ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 4, pp. 393–399.
Pleiades Publishing, Inc., 2009.
Original Russian Text
S.S. Marchenkov, S.I. Krivospitsky, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 4, pp. 107–114.
Superpositions of Continuous Functions
Deﬁned on a Baire Space
S. S. Marchenkov and S. I. Krivospitsky
Faculty of Computational Mathematics and Cybernetics,
Lomonosov Moscow State University
Received June 17, 2009; in ﬁnal form, September 10, 2009
Abstract—We consider uniformly continuous functions on a Baire space and introduce the
notion of a continuity modulus of a function. We formulate a condition on the growth of
the continuity modulus ϕ guaranteeing that superpositions of n-ary functions with continuity
modulus ϕ do not exhaust all (n + 1)-ary functions with continuity modulus ϕ for any n.
Moreover, negating this property leads to the inverse eﬀect.
Research in the ﬁeld of representing continuous functions of many variables as superpositions
of continuous functions of fewer variables was initiated by Hilbert’s 13th problem. In 1956–57,
A.N. Kolmogorov [1,2] and V.I. Arnold  proved that every continuous function of several variables
can be represented as a superposition of continuous functions of two variables. Moreover, it is shown
in  that this representation can be restricted to unary functions and addition.
However, a few years earlier A.G. Vitushkin [4,5] showed that under certain smoothness condi-
tions not every function of n + 1 variables can be represented as a superposition of functions of n
variables (concerning further research in this direction, see [6–8]). In , results were proved using
the multidimensional variation techniques .
In , the idea of getting similar results with methods of discrete mathematics was put forward.
The clearest exposition of this idea was given in the study of the so-called deterministic functions, a
relatively small subclass of continuous functions on a Baire space. This idea was further developed
in . Here, the notion of a continuity modulus allowed ﬁnding the border line between n-ary
continuous functions that cannot generate all (n + 1)-ary continuous functions by compositions
and n-ary functions that can. The main results of  were proved with information-theoretic and
combinatorial considerations based on Boolean functions.
In this paper, the idea of  is brought to the conclusion. In Theorem 2, proved below, we give
a condition on the continuity modulus ϕ (which is a substantial weakening of the corresponding
condition in ) such that under this condition, superpositions of n-ary functions with continuity
modulus ϕ do not exhaust all (n+1)-ary functions with continuity modulus ϕ for every n.Moreover,
negating this property leads to the inverse eﬀect: superpositions of binary functions with continuity
modulus ϕ can generate all continuous functions with continuity modulus ϕ
(this fact was proved
We begin with the necessary deﬁnitions. In deﬁning formulas and superpositions we mainly
Let M be a class of functions. By induction, we deﬁne a formula over M .Iff denotes an n-ary
function from M,andx
are distinct variable symbols, then we call f (x
Supported in part by the Russian Foundation for Basic Research, project no. 09-01-00701.