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quantum

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(A) For a given morphism (computable map) f : U V, estimate the smallest
→
size K (f) of the program p such that f = f .
S p
Kolmogorov, Solomonoff and Chaitin proved that there exists an optimal uni-
versal model of computations such that, with P = N and the bit size function,
U
for any other model there exists a constant c such that for any f
S
K (f) K (f)+c.
U S
≤
When is chosen, K (f) is called Kolmogorov’s complexity of f. With a different
U
U
choice of we will get the same complexity function up to O(1)–summand.
U
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This complexity measure is highly non–trivial (and especially interesting) for
an one–element world U and infinite V. It measures then the size of the most
compressed description of a variable constructive object in V. This complexity is
quite “objective” being almost independent of any arbitrary choices. Being un-
computable, it cannot be directly used in computer science. However, it furnishes
some basic restrictions on various complexity measures, somewhat similar to those
provided by the conservation laws in physics.
On N we have K (n) n + O(1) = log n + O(1). The first inequality
U ≤ | | 2k k
“generically” can be replaced by equality, but infinitely often K (n) becomes much
U
smaller that n .
| |
(B) For a given morphism (recursive map) f : U V, estimate the time needed
→
to calculate f(u),u D(f) using the program p and compare the results for different
∈
p and different models of computations.
(C) The same for the function “maximal size of intermediate configurations in
the protocol of the computation of f(u) using the program p” (space, or memory).
In the last two problems, we have to compare functions rather than numbers:
time and space depend on the size of input. Here a cruder polynomial scale appears
naturally. Let us show how this happens.
Fix a computational model with the transition function s computing func-
S
tions U U, and choose a bit size function on U satisfying the following crucial
→
assumption:
( ) u c s (u) u +c where the constant c may depend on p but not on
p
• | |− ≤ | | ≤ | |
u.
In this case we have sm(u) u +c m: the required space grows no more than
| p | ≤ | | p
linearly with time.
Let now ( ′,s′) be another model such that s = s′ for some q. For example,
S p q
such q always exists if ′ is universal. Assume that s′ satisfies ( ) as well, and
S •
additionally
( ) s can be computed in the model ′ in time bounded by a polynomial F in
•• S
the size of input.
This requirement is certainly satisfied for Turing and Markov models, and is
generally reasonable, because an elementary step of an algorithm deserves its name
only if it is computationally tractable.
Then we canreplace one applicationofs to sm(u)by F( u +cm) applications
p p ≤ | |
of s′. And if we needed T(u) steps in order to calculate f (u) using , we will need
q p S
no more than T(u) F( u +cm) steps to calculate the same function using ′
≤ m=1 | | S
and q. In a detailPed model, there might be a small additional cost of merging two
protocols. This is an example of the translation morphism (4) lifted to the worlds
of protocols.
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Thus, from ( ) and ( ) it follows that functions computable in polynomial time
• ••
by have the same property for all reasonable models. Notice also that for such
S
functions, f(u) G( u ) for some polynomial G and that the domain D(f) of
| | ≤ | |
such a function is decidable: if after T( u ) s –steps we are not in a final state, then
p
| |
u / D(f).
∈
Thus we can define the class PF of functions, say, Nk N computable in
→
polynomial time by using a fixed universal Turing machine and arguing as above
that this definition is model–independent.
If we want to extend it to a constructive universe however, we will have to
C
postulate additionally that any constructive world U comes together with a natural
class of numberings which, together with their inverses, are computable in polyno-
mial time. This seems to be a part of the content of the “polynomial Church thesis”
invoked by M. Freedman in [Fr1]. If we take this strengthening of the Church thesis
for granted, then we can define also the bit size of an arbitrary constructive object
as the bit size of its number with respect to one of these numberings. The quotient
of two such size functions is bounded from above and from zero.
Below we will be considering only the universes and worlds U with these prop-
C