Re: Noncontinuous examples
Dmitry
On 28 Dec 2012, at 13:21, Dmitry Nadezhin wrote:
> I want to confirm if I understand "common evaluation" correctly.
> So I describe two examples. Please say me if I misunderstand.
>
> The definition in 5.2 says:
> "The common operation instances are those (x1 , x2 , . . . , xk ; y) such that the point
> function phi is everywhere defined and continuous on the closed, bounded, nonempty box
> (x1 , x2 , . . . , xk ), and y equals the range of phi over this box."
> It says "phi is continous" not "restriction of phi is continous".
> I guess that this is reserved for possible cset flavour.
You make a very good point. I thought about this distinction when first designing the flavor concept but forgot about it when writing the text. Yes, your interpretation is the one I support, mainly to allow a cset flavor as you say. To make it clearer I suggest
> phi is everywhere defined and continuous on
should change to
> phi is defined and continuous at each point of
> I try to interpret this definition for floor(x) and case(c,g,h).
<<Typo: you wrote "float">>
> 1) The only common evaluations of floor(x) are
> ([a,b] ; [k,k] ) where k < a <= b < k + 1 , k \in Z .
> Pair ([0,0]; [0,0]), is not common evaluation because floor(x) is discontinous at 0 .
Yes
> 2) The only common evaluations of case(c,g,h) are
> i) ([a,b],[c,d],[e,f]; [c,d] ) where 0 < a <= b, c <= d, e <= f
> ii) ([a,b],[c,d],[e,f]; [e,f]) where a <= b < 0, c <= d, e <= f
Shouldn't the result be [e,f] in (i) and [c,d] in (ii), instead of what you have? Because case(c,g,h) is g if c<0 and h otherwise.
> iii) ([a,b],[c,c],[c,c]; [c,c]) where a <= b
> Pair ([0,1],[2,2],[2,3]; [2,2]) is not common evaluation because case(c,g,h) is discontinous at (0,2,3) .
([0,1],[2,2],[2,3]; [2,3]) I think, see above.
But yes, it seems the points (c,g,h) where case() is continuous are precisely the set
{c \ne 0; g,h arbitrary} union {c = 0; g = h}.
And the boxes contained in the set are just those specified by your (i), (ii), (iii) -- corrected as above.
> Is definition of common evaluation for case(c,g,h) is different from other point functions ?
The standard definition, applied to this function, does give a rather strange result. What do people think? Is there a justification for an alternative definition?
John Pryce