m-sophistication

TL;DR

The paper introduces m-sophistication as a new measure of string complexity, relating it to existing concepts and showing its non-approximability by semicomputable functions.

Contribution

It defines m-sophistication, establishes bounds with coarse and standard sophistication, and proves its non-approximability by semicomputable functions.

Findings

m-sophistication is bounded by coarse sophistication and standard sophistication.

It provides a probabilistic near sufficient statistic for strings based on m-sophistication.

m-sophistication cannot be approximated by semicomputable functions, even with large errors.

Abstract

The m-sophistication of a finite binary string x is introduced as a generalization of some parameter in the proof that complexity of complexity is rare. A probabilistic near sufficient statistic of x is given which length is upper bounded by the m-sophistication of x within small additive terms. This shows that m-sophistication is lower bounded by coarse sophistication and upper bounded by sophistication within small additive terms. It is also shown that m-sophistication and coarse sophistication can not be approximated by an upper or lower semicomputable function, not even within very large error.