If we have a #program, generating a #stream of symbols from an alphabet (say, 0's and 1's), then there is a (finite) natural #number of steps, after which the output will repeat?
And thus there is a #map from such programs onto the real numbers in [0, 1]? Not invertible, because e.g. 1.0… and 0.9… are mapped to the same real number.
May we introduce e.g. a meaningful #topology or #measure on the programs this way?
Any references, besides the whole theory of #computable #functions or #automata?
@JensE @feonixrift Thank you. I wasn't thinking enough before asking…
OTOH, even though the stream may never repeat, the map to the real numbers may still be defined. — It just means that *rational* numbers are not enough.
What I have on my mind here is something along the lines of a #p-adic #analysis of #algorithms.
#2-adic (#dyadic), more specifically, since base 2 is easier to interpret in logic.
The point is to (extensionally) identify programs with some numbers, and apply all sorts of "continuous" math to them.
And the p-adic expansion seem to be more appropriate for this approach than the usual "decimal" expansion.
