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C
We now turn to the computability properties of the sets of morphisms (U,V).
C
Again, it is a matter of principle that (U,V) itself is not a constructive world if
C
U is infinite. To describe the situation axiomatically, consider first any diagram
ev : P U V (3)
× →
6
in . It defines a partial map P (U,V), p p, where p(u) := ev(p,u). We will
C → C 7→
say that the constructive world P = P(U,V) together withthe evaluation map ev is
a programming method (for computing some maps U V). It is called universal,
→
if the following two conditions are satisfied. First, the map P (U,V) must
→ C
be surjective. Second, for any programming method Q = Q(U,V) with the same
source U and target V, (Q,P) contains translation morphisms
C
trans : Q(U,V) P(U,V) (4)
→
which are, by definition, everywhere defined, computable maps Q P such that if
→
q p, then q = p.
7→
We now complete the Definition 1.2 by adding the last axiom forming part of
the Church Thesis:
(d) For every two constructive worlds U,V, there exist universal programming
methods.
The standard examples of P for U = V = N are (formalized descriptions of)
Turing machines, or recursive functions.
From (d) it follows that the composition of morphisms can be lifted to a com-
putable function on the level of programming methods. To be more precise, if Q
(resp. P) is a programming method for U,V (resp. V,W), and R is a universal
programming method for U,W, there exist computable composition maps
comp : P(V,W) Q(U,V) R(U,W), (p,q) r (5)
× → 7→
such that r = p q.
◦
Concrete P(U,V) are furnished by the choice of what is called the “model of
computations” in computer science. This last notion comes with a detailed de-
scription not only of programs but also of all steps of the computational process.
At this stage the models of kinematics and dynamics of the process first emerge,
and the discussion of quantization can start.
Aformalizeddescriptionofthefirstnstepswillbecalleda history of computation
or, for short, a protocol (of length n.) For a fixed model, protocols (of all lenghts)
form a constructive world as well. We will give two formalized versions of this
notion, for functions with infinite and finite domains respectively. The first will be
well suited for the discussion of polynomial time computability, the second is the
base for quantum computing.
1.3. Models of computations I: normal models. Let U be an infinite
constructive world. In this subsection we will be considering partial functions U
→
U. The more general case U V can be reduced to this one by working with
→
U V.
`
7
A normal model of computations is the structure (P,U,I,F,s,)consisting of four
sets and a map:
I U, F P U, s : P U P U . (6)
⊂ ⊂ × × → ×
Here s is an everywhere defined function such that s(p,u) = (p,s (u)) for any
p
(p,u) P U. Intuitively, p is a program, u is a configuration of the deterministic
∈ ×
discrete time computing device, and s (u) is the new configuration obtained from
p
u after one unit of time (clock tick). Two additional subsets I U (initial config-
⊂
urations, or inputs) and F P U (final configurations) must be given, such that
⊂ ×
if (p,u) F, then s(p,u) = (p,u) i.e. u is a fixed point of s .
p
∈
In this setting, we denote by f the partial function f : I U such that we
p p
→
have
u D(f ) and f (u) = v iff for some n 0, (p,sn(u)) F and sn(u) = v. (7)
∈ p p ≥ p ∈ p
The minimal such n will be called the time (number of clock ticks) needed to
calculate f (u) using the program p.
p
Any finite sequence
(p,u,s (u),...,sm(u)), u I, (8)
p p ∈
will be called a protocol of computation of length m.
We now add the constructivity conditions.
We require P,U to be constructive worlds, s computable. In addition, we assume
that I,F are decidable subsets of U, P U respectively. Then f are computable,
p
×
and protocols of given length, (resp. of arbitrary length, resp. or those stopping at
F), form constructive worlds. If we denote by Q the world of protocols stopping
at F and by ev : Q U U the map (p,u) smax(u), we get a programming
× → 7→ p
method.
Such a model is called universal, if the respective programming method is uni-
versal.