Hi Stephen,
That is a fascinating claim! "...we could argue the UD is 0 dimensional: it computes an undefined function with 0 arguments."
What is the quantity of computational resources required for such a computation?
Probably null. Provably so with the QM hypothesis or the comp hyp.
A new question is born from your comment: Is your notion of a "dimension" flow from linear independence, like that of vectors? How does one define the notion of a "basis" in this computational dimension?
Yes, it is. Actually in computer science the notion of dimension is not globally relevant.
See the "parametrisation theorem" or the "SMN theorem" in my "amoeba, planaria and dreaming machines", or see a textbook like Cutland's one (ref in my thesis). Such theorems make it possible to code m-variable function by effective collection of n-variables function.
The combinators exploit this at the start(*)
Onward! ;)
Bruno
(*) see:
<x-tad-bigger> COMBINATORS I is
</x-tad-bigger><x-tad-bigger>http://www.escribe.com/science/theory/m5913.html</x-tad-bigger><x-tad-bigger>
COMBINATORS II is
</x-tad-bigger><x-tad-bigger>http://www.escribe.com/science/theory/m5942.html</x-tad-bigger><x-tad-bigger>
COMBINATORS III is
</x-tad-bigger><x-tad-bigger>http://www.escribe.com/science/theory/m5946.html</x-tad-bigger><x-tad-bigger>
COMBINATORS IV is
</x-tad-bigger><x-tad-bigger>http://www.escribe.com/science/theory/m5947.html</x-tad-bigger><x-tad-bigger>
COMBINATORS V is
</x-tad-bigger><x-tad-bigger>http://www.escribe.com/science/theory/m5948.html</x-tad-bigger>
http://iridia.ulb.ac.be/~marchal/
