**From:** Ben Goertzel (*ben@goertzel.org*)

**Date:** Sat Oct 08 2005 - 19:41:12 MDT

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Richard,

*> > Consider also a set F of function mapping E into E. Elements
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*> of E may be
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*> > mapped into elements of F via considering them as “constant functions.”
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*> > Define an operation * on F as function composition. Of course,
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*> * is neither
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*> > commutative nor associative.
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*> >
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*> > The operation + may be extended from E onto F in an obvious
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*> way, so that we
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*> > may now think about {+,*} as algebraic operations on F.
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*>
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*> Wait! You can extend + from E into F only if there is a unique x
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*> in E that
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*> is associated with each f in F. To do this you need to map elements of F
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*> into elements of E ... and the mapping has to be injective if you want to
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*> extend + as you suggest.
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*>
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*> But in the previous paragraph you talked about mapping in the opposite
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*> direction, from E to F, and that would not, in general, buy you an
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*> injective mapping.
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*>
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*> I think you must have meant something a lot more specific (a restricted
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*> notion of what is supposed to be in F). Could you clarify what
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*> you mean by
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*> "constant functions" and what F is (surely it cannot be the set of ALL
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*> possible functions from E to E?), and how this allows you to
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*> extend + to F?
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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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