Yes. There are lots of different settings one could consider, e.g.:
Finite strings
Infinite strings
Functions
LSCSMs
For all of these cases, one can compare different notions of complexity (plain K-complexity, prefix complexity, monotone complexity, if applicable) with algorithmic probability. My sense is that the correspondence is only exact for universal prefix machines and finite strings, but I didn’t consider all settings.
Yes. There are lots of different settings one could consider, e.g.:
Finite strings
Infinite strings
Functions
LSCSMs
For all of these cases, one can compare different notions of complexity (plain K-complexity, prefix complexity, monotone complexity, if applicable) with algorithmic probability. My sense is that the correspondence is only exact for universal prefix machines and finite strings, but I didn’t consider all settings.
I think the proof in the Gacs paper can be adapted to LSCSMs and functions but haven’t checked super carefully