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Let be a category whose objects are countable or finite sets U. Elements x of
C
these sets will generally be finite sets with additional structure. Without waiting
for all the necessary axioms to be introduced, we will call x U a constructive
∈
object of type U (an integer, a finite graph, a word in a given alphabet, a Boolean
expression, an instance of a mass problem ...) The set U itself will be called the
constructive world of objects of fixed type, and the constructive universe. The
C
category , which will be made more concrete below, will contain all finite products
C
and finite unions of its objects, and also finite sets U of all cardinalities.
Morphisms U V in are certain partial maps of the underlying sets. More
→ C
precisely, such a morphism is a pair (D(f),f) where D(f) U and f : D(f) V
⊂ →
is a set–theoretic map. Composition is defined by
(g−1D(f),g
(D(g),g) (D(f),f) = f).
◦ ◦
We will omit D(f) when it does not lead to a confusion.
The morphisms f that we will be considering are (semi)computable functions
U V. An intuitive meaning of this notion, which has a very strong heuristic
→
potential, can be explained as follows: there should exist an algorithm ϕ such that
if one takes as input the constructive object u U, one of the three alternatives
∈
holds:
(i) u D(f), ϕ produces in a finite number of steps the output f(u) V.
∈ ∈
(ii)u / D(f),ϕproducesinafinitenumberofstepsthestandardoutputmeaning
∈
NO.
(iii) u / D(f), ϕ works for an infinitely long time without producing any output.
∈
Thenecessityofincludingthealternative(iii)inthedefinitionof(semi–)computa-
bility was an important and non–trivial discovery of the classical theory. The set
of all morphisms U V is denoted (U,V).
→ C
The sets of the form D(f) U are called enumerable subsets of U. If both E U
⊂ ⊂
and U E are enumerable, E is called decidable.
\
The classical computation theory makes all of this more precise in the following
way.
5
1.2. Definition. A category as above is called a constructive universe if
C
it contains the constructive world N of all integers 1, finite sets , 1 ,...,
≥ ∅ { }
1,...,n ,... and satisfies the following conditions (a)–(d).
{ }
(a) (N,N) is defined as the set of all partially recursive functions (see e.g.
C
[Ma1], Chapter V, or [Sa]).
(b) Any infinite object of C is isomorphic to N.
(c) If U is finite, (U,V) consists of all partial maps U V. If V is finite,
C →
(U,V) consists of such f that inverse image of any element of V is enumerable.
C
Before stating the last condition (d), we make some comments.
Statement (b) is a part of the famous Church Thesis. Any isomorphism (com-
putable bijection) N U in is called a numbering. Thus, two different number-
→ C
ings of the same constructive world differ by a recursive permutation of N. We will
callsuch numberings equivalentones. Noticethatbecause of(c) twofiniteconstruc-
tiveworldsareisomorphic iffthey have thesame cardinality, and theautomorphism
group of any finite U consists of all permutations of U.
As a matter of principle, we always consider as an open category, and at any
C
moment allow ourselves to add to it new constructive worlds. If some infinite V is
added to , it must come together with a class of equivalent numberings. Thus,
C
any finite union of constructive worlds can be naturally turned into the constructive
world, so that the embeddings become computable morphisms, and their images
are decidable. As another example, the world N∗ of finite sequences of numbers
from N (“words in alphabet N”) is endowed with G¨odel’s numbering
2n1−13n2−1...pnk−1...
(n ,n ,...,n ,...) (1)
1 2 k 7→ k
where p is the k–th prime number. Hence we may assume that is closed with
k
C
respect to the construction U U∗. All natural functions, such as length of the
7→
word U∗ N, or the i–th letter of the word U∗ U are computable.
→ →
Similarly, can be made closed with respect to the finite direct products by
C
using the (inverse) numbering of N2:
1
(m,n) m+ (m+n 1)(m+n 2). (2)
7→ 2 − −
Projections, diagonal maps, fiber maps V U V,v (u ,v) are all computable.
0
→ × 7→
Decidable subsets of constructive worlds are again constructive.
Church Thesis is often invoked as a substitute for an explicit construction of a
numbering, and it says that the category is defined uniquely up to equivalence.