Upper semicomputable sumtests for lower semicomputable semimeasures

TL;DR

This paper characterizes upper semicomputable sumtests for lower semicomputable semimeasures using Kolmogorov complexity, establishing bounds on their size and their relation to the Halting problem, with implications for understanding their pathological behavior.

Contribution

It provides a characterization of upper semicomputable sumtests relative to lower semicomputable semimeasures and analyzes their size bounds and complexity-theoretic properties.

Findings

Logarithm of such tests is bounded by   |x| + O(  |x|)

Existence of tests exceeding  |x| - O(  |x|) infinitely often

Tests are large only for strings with high mutual information with the Halting problem

Abstract

A sumtest for a discrete semimeasure $P$ is a function $f$ mapping bitstrings to non-negative rational numbers such that \[ \sum P(x)f(x) \le 1 \,. \] Sumtests are the discrete analogue of Martin-L\"of tests. The behavior of sumtests for computable $P$ seems well understood, but for some applications lower semicomputable $P$ seem more appropriate. In the case of tests for independence, it is natural to consider upper semicomputable tests (see [B.Bauwens and S.Terwijn, Theory of Computing Systems 48.2 (2011): 247-268]). In this paper, we characterize upper semicomputable sumtests relative to any lower semicomputable semimeasures using Kolmogorov complexity. It is studied to what extend such tests are pathological: can upper semicomputable sumtests for $m(x)$ be large? It is shown that the logarithm of such tests does not exceed $\log |x| + O(\log^{(2)} |x|)$ (where $|x|$ denotes the length of $x$ and $\log^{(2)} = \log\log$) and that this bound is tight, i.e. there is a test whose logarithm exceeds $\log |x| - O(\log^{(2)} |x|$) infinitely often. Finally, it is shown that for each such test $e$ the mutual information of a string with the Halting problem is at least $\log e(x)-O(1)$; thus $e$ can only be large for ``exotic'' strings.