Re: Proposal: 6NF
Date: 19 Oct 2006 20:40:21 -0700
Message-ID: <1161315621.083006.191420_at_e3g2000cwe.googlegroups.com>
Marshall wrote:
> On Oct 19, 4:36 pm, "vc" <boston..._at_hotmail.com> wrote:
> > Marshall wrote:
> > > On Oct 19, 7:06 am, "vc" <boston..._at_hotmail.com> wrote:
> > > > Jan Hidders wrote:
> > > > > vc wrote:
> > > > > > Jan Hidders wrote:
> > > > > > [...]
> >
> > > > > > A much simpler example. Let {0, 1, 2, 3} be a set of four integers
> > > > > > with addition modulo 4. Then, none of its subsets, except {0} and
> > > > > > {0, 2}, retains the addition mod 4 operation which makes the idea of
> > > > > > 'subtype as subset' utterly silly, [....].
> >
> > > > > You keep on making the same mistake. The expression a +[mod 4] b has a
> > > > > well defined result if a and b are from any subset of {0, 1, 2, 3}.
> >
> > > > Consider the subset {2, 3}. What is the result of (2+3) mod 4 ? If
> > > > you say it's '1', what is '1'? There is no such element in {2, 3}.
> >
> > > That doesn't matter. The answer is 1. The fact that 1 is not a member
> > > of {2,3} is irrelevant. The function +[mod4] still exists, unchanged,
> > > even though you took a subset of its domain.
> > The original function maps pairs of values from {0,1,2,3} to the values
> > in the same set. In particular, (2,3) -> 1. Sunce we do not have '1'
> > in the subset {2,3}, the binary operation is not defined any longer.
>
> Not so; the function remains defined. However, it cannot be considered
> "binary" relative to the subset {2,3}. That doesn't mean that it
> suddenly
> zapped out of existence when you typed in that subset.
It means precisely that: if the binary operation is not defined for the entire SxS domain it's not a binary operation, read some algebra book or at least the reference I gave earlier.
> >
> > I am not surer how anyone can say, with a straight face too, that
> > there is '1' in {2, 3}.
>
> No one said that. {0,1,2,3} is closed over AddModFour(), but
> {2,3} is not closed over AddModFour().
>
> *** "Not closed" is something altogether different from "not defined."
Not closed means not defined for the entire SxS domain which means that is not an operation and that the LSP does not hold (obviously) because in order for LSP to hold, the function must be defined for the entire domain. Take another example, the set of odd numbers as a subset of the natural numbers with respect to addition. The addition 'operation' is defined nowhere in the domain. Does the example convince you that you cannot use the idea of 'subtype as subset' and that subsetting can lose an operation or two ?
> ***
> >> > Are you saying that the OOPL foundation is voodoo rather than logic and
> > > You're conflating math with type issues from OOPLs.
> > math ?
>
> Maybe not voodoo, but not strict math, either. Math doesn't
> have mutator methods that I'm aware of; it does have
> fuctions defined over values however.
> >> > is a subset of the natural numbers implemented by the computer) to
> > > > Objects in subclass B = {2, 3} won't understand 'plusMod4' because the
> > > > operation does not exist in B, it was lost as soon as you've subsetted
> > > > A = {0,1,2,3}. That is of course if you understand 'operation' in the
> > > > ordinary mathematical sense such that the set is closed under it. If
> > > > you have to go outside the set in question, it ain't no operation no
> > > > more in the accepted mathematical sense.
> >
> > > Certainly they will understand it. Behold:
> >
> > > class ModFour {
> > > private final int val;
> > > ModFour(int i) {
> > > if (i < 0 || i > 3) throw new RuntimeException();
> > > val = i;
> > > }
> >
> > > public ModFour addModFour(ModFour a) {
> > > return new ModFour((val+a.val)%4);
> > > }
> >
> > > public String toString() { return "" + val; }
> >
> > > }
> >
> > > public class TwoOrThree extends ModFour {
> > > TwoOrThree(int i) {
> > > super(i);
> > > if (i < 2 || i > 3) throw new RuntimeException();
> > > }
> >
> > > public static void main(String[] args) {
> > > TwoOrThree v2 = new TwoOrThree(2);
> > > TwoOrThree v3 = new TwoOrThree(3);
> > > System.out.println(v2.addModFour(v3));
> > > }
> > > }
> >
> > > If you run this program it prints "1". Notice that the
> > > code doesn't care that addModFour() isn't automatically
> > > covariant.
> >
> > "return new ModFour((val+a.val)%4)" is called a bug or cheating
> > because you used a function defined on the entire domain NxN (where N
> > generate a result which is undefined for the (2,3) pair.
>
> It's not a bug, since the program produces the correct result, which is
> 1. It's not "cheating" because WTF does cheating even mean in
> this context. What it is, is the absence of closure over the subclass.
> Which is no big deal.
So producing an absurd result is no big deal ? You seem to have agreed that in the {2,3} subset there is no '1', how come your java implementation manages to extract '1' out of nowhere ?
> >
> > At thispoint, it may be useful to recall what a function is and
> > apply the memory, if any, to the addMod4 operation.
>
> Indeed. A function is a mapping from one set to another.
> In this case, the domain is pairs of values from {0,1,2,3}
> and the range is that same set. This particular
> function qualifies as a binary function, and also
> exhibits closure over the set {0,1,2,3}. Since {2,3} is a
> subset of {0,1,2,3}, the function is fully defined over
> that subset. However, since that function is not closed
How can it be 'fully defined' over {2,3} and 'not closed' at the same time ? What does (2,3) map to ? If you say to one, then you exctract '1' out of nowhere again.
> over the subset, it does not qualify as a binary function
> on that subset. It remain a closed binary function over
> {0,1,2,3} however.
> > >> > > seen any math books that give functions the same
> >
> > > The *only* trouble anywhere in here is if you want to
> > > have methods which are mutator methods; that is,
> > > methods which modify the distinguished object in place.
> > > And yes, in that case, closure is an issue. I have not
> > > semantics as inherited OOPL mutator methods however.
>
> I note you did not respond to this important paragraph. Yes,
> in OOPLs, there is a problem with closure, subtyping,
> and mutator methods. Outside of OOPLs (in math, that
> is) this problem does not apply.
We may tackle those issues later after we come to some agreement re.
the trivial stuff like functions and binary operations.
>
>
> > You may want to find out how integers and real numbers are actually
> > > Furthermore, 2 = 2.0.
> > constructed before stating that.
>
> How they are constructed is not an issue; this is simple arithmetic.
That ("this is simple arithmetic") does not make any obvious sense, sorry.
> >> > The equation looks like a random collection of characters. What do
> > > Also, I found this equation in a math book on page 3:
> >
> > > P ⊆ N ⊆ Z ⊆ Q ⊆ R
> >
> > you suppose it means ?
>
> I suppose your browser doesn't have the necessary font.
>
> In prose, it means:
>
> The set of positive integers is a subset of the set of nonnegative
> integers
> which is a subset of the integers which is a subset of the rationals
> which
> is a subset of the reals.
That's truly a trivial observation which is not strictly speaking correct (see the earlier discussion re. real construction).
>
>
> Marshall
Received on Fri Oct 20 2006 - 05:40:21 CEST