Re: A Simple Notation

From: David Cressey <cressey73_at_verizon.net>
Date: Thu, 05 Jul 2007 16:56:00 GMT
Message-ID: <Ao9ji.4559$wu5.725@trndny03>

"paul c" <toledobythesea_at_oohay.ac> wrote in message news:Rl7ji.89241$xq1.85813_at_pd7urf1no...
> David Cressey wrote:
> > In Boolean algebra, you could, if you wanted to, express everything by
just
> > using brackets, as follows:
> >
> > [A B] means NOT (A AND B)
> >
> > This notation can be extended to 3 or more operands, as follows:
> >
> > [A B C] means NOT (A AND B AND C)
> >
> > "AND" is associative, so there's no confusion.
> >
> > You can reduce the notation to 1 operand as follows:
> >
> > [A] means NOT (A)
> >
> > And to zero operands as follows:
> >
> > [] means TRUE
> > [[]] means FALSE
> >
> > You can build up everything else from there. For example,
> >
> > [[A B]] = A AND B
> > [[A] [B]] = A OR B
> >
> > Now my question is, can you do the corresponding thing in the RA,
using
> > <NOT> and <AND>? I don't see why not.
> >
> > So you would get (for example)
> >
> > [[A B]] = A <AND> B
> > [[A] [B]] = A <OR> B
> >
> > As written text, this notation is rather unwieldy, but you can
represent it
> > fairly tightly in internal data structures. And its simplicity does
make
> > some things easier.
> >

>

> To me, it suggests an algebra that is based solely on NAND (ignoring
> projection for the moment).

>

Yeah. I couldn't figure out, off the top of my head, whether NOT (A AND B) was NAND or NOR. Whatever.

> Just playing with eg., De Morgan:
> <NOT> (A <AND> B) = (<NOT> A) <OR> (<NOT> B)
> ->
> [[[A B]]] = [[A] [B]].

>

> Mixing things up a bit, if [] is TRUE, it is the identity value for
> <AND>, so:

>


> A = A <AND> [] ->

> A = A [] ->
> A = [[A []]].

>

> Have to admit I like brackets because on my keyboard, I can type them
> without a shift key. At first glance, I imagine that a practical engine
> operating on "tuples" could simply reverse its tests whenever it
> encountered a leading bracket, eg. test 'not equal' instead of 'equal',
> assume <OR> whenever a leading bracket immediately follows a trailing
> bracket and so forth. I know that the electronic engineers like NAND
> because transistor-like circuit devices emulate it precisely. Does a
> language like Lisp make the same emulation easy? Another thing I wonder
> is what a debugging traceback of intermediate results would look like!
> Also, is it the case that one would never need more than three leading
> or trailing brackets in succession?

>

> (I think this kind of topic is always interesting. Personally, I was
> thinking of making a fairly sophisticated memory engine to test some of
> my "opinions" regarding view updating, transactions and so forth but I
> think I would like to also make a very sub-optimal parallel engine to
> validate the more sophisticated one, so the idea of using a different
> algebra to test an implementation appeals to me as it might have less
> chance of echoing the same bugs in the other one! Should mention I
> didn't have Lisp in mind but IIRC it is a favourite of David C.)
>
You got me, Paul C!

Lisp is my favorite language that I'm not proficient in. But I understand the data structures of Lisp better than those of many other languages. That's only because there is so little to understand and so much value in understanding it!
> p

> Received on Thu Jul 05 2007 - 11:56:00 CDT