19 Nov
2003
19 Nov
'03
3:14 p.m.
On Wednesday, November 19, 2003, at 10:17 PM, Vintage
Computer
Festival wrote:
A good example of closure comes from elementary arithmetic. It is known
that
the sum of two integers is also an integer. This feature of mathematics is
known
an closure, and the specific example is, Addition is Closed for the
Integers. It is
also true that Addition is Closed for the Real Numbers.
In the case of Turing closure, the notion is much broader. Turing closure
refers
to the ability of a system to perform any and all computations that can be
expressed. Now, there are problems with this notion, since Godel has shown
that some expressible computations in fact can not be computed. Still, the
general notion is: all that can be computed is computable upon a TM, and a
TM
is capable of computing all computations.
The real key to understanding the theory of computation is to understand the
works of the mathematician David Hilbert, the logicians Kurt Godel, Alonzo
Church, Stephen Cole Kleene, and Emil Post, the linguist Noam Chomsky,
and the mathematician Alan Mathison Turing.
On Wed, 19 Nov 2003, William R. Buckley wrote:
What does "closure" mean? Moreover, there is the _analog computer_ with
programming very
similar to the unit record equipment, and such machines have always
been known as computers.
Hardly. That's like saying French and Spanish are the same language
because they share a common character set. They are computers in a
wholly
different sense of the word and have nothing at all to do with a Turing
Machine, and thereby this discussion has suddenly drifted off into
bizarre
and meaningless abstracts.
> The important point for computation is closure, and Turing is the >> ideal model. It is not efficient, it is not
pretty but, all systems
>> that
>> exhibit computational closure are Turing machine equivalents, and
>> this is the foundation of computer science.
>
> Including analog computers and unit record equipment?
William R. Buckley