On 10/20/2012 5:45 PM, Russell Standish wrote:
A UD generates and executes all programs, many of which are
equivalent. So some programs are represented more than others. The
COMP measure is a function over all programs that captures this
variation in program respresentation.
Why should this be unique, independent of UD, or the universal Turing
machine it runs on? Because the UD executes every other UD, as well as
itself, the measure will be a limit over contributions from all UDs.
I worry a bit about the use of the word "all" in your remark. "All"
is too big, usually, to have a single constructable measure! Why not
consider some large enough but finite collections of programs, such as
what would be captured by the idea of an equivalence class of programs
that satisfy some arbitrary parameters (such as solving a finite NP-hard
problem) given some large but finite quantity of resources?
Of course this goes against the grain of Bruno's theology, but
maybe that is what it required to solve the measure problem. :-) I find
myself being won over by the finitists, such as Norman J. Wildberger!
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