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JOURNAL OF COMPUTER AND SYSTEM SCIENCES 49, 605--619 (1994)
On Hausdorff and Topological Dimensions of the
Kolmogorov Complexity of the Real
Line*
JIN-YI CAI t
Department of Computer Science, State University of New York at Buffalo,
Buffalo, New York 14260
AND
JURIS HARTMANIS $
Department of Computer Science, Cornell University,
Ithaca, New York 14853
Received July 21, 1991; revised July 21, 1992
We investigate the Kolmogorov complexity of real numbers. Let K be the Kolmogorov
complexity function; we determine the Hausdorff dimension and the topological dimension of
the graph of K. Since these dimensions are different, the graph of the Kolmogorov complexity
function of the real line forms a fractal in the sense of Mandelbrot. We also solve an open
problem of Razborov using our exact bound on the topological dimension. © 1994 Academic
Press, Inc.
1. INTRODUCTION
We investigate the Kolmogorov complexity of real numbers. We show that, from
a computational point of view, the real line is not a set of points without individual
distinguishing
characteristics, but rather, the real numbers and their complexity
form a very complex object.
We consider a
well-defined function that assigns to each real number its
Kolmogorov complexity
and determine the Hausdorff dimension and the topological
dimension of the graph of the Kolmogorov complexity of the real line. In par-
ticular, the Hausdorff dimension is strictly greater than the topological dimension.
Thus the graph of the Kolmogorov complexity of the real line forms a
fractal
in the
sense of Mandelbrot
[M].
* A preliminary version of the paper titled "The Real Line is a Fractal" has appeared in "The
Structure in Complexity Theory Conference 1989."
t Research supported by NSF Grant CGR-8709818, while the author was at Yale Unversity.
Research supported by NSF Grants CCR-8520597 and CCR-8823053.
6O5
0022-0000/94 $6.00
Copyright © 1994 by Academic Press, Inc,
All rights of reproduction in any form reserved.
606 c~a AND HARTMANIS
In Section 2, we give some motivations for our results and state all the necessary
definitions. In Section 3, we deal with the Hausdorff dimension of the Kolmogorov
complexity function of the real line. We prove matching upper and lower bounds
for the "fibre set" of points with Kolmogorov complexity equal to a. It has
Hausdorff dimension exactly a. We then determine the Hausdorff dimension of the
graph of K to be exactly 2.
In Section 4, we deal with the topological dimension of the same graph. We show
that the topological dimension is exactly 1. Our determination of the topological
dimension of the graph of the Kolmogorov complexity of the reals solves an inter-
esting open problem by Razborov IRa] who asked what relationship can there be
between
K(x)
and
K(K(x))?
He asked, for instance, is it true that there exist easily
computable functions f from [0, 1] to [0, 1], such that,
K(K(x))<~f(K(x)),
for all
0 ~< x ~< 1 ? In particular, is it true that
K(K(x)) <~ K(x)
for all 0 ~< x ~< 1 ?
We show that the answer to Razborov's question is negative, and it follows easily
from our exact bound on the topological dimension. In fact we give a stronger
statement in Theorem 4.4.
Our work grew out of an attempt to formulate a theory of computational
complexity over the reals. In our approach, for a wide class of functions such as any
non-constant polynomial with rational coefficients, or, analytic functions with
uniformly computable coefficients, the complexity of a real number is invariant
under the transformation. Thus, the nature of the complexity graph is preserved by
finite iterations of such computable functions.
Previous work on recursive real analysis has adopted a different approach, where
the objects one deals with are necessarily countable. This has the advantage of
being more true to recursion theory, but much of "continuous mathematics" is
rendered inapplicable. Recently Blum, Shub, and Smale considered a complexity
theory of real numbers [BSS], where the emphasis is on the algebraic operational
costs. They make the assumption that every real number has unit complexity. On
the other hand, their approach brings more classical mathematics into the picture.
We believe an approach similar to the finitary Kolmogorov complexity should be
pursued. Such an approach has both the advantage of being more realistic in
differentiating the complexity between easily computable numbers, on the one
hand, and intractable ones, on the other, and much of classical mathematics,
especially non-trivial analysis and topology, are inherently applicable.
The search for appropriate complexity theories for real number computations is
a challenging task. We do not claim to have found the ultimate approach; it is
doubtful whether such a single approach exists that is "right" for all. However, we
hope that this paper may help to stimulate a systematic investigation to the nature
of the underlying computational domains in real computations.
Some of our results in Section 3 were independently obtained by Staiger and
others. As one referee pointed out, although motivated by different issues,
Besicovitch had pursued a highly related notion that can in fact be used to give
separate proofs of some of our results in Section 3. We will discuss this in more
detail at the end of that section. For related work please see [L2, Ry, Stl, St2, Be].
HAUSDORFF AND TOPOLOGICAL DIMENSIONS 607
2.
DEFINITIONS AND PRELIMINARIES
What is randomness ? And what is a random object? Surely a large object with
any easily distinguishable patterns, or one which can be generated by any well
specified short procedure, should not be considered random. The
Kolmogorov
complexity K(x)
of a binary string x is defined to be the information content of x,
i.e., the size in bits of the smallest input string---program--which will cause a fixed
universal Turing machine to produce x. (The choice of the fixed universal Turing
machine introduces at most an additive constant in the value of
K(x),
which
asymptotically can be ignored. We shall fix one universal machine once and for all.)
The notion of Kolmogorov complexity was due to Solomonoff, Kolmogorov, and
Chaitin IS, K, C]. There have been quite a few variations of the original notion of
Kolmogorov complexity, most notably by Chaitin and Levin on self-delimiting
Kolmogorov complexity [C, L], and the resource-bounded versions, such as poly-
nomial time/space bounded Kolmogorov complexity [Ba, HI. However, the result
of this paper is
robust,
in the sense that any and all such definitions lead to the
same conclusion--the computational line is a fractal. For definiteness, we adopt the
classical definition through out this paper.
To define the complexity of a real number x, we consider any reasonable
representation of x, such as its binary expansion. We take the n-bit prefix xn of x
and consider its normalized Kolmogorov complexity
K(xn)/n.
It should be clear
that the choice of which particular enumeration scheme to represent x (for instance,
ternary, decimal, or continued fraction) is of no significance, asymptotically
speaking, as long as the conversion between them is computable. (ff we are using
polynomial time bounded Kolmogorov complexity, then we should require polyno-
mial time conversion algorithms, which certainly exist for those we mentioned.)
Now we define the complexity of x as
K(x)
= lim
K(x~)/n.
n~oo
We denote the graph of the function K by F~.
A technical note. When the limit does not exist, we may take any reasonable
value, such as the arithmetic mean of upper and lower limit ~ :
K(x)
= ½ (lim inf
K(x,,)/n +
lira sup
K(x,)/n).
tl --~. oo n~-~ co
It follows from the definition that
Fx
has perfect scaling properties:
K(rx+s)=K(x),
VxeR,
r,s~Q, r~O.
In fact this scaling property can be significantly strengthened to arbitrary polyno-
mials (or even analytic functions) with (uniformly) computable coefficients. To see
1 Such arbitrariness is perhaps disquieting; however, the reassuring fact is that it leads to the same
theory no matter how one extends the definition, as we will see.
608 CAI AND HARTMANIS
this, we first note that the zero set of any such function f (and therefore that of its
derivative f') is discrete in the domain of its definition and consists of computable
numbers (in the sense of Turing). Thus, modulo a discrete set of points, where
K(x) = 0 = K(f(x)),
the function f is locally monotonic with a non-zero derivative.
This enables us to prove
K(x) = K(f(x)),
for all x. As a consequence of this sealing
property, we will only consider the function K as defined on the unit interval
Z=[O, I].
We will investigate the Hausdorff and topological dimensions of the graph of K.
A general reference on dimension theory can be found in VHW].
DEFINITION 2.1. Given a set S in a metric space X, and any real number p ~> 0,
let e > 0 and
mp(S)=inf ~ 6(Sy,
i>~l
where S = U ~=1 S~ is any decomposition of S in a countable number of subsets of
diameter
6(S~)
less than e and the superscript p denotes exponentiation. Let
mp(S)
= lim m;(S);
8~0
rap(S)
is called the p-dimensional (Hausdorff) measure of S.
We observe that the limit in the definition exists (including infinity oe), since
m~p(S)
is monotonic non-decreasing as e~0. We also note that
p<q
and
mp(S)
< o(3
imply that mq(S) = O.
DEFINITION 2.2. Given a set S in a metric space X, the Hausdorff dimension of
S, dim~,(S), is the supremum of all real numbers p such that
rnp(S)>0.
Clearly the above definition of the Hausdorff dimension of S can be equivalently
stated in terms of the limit
lim inf ~
6(Di) p,
~0 i~>1
where the infimum takes over all countable coverings of S by open (or closed) discs
(9 =-{Di[i>~
1 }, with supi~>16(Di)~< ~. In what follows, we will use the notion of a
covering to compute the Hausdorff dimension.
As an example, it is well known that the (classical) Cantor set cg has Hausdorff
dimension log 2/log 3. This can be seen intuitively by the following family
of finite
coverings for cg inductively defined. (91 consists of a single interval [0, 1]; (9 k
consists of all the intervals that are the first or the last third of any interval in (gk- 1.
(Although a rigorous proof of equality can be given along this line, the existence
of such a cover only shows that dimx/(<g) ~< log 2/log 3. Note also that in general a
countable
cover is used instead of a finite one.)
HAUSDORFF AND TOPOLOGICAL DIMENSIONS 609
We now define the notion of the topological dimension of a space X. It turns out
that there are three commonly used concepts of dimension in the literature.
Although for more general spaces they do not necessarily agree, they do agree on
all separable metric spaces (spaces with a countable dense subset.) Since this is the
case for our investigation (subspaces of Euclidean space) we will give just one
definition of the topological dimension, also known as the Urysohn-Menger (small
inductive) dimension.
DEFINITION 2.3. Given a metric space X,
1. dimr(X)= -1, ifX=~;
2. direr(X) ~< n, if for every p e X and open set U containing p there is an
open set V satisfying
pc V~ U and dimT(~V )~<n- 1.
3. dimr(Z) =n, if dimr(X)~<n and dimr(X) 4; n-1.
4. dimr(X) = 0% if direr(X) 4; n for all n.
We note that when X is a subspace, say of an Euclidean space, the topology on
35 is the induced topology. If X is everywhere dense, then the boundary of an open
set in
X, ~x(O n X),
equals ~O ~ X.
As an example, any non-empty finite or countable space is zero-dimensional. Any
subset of the real line that does not contain any interval also has topological dimen-
sion zero. And as a consequence of the Brouwer fix-point theorem, the Euclidean
n-space has topological dimension n [Br]. (The non-trivial part is to show that
dimT(R n) 4; n-1).
Note that the topological dimension of a space is always an integer (if it is finite).
It is known that the Hausdorff dimension is always greater than or equal to the
topological dimension. Mandelbrot defined a space to be a fractal if they do not
agree; i.e., X is called a fractal if dimH(X ) > dimr(X). For our set FK, the graph
of K, we will establish just that:
dimH(FK)=
2 and dimr(FK)= 1.
3. THE HAUSDORFF DIMENSION OF /'K
The main theorem in this section is the following.
THEOREM 3.1.
For any numbers 0 <~ a < b <~ 1, the Hausdorff dimension of the set
rKr~ ([0, 1] × [a, b])
bl+b.
An immediate corollary is
COROLLARY 3.2.
The Hausdorff dimension of the graph
dimH (Fx) = 2.
610
CAI AND HARTMANIS
We first investigate the "fibre sets"
Fa = {xe
[0, 1] [K(x) =a}, for 0~<a~< 1. We
will show that dimn(Fa)= a, from which the main theorem will follow.
LEMMA 3.3.
Almost all points in
[0, 1 ]
have complexity
1;
i.e., F 1 has full
Lebesgue measure.
The proof is a simple counting argument, which we shall omit here. The next
lemma is in fact implied by the more general Lemma 3.6 (where the proof is inde-
pendent of this). But we include a separate proof sketch here since it introduces in
the simple setting a method which will be generalized later in the proof of
Theorem 4.1. It also has a corollary that will be used in Section 4.
LEMMA 3.4:
For all a, 0 <<. a <~
1,
F a is an uncountable infinite set.
Proof
For notational simplicity we assume that 0 < a < 1. (The case a = 1 is
implied by Lemma 3.3.) To exhibit a real number x with complexity a, we first take
a random string as the initial segment of x, so long that the normalized complexity
is "pushed" above a. Then we append any "simple" string such as all zeros, so long
that the normalized complexity is "pushed" below a. Now we repeat the process,
with ever smaller oscillation. The number x defined by this infinite sequence of bits
clearly has complexity a. Moreover, if we used simple strings such as all ones, in
addition to all zeros, it is clear there are uncountably many points in Fa. |
A consequence of this lemm and the scaling property noted in Section 2 is
Corollary 3.5.
COROLLARY 3.5.
The graph F K is everywhere dense in the unit square.
Consider the Cantor set cg again. We claim that the fibre set Fc, where
c = log 2/log 3 contains "almost all" points of cg. It follows that dim~(Fc) ~> c, for
c = log 2/log 3. First we have to clarify the meaning of "almost all" here, as the
Cantor set itself has Lebesgue measure zero. Intuitively the notion of a "random"
Cantor set point should be clear, as points in cg are represented by ternary numbers
with 0 or 2 as its bits. This can be formalized as follows: Define a map e from the
Cantor set cg onto the unit interval [0, 1 ] that is one-to-one, except on a countable
subset of cg. Furthermore, modulo a countable subset the map e is an isomorphism
between the measure space cg endowed with the c-dimensional Hausdorff measure
and the unit interval with the Lebesgue measure. The map can be defined by a
sequence of "expansion" as follows : first map the points ½ and 2 to ½ and expand the
two intervals [0, ½] and [z,3 13 linearly onto [0, ½] and [½, 1], respectively. Then
recursively expand the remaining two intervals exactly the same way, ad infinitum.
It can be shown rigorously that all claims of the map e are satisfied. Now every
x ~ ~ certainly has complexity no more than c = log 2/log 3; in order to obtain
[log2 3 • n_] bits in a binary expansion we need no more than n bits asymptotically.
On the other hand, just as in Lemma 3.3, a "random" point of the Cantor set (i.e.,
"almost all" under the c-dimensional Hausdorff measure) has complexity exactly c.
HAUSDORFF AND TOPOLOGICAL DIMENSIONS
611
The above discussion is capable of generalization to an arbitrary a.
LEMMA 3.6. For any a, 0 <~ a <~ 1, the fibre set F, has dimension at least a.
We observe that there is nothing special about ½ and ~ in the Cantor set construc-
tion. One can easily construct generalized Cantor sets. Let {p,/q,} be a recursive
sequence of rational numbers so that 0 < p, < q, and log p,/log q, ~ a, for the
given real number a. Such a sequence certainly exists. One constructs a generalized
Cantor set where in the n th step, we delete the middle q,- p, subintervals each of
length 1/q, of the length of intervals obtained in the (n- 1)th step. It can be shown
that almost all points (under the a-dimensional Hausdorff measure) of the
generalized Cantor set are contained in F a, and thus the latter has dimension at
least a. The lemma follows.
On the other hand, we claim that for 0 ~< a ~< 1, and any e > 0, dim/dF,) ~< a + ~.
And, hence, taking the limit, we have
LEMMA 3.7. For any a, 0 <~ a <<. 1, the fibre set Fa has dimension at most a.
Proof. Let x ~ F, and 1/k < e. Consider the family of closed intervals,
{[m/2",(m+l)/2n][K(m)<<.(a+l/k)n}, n=l, 2 .... ,
where K(m) is the Kolmogorov complexity of the binary number m. Observe that
for x with lira inf,_~ o~ K(x(n))/n < a + 1/k, where x(n) is the n-place binary expan-
sion of x, x is covered by infinitely many intervals in the above family. However,
the number of intervals of length 1/2" in the above family is bounded by
2 (a+l/k)n+l, and thus the series
.
=,
\~I
converges. Therefore its tail can be made arbitrarily small, and the tail corresponds
to a countable covering of the set F~ with arbitrarily small diameter. |
We note that the preceding proof actually proved more, namely that
dimz~(Uo<~y~aFy)<<.a, for all a. Combining the above two lemmas, we have
Theorem 3.8.
THEOREM 3.8. For any a, 0 <~ a ~ 1, the fibre set Fa has dimension exactly a.
We now turn to Theorem 3.1. We need the following technical result about
Hausdorff dimension. For completeness we will include a proof here. See also
Corollary 7.12 in [F].
THEOREM 3.9. If for any y, 0 ~ y <~ 1, a "fibre set" Fw ~ I is defined and has
Hausdorff dimension at least h, then dim/~(U0,< y_< 1 (Fy x {y}))/> 1 + h.
612 CA~ AND HARTMANIS
Proof
Without loss of generality, we consider any countable covering of the set
Uo ~< y ~<1 (Fy x { y }) by squares, 5 e = { [a/, a/+ 6;] x [b/, b; + 6/] [i >~ 1 }. We note
that the covering 5 ~ naturally induces a covering for each fibre set
Fy,
{ [a/, a~ + 6i]
L i ~> 1 and b/~< y ~< bi + 6/}. Fix any e > 0, define a modified "charac-
teristic" function for each square,
{60h-~ if b/<-.y<'-.b/+6i
)~i(Y) = i
otherwise.
Since each Zi is non-negative, it follows from the monotone convergence theorem
that
Z,(y)dy= Z,(y)dy= 6~ +h-'.
i=1 i=1 i=1
We only need to show that the integral on the left approaches infinity (uniform
over all coverings) as 6=sup/~>a 6/~0. This follows from Ergorov's theorem.
We can show directly as follows: For any M large and integer n, define
S n = {y[inf~/6~-'>~ 2M, where the infimum takes over all countable coverings of
Fy
by intervals of lengths 6/, and sup
6e <~ l/n}.
Since each
Fy
has Hausdorff dimen-
sion at least h, Sn forms a monotone non-decreasing sequence of sets with limit
Un S~ = [0, 1]. Then by the continuity of Lebesgue measure, lim .... #(S,) = 1. We
choose n sufficiently large such that/~(S,) > 1 and
2,
Z,(Y)
dy >1 Z,(Y) dy >/M. |
i~l ni=l
We note that in Theorem 3.9, one can replace the interval 0 ~< y ~< 1 by any other
non-trivial interval. It follows that
dim~(FKn ([0, 1] x [a, b])) >~ dimH(FKn ([0, 1] x
[b-e,
b]))>~ 1 +b-e,
for all e > 0. On the other hand, it follows from the remark after Lemma 3.7,
dim.(Fxc~([O, 1])x[a,b]))<~dim~((
U
Fy) x[a,b])<~l+b.
\',O<~y<<.b
Theorem 3.1 follows.
As one referee pointed out, although motivated by different issues, Besicovitch
[Be] had some remarkable resuls that are highly related to what has been
presented in this section, in particular, Lemma 3.6.
Let us define the set
Ep
= {x: lira supn~ co [ # of ones in n-bits expansion of
x]/
n=p}.
Then Besicovitch proved that
Ep
has Hausdorff dimension H(p)=
-p log2 p- (1- p)log2(1-p). Although the Besicovitch set
Ep
is not a subset of
some fibre set Fa, "almost all" points of
Ep
indeed are, for a =
H(p).
This follows
exactly the same reasoning as in Lemma 3.6, where the Cantor set was used.
HAUSDORFF AND TOPOLOGICAL DIMENSIONS 613
Instead of using the Cantor set, it is now possible to use Besicovitch's result.
Consider the subset of the Besicovitch set Ep consisting of only those points with
Kolmogorov complexity that are equal to H(p). "Almost all" points of Ep belong
to this subset; thus it has Hausdorff dimension H(p). As the function
H(p) = -p log 2 p- (1 -p) log2(1 -p) is a one to one mapping from I-0, ½] onto
[-0, 1 ], it follows that our fibre set F a has Hausdorff dimension at least a, for all a,
namely the lower bound in Lemma 3.6.
Addendum
The referee in his detailed report suggested that we look for "a simultaneous
generalization of the Besicovitch and Cai-Hartmanis theorems, which preserves the
spirit and the technical content of both." He gave the following definition A, and
then he stated the following theorem B and conjecture C. 2
DEFINmON A. A real function f on A is r-expansive at x if there is a function
g from strings to strings: (1) if a ~ ~, where a and ~ are initial segments of the
binary expansion of some y E A, then g(o-)c g(r); (2) for all y ~ A, limo ~ y g(a)=
f(y); and (3) lim sup~ _~ x ([ a[/[ g(a)[ ) = r. f is r-expansive on A iff is r-expansive at
x for all x ~ A.
THEOREM B. Let A be a set of positive Lebesgue measure, and let f be r-expansive
on A. Then the Hausdorff-Besicovitch dimension off(A) is at most r; moreover, this
dimension is attained if f and g are one-one functions.
Conjecture C. Let A be a set of Hausdorff-Besicovitch dimension s, and let f be
r-expansive on A. Then the Hausdorff-Besicovitch dimension off(A) is at most rs;
moreover, this dimension is attained iff and g are one-one functions.
He outlined how one can derive the results in Theorem 3.8 using his notion of
r-expansiveness. Essentially the derivation goes as follows. For y ~ A, we will consider
the initial segments of the binary expansion of y as "programs" used in the
Kolmogorov complexity. We will consider only programs that are "extensional" in
the sense that the requirement (1) in Definition A is satisfied. Note that for any x
with K(x)= r, there exist encodings (e.g., using relative Kolmogorov complexity of
successive segments) that satisfy this extensionality. Let Af, r be the set of those y
where the recursive function f is r-expansive. Then Fr---Ufrec {f(Af, r)}. It follows
that dimH(Fr) ~< r, since in this countable union each set AF, r has dimension at most
one and, thus, each set f(Ay,~) has dimension at most r. Moreover, the equality
dim~r(Fr)=r holds. This follows from a limit argument similar to that of
Lemma 3.6 and the second part of Theorem B and Conjecture C. Thus one can
construct an appropriate A and one-one functions f and g such thatfis r-expansive
on A, dim/~(A)= 1, andf(A)___ Ft. This completes the derivation of the estimate on
the Hausdorff dimension of the fibre sets as in Theorem 3.8 : dimn(F~) = r.
2 These statements are taken from the referee's report with minor modifications.
614
CAI AND HARTMANIS
Regarding his conjecture C the referee thinks "it would be nice to have such a
theorem." We prove the conjecture in the remaining part of this addendum.
For a finite string o-e {0, 1}*, let [al denote its length, let ~ denote the real
number between 0 and 1 corresponding to o-, i.e., ~= a/2 I<, let I(o-) denote the
interval [~, ~ + 1/21<), and let [II denote the length of the interval I, so II(rr)l =
1/2 t~l. Note that VxeI(a), the ]cr[th place binary expansion of x, denoted by
al<(x ),
is just a.
We fix any r'> r, s'> s, and any e > 0. Our goal is to construct a covering N of
f(A)
with arbitrary small diameter 6(N), such that
Since for
x~A,
lira sup(Io-l/[g(a)l ) = r, where a=a,(x) and the limit is with
n ~ m, we have
n(x),
such that for all
n >t n(x),
I g(~r, (x))[ > [a, (x)[/r'
= n/r'.
Define an infinite sequence nl < n2 < -.. < nk < "", such that Vk, a cover cg k of
A exists, cg k = {I(akj'): j~> 1}, satisfying
I%1 > nk
and
E s'
II(%)1 <~.
j)l
We claim that such a covering exists because dim~(A)<
s',
and we can assume
that the cover is of this form. In fact, since dimLr(A)<
s',
for the given e, k, and nk,
there exists a covering ~; = { [ekj,
~3kS)
: J >~ 1 }, such that,
~3kS
-- C% < 1/2 ~ and,
g
E (/31cJ--Otkj)S'<2k+s'+l"
j>>- I
To obtain our cover cg k from cg;, we will replace each [e,/3) = [ekj, /3kj) by at
most two intervals which cover it and which have a combined length at most twice
that of [cq/3).
Let N be the least integer such that there exists an integer
u, ~ <, u/TV</3.
For
the least N such a u is unique; thus (u - 1)/2
N
< e and (u + 1
)/2 N >1/3.
Let NI (and
N2, respectively) be the largest integer such that
u/2N--e<l/2 N1
(and
/3 -- U/2 N ~
1/2 N2, respectively.) Then clearly N1,
N 2 >~
N, and
1/2 Nl + 1 <~
u/2 N_
c~
and
1/2 N2 + 1 </3 _ u/2 N,
by the maximality of N1 and N2.
Thus, the length of the interval I=
[(2N1-N.u -
1)/2 ~1, 2N1--N-U/UV~) is 1/U vl,
which is at most
2(u/2 N-
~); i.e., it is at most twice the length of [c~,
u/UV).
Call this
length A. Similarly the length of J= [2 u2-u.
u/2 u2,
(2 N2- u. U + 1
)/2 N2)
is at most
twice the length of
[u/2 N, 8).
Call this length B. Note that A + B =/3-- e. Now,
I11"+ IJl~'~ 2S'(A ~' + B")
~<2 s'+l max(A", B ¢)
~<2s'+t.(A+B) ''.
HAUSDORFF AND TOPOLOGICAL DIMENSIONS
615
Let c~ k consist of Ij and J1 for each [ekj, flkj) in cg;, we obtain
(IZjl~'+ IJjlS)< ~.
j>~l
The claim is proved.
Now we define a covering ~ off(A)"
~= {I(g(a))13k, I(a)eC-gk,
and
3x~A, xeI(a),
and
nk >~n(x)}.
We claim that @ is a covering off(A).
Let
x~A,
then
3k, nk>~n(x).
Consider cg k which covers A. So
3I(a)e~k
and
x ~ I(a).
Now
I(g(a))~ ~
and
f(x)e I(g(a)).
The first part is clear, as witnessed by
k and x. Since
xsI(a), a=alaL(x)
and
f(x)EI(g(a,(x)))
for all n by the exten-
sionality property of g. By setting n = lal,
f(x) ~ I(g(ala p
(x))) =
I(g(a)).
We now estimate
[I(g(a))l
for any interval in ~. For
I(g(a))~N, 3k
and
xsA,
nk>.n(x), xeI(a),
and
I(a)~f k.
By the definition of c~ k,
]a[>>.nk>~n(x).
As
a= al,l(x ),
since
xeA
and by the definition of
n(x),
Jg(a)]
= Ig(ara I (x))l > ]ajaj (X)[ laf
?.r r r
Hence,
1 1
II(g(a))[
= ~ < ~ = II(a)] l/r,.
Therefore,
[l(g(a))l ''~'< ~ Z II(al, J)t s'<e"
k=l j>~l
As we can make n~ arbitrarily large, so that the diameter 6(~) is arbitrarily
small, this proves that dimn(f(A))~<
r's'.
But as r'> r, s'> s are arbitrary, we have
shown that dimH(f(A)) ~<
rs.
In the case of one-one functions, we just apply the above theorem to f-l, and
equality follows. This completes the proof of Conjecture C and concludes this
addendum.
4. THE TOPOLOGICAL DIMENSION OF F K
In this section we prove the following theorem.
THEOREM 4.1.
The topological dimension of the set FK is 1.
571/49/3-14
616 CAI AND HARTMANIS
\
\ /
FIG. 1. The function I.
COROLLARY 4.2.
The graph Fx of the Kolmogorov complexity function K is a
fractal in the sense of Mandelbrot.
The proof of Theorem4.1 has two parts; we show that dimT(f'K)~<l and
dimr (FK) ~ 0.
It is easy to show that dimr(F,:) ~< 1. Given any point
p ~ FK,
we need to find
an arbitrarily small neighborhood of p such that its boundary has topological
dimension zero. This can be accomplished by a square (a, a')x(b,
b')~p,
where
K(a), K(a') (~ [b, b'].
Thus the boundary of the square in the subspace/'K is a part
of the fibre sets
Fb
and Fb,, which certainly has dimension zero, for it does not
contain any interval.
We show next that the topological dimension of FK is not zero. In fact, we show,
for all p ~ F K and any sufficiently small open neighborhood O of p, that the
boundary
OrKOva(25.
Recall that, by Lemma3.5,
~rKO=OOc~FK
as FK is
everywhere dense in [0, 1 ]z.
Suppose that
p = (Px, Py) ~ O.
Either
py
< 1 or
py
> 0. Without loss of generality
we assume that
py<l,
and O~[0, 1]x[0, 1). Take a small square [a,a']×
[b, b'] c O centered at p. We define a function h
l(x)=inf{yly>pyand(x,y)~O}
for
a<~x<~a'.
Surely,
py < b' < l(x)
~< 1 (see Fig. 1).
LEMMA 4.3.
Except on a countable subset of
[a, a'],
the function I satisfies
lira inf
I(z) = I(z).
z~x
Proof
We first observe that for all
x~(a,a'],
since ~O is closed,
lira infz~x-
l(z) >~ l(x).
Similarly, for all x ~ [a, a'), lira influx+
l(z) >~ l(x).
Let
J~ = {a < x ~< a' I lira inf
l(z) > l(x) + l/n},
z---* x
HAUSDORFF AND TOPOLOGICAL DIMENSIONS
617
and J- = U, > ~ J2. Similarly,
J+ -- {a ~< x < a' I lim !nf
l(z) > l(x) + 1In },
Z~
and J + = U, > 1 J+. Finally,
J = J- u J + ~_ {a ~< x ~< a'l lim inf
l(z) > l(x) }.
z-+ x
We claim that J is a countable set. Clearly, since a dual argument applies,
it suffices to show that for each n and l~<m~<n, the set
J,,m=
J~- ~ l-l(((m --
1)/n, m/n])
is countable.
For all
x ~ J~,m,
since (m -- 1
)/n < l(x)
and lira inf,
~ x- l(z) > l(x) + l/n, 3ex > O,
such that
inf{l(z) Ix - e x < z < x} >~ l(x) + 1In > m/n.
Thus
(X-~x,X)nJ2,m=;25.
It follows that
{(X--ex, X)lXeJn,m}
is a pairwise
disjoint class of open intervals. Since
ex<~a'--a<~l,
J£m must be countable, and hence so is the set J. I
We write J--{al, a2 .... }. Now, to complete the proof of Theorem4.1, we can
exhibit a point on the intersection of I) and ~O. The idea is to construct binary
sequence in stages as in Lemma 3.4, approximating a "moving target" value which
converges. Specifically, at stage i, we take the value inf/(z), where the infimum
takes over the small interval [m/2 n, (m + 1)/2 n] defined by the binary number m
which, as a binary string, was constructed up to the previous stage i-1. Then we
"push" the normalized Kolmogorov complexity closer (up or down) to this
infimum, by appending hard or easy strings. Meanwhile, we avoid one more excep-
tional point ai from J by a positive distance (starting with 00 or 11). As the nested
intervals shrink, it defines a unique number x ¢ J. Therefore, lim infz_~x
l(z)= l(x).
On the other hand, the "moving target" clearly converges to lim infz~x
l(z).
Thus
the construction yields K(x)=lim infz~ x
l(z)=l(x).
The proof of Theorem4.1 is
completed.
We remark that Theorem 4.1, as well as Theorem 3.1, are valid no matter how
one extends the definition of
K(x)
for x, where lira, ~ ~
K(xn)/n
does not exist. For
instance, for the point x we exhibited in the proof above, the limit lim, ~ ~
K(x,)/n
in fact exists.
Theorem 4.1 can be quite a powerful tool in the study of Kolmogorov complexity
of the reals. We indicate this by a simple solution to a problem of Razborov: What
relationship can there be between
K(x)
and
K(K(x))?
He asks in particular, for
instance, is it true that
K(K(x)) <~ K(x)
for all 0 ~< x ~< 1 ?
618 CAI AND HARTMANIS
The answer is negative. We simply consider the line Y= X+ e within the unit
square [0, 13 x [0, 13. As F,r intersects both triangular regions formed by the line
Y= X+ e (FK is everywhere dense in the unit square), it is impossible to have the
boundary of these triangular open sets, namely the line, disjoint from FK, by
Theorem 4.1. Hence, there exists t, such that K(t)= t+ e > t. Then, since the fiber
set Ft is nonempty, there exists x, such that K(x) = t and; thus, K(K(x)) > K(x). Of
course this argument can be generalized.
THEOREM 4.4. For any differentiable function f from [0, 1] to (0, 1) neither
K(K(x) ) <~ f(K(x)) nor K(K(x)) >~ f(K(x)) is true for all x.
ACKNOWLEDGMENTS
The authors thank Professor Wolfgang Fuchs for pointing out a simplification of a proof in an earlier
draft. We thank Professors Allen Back, David Henderson, and Tom Rishel for interesting discussions on
dimension theory and analysis. The authors also thank Sasha Razborov for discussions on his questions
regarding the relationship of K(x) and K(K(x)). Last but not least, we wish to thank the anonymous
referee for providing a detailed report, for telling us his notion of r-expansiveness, his theorem, and his
conjecture. We also wish to thank him for pointing out the related work of Besicovitch [Be] and the
reference in Falconer's book [F].
REFERENCES
[Ba] Y.M. BARZDIN, Complexity of programs to determine whether natural numbers not greater
than n belong to a recursively enumerable set, Soviet Math. Dokl. 9 (1968), 1251-1254.
[Be] A.S. BESICOVrrCH, On the sum of digits of real numbers represented in the dyadic systems,
Math. Ann. 34 (1934), 321-330.
[B1] P. BLANCHARD, Complex analytic dynamic systems on the Riemann sphere, Bull. Amer. Math.
Soc. 11 (1984), 85-141.
[Br] J. BROUWER, Uber den natiirlichen Dimensionsbegriff, J. Math. 142 (1913), 146-152.
[C1] G.J. C~ITIr~, On the length of programs for computing finite binary sequences, J. Assoc.
Comput. Math. 13 (1966), 547-569.
[C2] G.J. CHAITIN, A theory of program size formally identical to information theory, J. Assoc.
Comput. Math. 22 (1975), 329-340.
[F] K. FALCONER, "Fractal Geometry--Mathematical Foundations and Applications," Wiley,
New York, 1990.
[HI J. HagTMANIS, Generalized Kolmogorov complexity and the structure of feasible computation,
in "Proceedings, 24th Annual Symposium on Foundations of Computer Science, 1983,"
pp. 439-445.
[HW] W. Hul~wlcz AND H. WALLMAN, "Dimension Theory," Princeton Univ. Press, Princeton, NJ,
1974.
[K] A.N. KOLMOGOROV, Three approaches to the quantitative definition of information, Problems
Inform. Transmission 1, No. 1 (1965), 1-7.
ILl] L.A. LEVIN, Laws of information conservation (non-growth) and aspects of the foundation of
probability theory, Problems Inform. Transmission 10 (1974), 206-210.
[L2] L.A. L~vIN, Homogeneous measures and polynomial time invariants, in "Proceedings, 29th
Annual Symposium on Foundations of Computer Science, 1988," pp. 36-41.
HAUSDORFF AND TOPOLOGICAL DIMENSIONS
619
[LV] M. LI AND P. M. B.
VITANYI,
Two decades of applied Kolmogorov complexity: in memoriam
Andrei Nikolaevich Kolmogorov 1903-1987,
in
"Proceedings, Structure in Complexity Theory,
3rd Annual Conference, 1988," pp. 80-101.
[M] B. MANDELBgOT, "The Fractal Nature of Geometry," Freeman, San Francisco, 1983.
IRa] A.A. RAZBOROV, personal communication.
[Ry] B. YA. RYABI(O, Noiseless encoding of combinatorial sources, Hausdorff dimension and
Kolmogorov complexity,
Problems Inform. Transmission
22, No. 3 (1986), 16-26. [Russian]
[Sm] S. SMALE, Differentiable dynamic systems,
Bull. Amer. Math. Soe.
73 (1987), 747-817.
[So] R.J. SOLOMONOFF, A formal theory of inductive inference, Parts 1, 2,
Inform. and Control 7
(1964), 1-22, 224-254.
[Stl] L. STAIGER, "Complexity and Entropy, MFCS81," Lecture Notes in Computer Science, Vol. 118,
pp. 508-514, Springer-Verlag, New York/Berlin, 1981.
[St2] L. STAIGER, Kolmogorov complexity and Hausdorff dimension, manuscript.
