Re: A Logical Model for Lists as Relations

From: Jay Dee <ais01479_at_aeneas.net>
Date: Fri, 12 May 2006 02:09:12 GMT
Message-ID: <c7S8g.23865$YI5.3580@tornado.ohiordc.rr.com>

vc wrote:
> Jay Dee wrote:
>

>>vc wrote:

>
> [...]
>

>>>data Nat = Zero | Succ Nat
>>
>>Others might write `0, nat+1: nat'

>
> It depends on what 'others' might have meant. If they had in mind
> initial algebra mumbo-jumbo, they would have been wrong. The 'others'
> should have rather written [zero, succ]: 1 + nat ->nat. But the
> initial algebra mumbo-jumbo hardly belongs here, being just a
> variation on the Peano theme in the case of naturals.

You're making my point...

[snip]

> I am sorry but the 'bunch' vs. set juxtaposition just does not make any
> obvious sense. As soon as you talk about a 'bunch', 'herd', 'pack' of
> 'set', the intuition is the same: a collection of some elements. It's
> not important whether or not you use the pretty curly brackets.
>

>>The empty bunch is null and the empty set is {null}.

>
> See above. As soon as you imagine an empty collection, it does not
> matter how you label it. Besides, {null} is not an empty set, {} is,
> in the traditional math at least.

>>Strings consist of items which may be any boolean, number, character,
>>non-empty bunch, or set.  Strings are catenated with ; (semicolon).
>>So
>>   17; 42; A; 17
>>is a string of length 4.

>
> So what's the difference between the traditional list and the 'string'
> you've just described ?

As above, ; and [] are operators, not merely punctuation, and when items are packaged, more operators can be defined.

>> Items in a string can be referred to by
>>their 0-origin ordinal position.  Ordering is defined as lexical and
>>the < = > &c operators are defined.  Indexing?  You bet!  Slicing?
>>Of course!
>>
>>Lists are to strings as sets are to bunches.
>>   17; 42; A; 17  is a string
>>  [17; 42; A; 17] is a list

>
>
> Ah, ok. So there is no difference except the superfluos brackets ?

In the full description there are significant differences. {} is an operator that constructs sets; there is a symmetric operator that turns them into bunches. The point is that there are more operators available for sets than there are bunches -- just as there are more operators available on lists than there are on strings.

>>Catenation can be defined on lists:
>>  [A] + [B] = [A; B]
>>So can mapping, composition, &c -- all it takes is definition.
>>
>>Empty strings and empty lists?  Call 'em nil and [nil].

>
> When you explain the substantial, not notational, difference between
> 'bunches' and sets, we'll talk about the rest of the crowd, deal ?

Well, sure: once you've got sets you can have subsets and powersets and intersections and union on sets rather than union on elements.

Lists provide operators for composition and modification via indices -- what the perl crowd likes to call slicing.

There is more to all these -- and I'm not trying to convert anyone or preach my religion. The point is that there are other welldeveloped  theories that use the same word to mean different things.

Remember Kenneth Iverson? I used his notation for quite a while -- still do, as a matter of fact -- and it ain't like very many other things. Received on Thu May 11 2006 - 21:09:12 CDT