ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 1, pp. 54–64.
Pleiades Publishing, Inc., 2009.
Original Russian Text
An.A. Muchnik, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 1, pp. 60–70.
Algorithmic Randomness and Splitting
An. A. Muchnik
Received September 10, 2008
Abstract—Randomness in the sense of Martin-L¨of can be deﬁned in terms of lower semicom-
putable supermartingales. We show that such a supermartingale cannot be replaced by a pair
of supermartingales that bet only on even bits (the ﬁrst one) and on odd bits (the second one)
knowing all the preceding bits.
1. RANDOMNESS AND LOWER SEMICOMPUTABLE SUPERMARTINGALES
The notion of algorithmic randomness (Martin-L¨of randomness) for an inﬁnite sequence of zeros
and ones (with respect to a uniform Bernoulli distribution and independent trials) can be deﬁned
using supermartingales. In this context, a supermartingale is a nonnegative real-valued function m
on binary strings such that
m(x0) + m(x1)
for all strings x. Any supermartingale corresponds to a strategy in the following game. In the
beginning we have some initial capital (m(Λ), where Λ is the empty string). Before each round,
we put part of the money on zero, some other part on one, and throw away the rest. Then the
next random bit of the sequence is generated, the correct stake is doubled, and the incorrect one
is lost. In these terms, m(x) is the capital after bit string x appears. (If the option to throw away
a portion of money is not used, then the inequality becomes an equality and the function m is a
We say that a supermartingale m wins on an inﬁnite sequence ω if the values of m on the initial
segments of ω are unbounded. The set of all sequences where m wins is called its winning set.
Algorithmic probability theory often uses lower semicomputable supermartingales. This means
that for each x the value m(x) is a limit of a nondecreasing sequence of nonnegative rational
numbers M(x, 0),M(x, 1),..., and the mapping (x, n) → M(x, n) is computable.
The class of all lower semicomputable supermartingales has a maximal element (up to a constant
factor). Its winning set contains winning sets of all lower semicomputable supermartingales; this
set is called the set of nonrandom sequences. The complement of this set is the set of random
This deﬁnition is equivalent to the standard deﬁnition given by Martin-L¨of (e. g., see ); some-
times, it is called the criterion of Martin-L¨of randomness in terms of supermartingales.
This work was done by Andrei Muchnik (1958–2007) in 2003. Soon after that, A. Chernov wrote a draft
version of the paper based on his talk with Muchnik. Then Muchnik looked it through; he planned to
edit the text but did not manage to do it. The present text was prepared by Chernov and A. Shen in
2007–2008; they are responsible for any possible errors and inaccuracies. A draft version of the paper was
published as arXiv:0807.3156.
Supported in part by the Russian Foundation for Basic Research, project nos. 06-01-00122-a and 05-02-
02803-CNRS-a, Sycomore (ANR, CNRS, France), and NAFIT ANR-08-EMER-008-01.