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The notion of normal model of computations generalizes both normal algorithms
and Turing machines. For their common treatment see e.g. [Sa], Chapter 4. In
broad terms, p P is the list of Markov substitutions, or the table defining the
∈
operation of a Turing machine. The remaining worlds U,I,F consist of various
words over the working alphabet.
1.3.1. Claim. For any U, universal normal models of computations exist, and
can be effectively constructed.
For U = N, this follows from the existence of universal Turing machines, and
generally, from the Church Thesis. It is well known that the universal machine for
calculating functions of k arguments is obtained by taking an appropriate function
8
of k+1 arguments and making the first argument the variable part of the program.
Hence P, in this case, consists of pairs (q,m), where q is the fixed program of the
(k+1)–variable universal function (hardware) and m is a word written on the tape
(software).
1.4. Models of computations II: Boolean circuits. Boolean circuits are
classical models of computation well suited for studying maps between the finite
sets whose elements are encoded by sequences of 0’s and 1’s.
Consider the Booleanalgebra B generated over F by a countable sequence ofin-
2
dependent variables, say x ,x ,x ,... This is the quotient algebra of F [x ,x ,...]
1 2 3 2 1 2
with respect to the relations x2 = x . Each Boolean polynomial determines a func-
i i
tion on ∞ F with values in F = 0,1 .
⊕i=1 2 2 { }
We start with the following simple fact.
1.4.1. Claim. Any map f : Fm Fn can be represented by a unique vector of
2 → 2
Boolean polynomials.
Proof. It suffices to consider the case n = 1. Then f is represented by
F(x ,...,x ) := f(y) (x +y +1) (9)
1 n i i
y=(yX i)∈Fm Yi
2
because the product in (9) is the delta function in x supported by y. Moreover, the
spaces of maps and of Boolean polynomials have the common dimension 2m over
F .
2
Now we can calculate any vector of Boolean polynomials iterating operations
from a small finite list, which is chosen and fixed, e.g. := x, 1, x+y, xy, (x,x) .
B { }
Such operators are called classical gates. A sequence of such operators, together
with indication of their arguments from the previously computed bits, is called a
Boolean circuit. The number of steps in such a circuit is considered as (a measure
of) the time of computation.
When the relevant finite sets are not Fm and perhaps have a wrong cardinality,
2
we encode their elements by finite sequences of bits and consider the restriction of
the Boolean polynomial to the relevant subset.
As above, a protocol of computation in this model can be represented as the
finite table consisting of rows (generally of variable length) which accommodate
sequences of 0’s and 1’s. The initial line of the table is the input. Each subsequent
line must be obtainable from the previous one by the application of one the basic
functions in to the sequence of neighboring bits (the remaining bits are copied
B
unchanged). The last line is the output. The exact location of the bits which are
changed in each row and the nature of change must be a part of the protocol.
Physically, one can implement the rows as the different registers of the memory,
or else as the consecutive states of the same register (then we have to make a
prescription for how to cope with the variable length, e.g. using blank symbols).
9
1.4.2. Turing machines vs Boolean circuits. Any protocol of the Turing
computation of a function can be treated as such a protocol of an appropriate
Boolean circuit, and in this case we have only one register (the initial part of the
tape) whose states are consecutively changed by the head/processor. We will still
use the term “gate” in this context.
A computable function f with infinite domain is the limit of a sequence of func-
tions f between finite sets whose graphs extend each other. A Turing program
i
for f furnishes a computable sequence of Boolean circuits, which compute all f in
i
turn. Such a sequence is sometimes called uniform.
1.5. Size, complexity, and polynomial time computability. The quan-
titative theory of computational models deals simultaneously with the space and
time dimensions of protocols. The preceding subsection focused on time, here we
introduce space. For Boolean (and Turing machine) protocols this is easy: the
length of each row of the protocol is the space required at that moment (plus sev-
eral more bits for specifying the next gate). The maximum of these lengths is the
total space required.
The case of normal models and infinite constructive worlds is more interesting.
Generally we willcall a size functionU N : u u any function such that for
→ → | |
every B N, there are only finitely many objects with u B. Thus the number
∈ | | ≤
of bits n = [log n]+1 and the identical function n = n are both size functions.
| | 2 k k
Using a numbering, we can transfer them to any constructive world. In these two
examples, the number of constructive objects of size H grows as expcH, resp.
≤
cH. Such a count in more general cases allows one to make a distinction between
the bit size, measuring the length of a description of the object, and the volume of
the object.
In most cases we require computability of size functions. However, there are
exceptions: for example, Kolmogorov complexity is a non–computable size function
with very important properties: see below and sec. 5.
Givenasizefunction(onallrelevantworlds)andanormalmodelofcomputations
, we can consider the following complexity problems.
S