Problems of Information Transmission, Vol. 41, No. 3, 2005, pp. 237–242. Translated from Problemy Peredachi Informatsii, No. 3, 2005, pp. 58–63.
Original Russian Text Copyright
2005 by Ustinov.
Nonreducible Descriptions for the Conditional
M. A. Ustinov
M.V. Lomonosov Moscow State University
Received December 6, 2004; in ﬁnal form, May 24, 2005
Abstract—Assume that a program p produces an output string b for an input string a:
p(a)=b. We look for a “reduction” (simpliﬁcation) of p, i.e., a program q such that q(a)=b
but q has Kolmogorov complexity smaller than p and contains no additional information as
compared to p (this means that the conditional complexity K(q | p) is negligible). We show
that, for any two strings a and b (except for some degenerate cases), one can ﬁnd a nonre-
ducible program p of any arbitrarily large complexity (any value larger than K(a)+K(b | a)
1. DEFINITIONS AND STATEMENTS
Let a and b be two binary strings. Consider all programs p such that p(a)=b, i.e., program p
produces output b for input a. The minimal length of such a program is close to K(b | a)ifthe
programming language is chosen in a reasonable way. Here K(b| a) is the conditional Kolmogorov
complexity, introduced by A.N. Kolmogorov in 1965 (see ).
In the sequel, we ignore additive terms of order O(log n), where n in the maximal length of strings
involved. Therefore, we may use any version of the Kolmogorov complexity (plain complexity, preﬁx
complexity, etc.) since their diﬀerence is of the order O(log n).
To avoid references to a speciﬁc programming language, we consider “descriptions” instead of
programs. A string p is called a (conditional) description of a string b given a if K(b | a, p) ≈ 0.
Hereandinwhatfollows,≈ means that the diﬀerence is O(log n), where n is the maximum length
of strings involved.
For given a and b, consider all descriptions p, i.e., all strings p such that K(b | a, p) ≈ 0. The
length of any such p is at least K(b |a)(uptoaO(log n) term), and this bound is tight. We say
that a description q is a simpliﬁcation of p if K(q | p) ≈ 0.
The string b itself is a description of b given a. It is shown in  that there exists a description
of minimal length that is a simpliﬁcation of b. Later it was shown  that this is not guaranteed for
an arbitrary description p instead of b:forsomea and b, there exists a description p of complexity
much larger than K(b | a) that cannot be simpliﬁed signiﬁcantly.
We will prove that, for all a and b (except for some degenerate cases) and for any k>K(a)+
K(b| a), there exists a program p such that K(p) ≈ k, p transforms a into b,andp cannot be
The degenerate cases (where simpliﬁcation is possible) are the following two: K(a) ≈ 0 (in this
case, b is a simpliﬁcation of p), and K(b | a) ≈ 0 (in this case, the empty string is a simpliﬁcation of p).
An exact statement is as follows.
2005 Pleiades Publishing, Inc.