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u,v u u,v v
≤
exist (see e.g. [Ma1], Theorem VI.9.2); in particular, they take all positive integer
values (however they certainly are not everywhere defined). Fix one such u and
call C (x) the (exponential) complexity of x. By definition, K = K rearranges N
u u
in the order of increasing complexity. In other words,
K(x) := 1+card y C (y) < C (x) . (24)
u u
{ | }
We first show that
K(x) = exp(O(1))C (x). (25)
u
Since C takes each value at most once, it follows from (24) that K(n) C (n).
u u
≤
In order to show that C (x) cK(x) for some c it suffices to check that
u
≤
card k N x, C (x) = k bN
u
{ ≤ |∃ } ≥
with some b > 0. In fact, at least half of the numbers x N have the complexity
≤
which is no less than x/2.
Now, VI.9.7(b) in [Ma1] implies that, for any recursive function f and all x
∈
D(f), we have C (f(x)) constC (x). Since C (x) and K(x) have the same order
u u u
≤
of growth up to a bounded factor, our claim follows.
5.2. Corollary. Denote by Srec be the group of recursive permutations of N.
∞
Then KSrecK−1 is a subgroup of permutations of no more than linear growth.
∞
Actually, appealing to the Proposition VI.9.6 of [Ma1], one can considerably
K
strengthen this result. For example, let σ be a recursive permutation, σ =
25
KσK−1. Then σK (x) cx so that (σK )n(x) cnx for n > 0. But actually the last
≤ ≤
inequality can be replaced by
(σK )n(x) c′ n
≤
for a fixed x and variable n. With both x and n variable one gets the estimate
O(xnlog(xn)).
In the same way as finite permutations appear in the quantum versions of
Boolean circuits, infinite (computable) permutations are natural for treating quan-
tum Turing machines ([Deu]) and our normal computation models. In fact, if one
assumes that the transition function s is a permutation, and then extends it to
the unitary operator U in the infinite–dimensional Hilbert space, one might be
s
interested in studying the spectral properties of such operators. But the latter
depend only on the conjugacy class. Perhaps the universal conjugation UK might
be a useful theoretical tool in this context. In the purely classical situation, (23)
may play a role in studying the limiting behavior of polynomial time algorithms,
as suggested in [Fr1] and [Fr2].
Finally, I would like to comment upon the hidden role of Kolmogorov complex-
ity in the real life of classical computing. The point is that in a sense (which is
difficult to formalize), we are interested only in the calculation of sufficiently nice
functions, because a random Boolean function will have (super)exponential com-
plexity anyway. A nice function, at the very least, has a short description and,
therefore, a small Kolmogorov complexity. Thus, dealing with practical problems,
we actually work not with small numbers, graphs, circuits, ... , but rather with an
initial segment of the respective constructive world reordered with the help of K.
We systematically replace a large object by its short description, and then try to
overcome the computational difficulties generated by this replacement.
Appendix
The following text is a contribution to the prehistory of quantum computing. It
is the translation from Russian of the last three paragraphs of the Introduction to
[Ma2] (1980). For this reference I am grateful to A. Kitaev [Ki].
“ Perhaps, for better understanding of this phenomenon [DNA replication], we
need a mathematical theory of quantum automata. Such a theory would provide us
with mathematical models of deterministic processes with quite unusual properties.
One reason for this is that the quantum state space has far greater capacity than
the classical one: for a classical system with N states, its quantum version allow-
ing superposition accommodates cN states. When we join two classical systems,
their number of states N and N are multiplied, and in the quantum case we get
1 2
exponential growth cN1N2.
26
These crude estimates show that the quantum behavior of the system might
be much more complex than its classical simulation. In particular, since there is
no unique decomposition of a quantum system into its constituent parts, a state
of the quantum automaton can be considered in many ways as a state of various
virtual classical automata. Cf. the following instructive comment at the end of the
article [Po]: ‘The quantum–mechanical computation of one molecule of methane
requires 1042 grid points. Assuming that at each point we have to perform only
10 elementary operations, and that the computation is performed at the extremely
low temperature T = 3.10−3K, we would still have to use all the energy produced
on Earth during the last century.’
The first difficulty we must overcome is the choice of the correct balance between
themathematicalandthephysicalprinciples. Thequantumautomatonhastobean
abstract one: its mathematical model must appeal only to the general principles of
quantum physics, without prescribing a physical implementation. Then the model
of evolution is the unitary rotation in a finite dimensional Hilbert space, and the
decomposition of the system into its virtual parts corresponds to the tensor product
decomposition of the state space. Somewhere in this picture we must accommodate
interaction, which is described by density matrices and probabilities.”
Bibliography