From: Ben Goertzel (ben@goertzel.org)
Date: Sat Oct 08 2005 - 19:41:12 MDT
Richard,
> > Consider also a set F of function mapping E into E. Elements
> of E may be
> > mapped into elements of F via considering them as “constant functions.”
> > Define an operation * on F as function composition. Of course,
> * is neither
> > commutative nor associative.
> >
> > The operation + may be extended from E onto F in an obvious
> way, so that we
> > may now think about {+,*} as algebraic operations on F.
>
> Wait! You can extend + from E into F only if there is a unique x
> in E that
> is associated with each f in F. To do this you need to map elements of F
> into elements of E ... and the mapping has to be injective if you want to
> extend + as you suggest.
>
> But in the previous paragraph you talked about mapping in the opposite
> direction, from E to F, and that would not, in general, buy you an
> injective mapping.
>
> I think you must have meant something a lot more specific (a restricted
> notion of what is supposed to be in F). Could you clarify what
> you mean by
> "constant functions" and what F is (surely it cannot be the set of ALL
> possible functions from E to E?), and how this allows you to
> extend + to F?
Sorry I wasn't explicit enough. We can extend + to F by defining
(f+g)(x) = f(x) + g(x)
This is how e.g. we define addition on the space of functions R^n --> R^m
As for F, in a computational context it suffices to consider the set of all
computable functions from E to E; or even more restrictively, the set of
all functions computable via programs of length less than N for some very
large number N.
-- Ben
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