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On two types of infinite sets of infinite sequences.
In the following x will stand for the infinite sequence (x(0), x(1), x(2), ...........) where the x(i) are taken from a finite alphabet of at least two characters, say {0, 1}
Let us consider the sets of sequences S1 and S2 , defined as the solution sets of the equations P1(x) and P2(x), respectively, with
Pl(x): | (E i: i ≥ 0: (A j: 0 ≤ j < i: x(j) = 0) and (A j: i ≤ j: x(j) = 1)) |
P2(x): | (A i: i ≥ 0: x(i) = 0 or (A j: j ≥ i: x(j) = 1)) . |
Because P1(x) ⇒ P2(x) , S1 is a subset of S2 , even a proper subset: there exists one solution of P2(x) and non P1(x) , viz. the sequence with x(1) = 0 for all i ≥ 0.
The set S2 has the property that for any non-member of S2 —i.e. any solution of non P2(x)— non-membership can be established on account of an initial segment of it. As a matter of fact, in this particular case only two elements suffice for this evidence, as follows from non P2(x):
(E i: i > 0: x(i) ≠ 0 and (E j: j > i: x(j) ≠ 1)) . |
The set S1 doesn’t have this property because there exists a non-member —viz. x(i) = 0 for all i ≥ 0— with the property that any initial segment of it is also the initial segment of a member of S1 . We call S2 “closed” and S1 “non-closed”, that is:
“The set S is closed” means “any non-member of S has an initial segment that is not the initial segment of any member of S”. | |
“The set S is non-closed” means “there exists a non-member of S such that any initial segment of it is also the initial segment of some member of S ”. |
1) that a set of sequences is either closed or non-closed,
2) that any finite set —in particular the empty set— is closed,
3) that the (uncountably infinite) universe of all possible sequences is closed.
* *
*
The notion of a closed set is of significance in connection with non-deterministic infinite computations. Consider the set of possible output sequences y corresponding to
“initialize; i:= 0; | (1) | |
do true → compute; print(y(i)); i:= i + 1 od” |
“Program (1) is a continuous machine” means “in program (1) ”initialize“ and ”compute“ are both of bounded non-determinacy”. |
We can now prove the following
Theorem 1. The possible output sequences of program (1) form a closed set or program (1) is not a continuous machine.
Proof. Let S be the set of possible output sequences of program (1). Either S is closed —in which case Theorem 1 holds— or S is non-closed. In the latter case, let x be a non-member of S such that each initial segment of x is also an initial segment of some member of S . Consider now the following program:
“initialize; i=: 0; | (2) | |
do ”y(0),...,y(i-1) is an initial segment of x “ → | ||
compute; print(y(i)); i:= i + 1 | ||
od; print(i)” |
Non-termination of (2) is excluded because x is not a member of S Because each initial segment of x is also the initial segment of some member of S , the final value of i is unbounded. Hence, program (2) is a (weakly) terminating program of unbounded non-determinacy; hence “initialize” or “compute” is of unbounded non-determinacy, i.e. program (1) is not a continuous machine. (End of Proof.)
Theorem 2. A set of sequences is non-closed or is the set of possible output sequences of some continuous machine.
Proof. Let S be the set of sequences. Either set S is non-closed —in which case Theorem 2 holds— or it is closed. In the latter case consider program (1) with “initialize” deterministic and “compute” only constrained by the requirement that the next y(i) leads to an initial segment of some element of S . Because the alphabet is finite, “compute” need not be of unbounded non-determinacy. By virtue of its construction program (1) may then generate any member of S , and, S being closed, it cannot generate any non-member of S . (End of Proof.)
Having identified closed sets of possible output sequences with continuous machines, we propose to ignore for the time being non-deterministic infinite computations to which non-closed sets correspond.
Plataanstraat 5 | 19th February 1980 |
5671 AL NUENEN | prof.dr.Edsger W.Dijkstra |
The Netherlands | Burroughs Research Fellow |
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