Universality, optimality, and randomness deficiency

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Abstract

A Martin-Löf test U is universal if it captures all non-Martin-Löf random sequences, and it is optimal if for every ML-test V there is a cω such that n(Vn+cUn). We study the computational differences between universal and optimal ML-tests as well as the effects that these differences have on both the notion of layerwise computability and the Weihrauch degree of LAY, the function that produces a bound for a given Martin-Löf random sequence's randomness deficiency. We prove several robustness and idempotence results concerning the Weihrauch degree of LAY, and we show that layerwise computability is more restrictive than Weihrauch reducibility to LAY. Along similar lines we also study the principle RD, a variant of LAY outputting the precise randomness deficiency of sequences instead of only an upper bound as LAY.

MSC

03D32
68Q30
03D30
03F60

Keywords

Universal Martin-Löf test
Optimal Martin-Löf test
Randomness deficiency
Layerwise computability
Weihrauch degrees

Cited by (0)

Research was partially completed while the second author was visiting the Institute for Mathematical Sciences of the National University of Singapore in April 2015.

1

Rupert Hölzl was supported by a Feodor Lynen postdoctoral research fellowship of the Alexander von Humboldt Foundation and is supported by the Ministry of Education of Singapore through grant R146-000-184-112 (MOE2013-T2-1-062).

2

Paul Shafer is an FWO Pegasus Long Postdoctoral Fellow. He was also supported by the Fondation Sciences Mathématiques de Paris, and he received travel support from the Bayerisch-Französisches Hochschulzentrum/Centre de Coopération Universitaire Franco-Bavarois.