces a broader definition of partial randomness and complexity, and proves a generalized Levin-Schnorr duality theorem, expanding understanding of randomness in arithmetic contexts.

Findings

Established a generalized Levin-Schnorr duality theorem

Extended the definition of partial randomness and complexity

Analyzed randomness using relativization to $ ext{Pi}^0_1$-classes

Abstract

In this paper, we will generalize the definition of partially random or complex reals, and then show the duality of random and complex, i.e., a generalized version of Levin-Schnorr's theorem. We also study randomness from the view point of arithmetic using the relativization to a complete $\Pi^0_1$-class.