Randomness notions and reverse mathematics

TL;DR

This paper explores the logical strength of various randomness notions within second-order arithmetic, establishing equivalences and implications among them, and analyzing their foundational significance in reverse mathematics.

Contribution

It proves the equivalence of 2-randomness and infinitely often C-incompressibility in RCA_0, and clarifies the relationships among multiple randomness notions in reverse mathematics.

Findings

Equivalence of 2-randomness and C-incompressibility in RCA_0

Hierarchy of randomness notions provable in RCA_0

Equivalence of balanced randomness and Martin-Löf randomness

Abstract

We investigate the strength of a randomness notion $\mathcal R$ as a set-existence principle in second-order arithmetic: for each $Z$ there is an $X$ that is $\mathcal R$-random relative to $Z$. We show that the equivalence between $2$-randomness and being infinitely often $C$-incompressible is provable in $\mathsf{RCA}_0$. We verify that $\mathsf{RCA}_0$ proves the basic implications among randomness notions: $2$-random $\Rightarrow$ weakly $2$-random $\Rightarrow$ Martin-L\"{o}f random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $\mathsf{RCA}_0$ the existence of computable randoms is equivalent to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-L\"{o}f randoms, and we describe a sense in which this result is nearly optimal.