Re: On Formal IS-A definition

On May 8, 9:44 pm, David BL <davi..._at_iinet.net.au> wrote:
> On May 9, 2:34 am, Nilone <rea..._at_gmail.com> wrote:
>
> > On May 8, 7:11 am, David BL <davi..._at_iinet.net.au> wrote:
>
> > > Values are immutable. Variables accessed by imperative programs are
> > > usually mutable. Sets are values. If a set contained a variable then
> > > it wouldn't be immutable.
>
> > We can generalize values and variables to elements of domains, where a
> > value is any element of a domain while a variable is an element of a
> > domain for which a homomorphism to another domain is defined.
> > Assigning to a variable would reduce to modification of the
> > homomorphism, so sets containing variables would not be modified by
> > assignment to a variable.

Mathematically there is no "modification" or "mutation" nor any such anthropomorphic passage of time sense. A variable is a symbol. That symbol might have a binding. That binding is also a relation whose key is the variable symbol and in the case of an imperative interpretation if the variable is "mutable" also the "time" or "program counter" or similar is part of the key. For example the variable X might have the following binding relation

   S T V
   X 0 0
   X 1 0
   X 2 5
   X 3 5
   ...

where S is the variable symbol, T is the "time", and V is the bound value. But note that nothing "changes" at T=2 from this meta perspective of math where "time" is just yet-another dimension.

> Wrong. You can't modify a homomorphism just like you can't modify a
> number or a set. Homomorphisms are values and are therefore
> immutable. You have invented a homomorphism variable to hold a
> homomorphism value. What you claimed were variables were just values
> intended to act as inputs to a homomorphism function.

Except his error is irrelevant. Variables are symbols and are representable by sets. Their bindings (regardless of extent ie dependence on "time") can be represented by relations which are sets just as their interpretations are relations which are sets.

KHD Received on Mon May 10 2010 - 08:06:06 CEST