Re: Creating a "toy" OO/AO language...

On 31 Oct, 17:57, "Dmitry A. Kazakov" <mail...@xxxxxxxxxxxxxxxxx>
wrote:
On Wed, 31 Oct 2007 17:00:12 -0000, Mark Nicholls wrote:

You understand specification wrongly. Specification is not an equivalent
of implementation, so it does not define the outcome completely. If you
knew the outcome you would not need to specify it.

Really? How do you provide testable requirements if you cannot predict
what the output will be for any set of acceptable inputs?

Can you do it for sine? I bet you don't.

Of course you can. A sine is mathematically defined.

Sure, but that definition is not testable on a finite machine. Moreover it
cannot serve as a definition there. What is an implementation on the
"hardware" of classical analysis is not on the hardware of a "computer
program'. Defined mathematically /= defined programmable.

Actually this is an interesting question (at least to me)....there are
obviously things that are definable but are not computable.

At the position of constructivism non-computable = undefined. Existential
proofs do not count.

It is a specification, not a definition....it is irrelevant whether
existence is provable or not, or in fact provable doesn't exist, the
specification is still 'well formed'....the onus is on the
implementation to constructively prove....but you didn't ask for am
implementation...you asked for a specification.


But the issue is different, it is the substrate used for the definition.
You may not change it without reconsidering all previously "defined"
things. Mathematical analysis is just a different substrate.

I don't understand this.

I am not changing anything, I am giving a sentence that is a well
formed specification...undeniably....and academically allegedly
categorically specifies sin.


ForAll real x: (x>0 and x<1)->((sin2(x) + cos2(x) = 1) AND (sin(x+y) =
sin(x)cos(y) + cos(x)sin(y)) AND (cos(x+y) = cos(x)cos(y) +
sin(x)sin(y)) AND (0 < x.cos(x) < sin(x) < x))

allegedly this does uniquely specify sin.

don't ask me to prove it

You need not, because already "real" is an illegal term. Reals are
incomputable.

you need to explicitly specify exactly what you want specified!....if
you write sin...then I don't think it's unreasonable for me to real
this as a function defined over the Reals....if it is over the Reals,
then the answer is correct in that context.

If the question is specify sin to within some sort of level of
accuracy over some sort of floating point construct, thats a different
question, but it's also clealy finitely computable, and specifiable
within a finite sentence using a finite expansion of the standard
series definition e.g. if we call the sum SIN(X,N)...where N is the
maximum index in the sum....then clealy over the reals

forall x,n: ((|x| < 1/2) AND (n>0)) -> (SIN(x,2n) < sin(x) < SIN(x,2n
+1))

you define the n, and thats your specification.

Floating-point numbers are intervals. Interval-valued
trigonometric functions is a very different beast than the standard sine.
So all that "nice" stuff about ranges, precision, accuracy, ideals comes
into the play. When already x+(y+z) /= (x+y)+z, what do you want from sine?


who said anything about floating point numbers?

What universe do you want sin defined on?.....!

(this page is going to get lots of hits from people interested in all
sort of strange things.)

P.S.

I don't think in any way changes your conversation with
Lahman...clearly using a finite expansion for the specification
doesn't satisfy his requirement that the specification uniquely
defines the implementation....the set of valid implementions is
unbounded.

--
Regards,
Dmitry A. Kazakovhttp://www.dmitry-kazakov.de- Hide quoted text -

- Show quoted text -


.

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