Re: A Logical Model for Lists as Relations


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.


See, I probably grok a successor operator, maybe a nil operator, but
I'm not tracking cons and am not sure why zero and nil differ.

You do not know what the difference between the number zero and the
empty list (Nil) is ?

I am aware of the difference between the number zero and an empty list.
Or set. Or bag. I was not aware that you attached the name Nil to
such. And my notation would use brackets, not parentheses.

From reading the other responses to this post, I can see that
we're stumbling all over each other's terminology.

I'll humbly follow the elders advice about not touching the Cons/Nil
subject.

What's 'bunch theory' ?

As for my own: scalars are boolean, numbers, and characters. Data
may be structured (Here we go down the rabbit hole!) as:
a bunch (unpackaged and unindexed),
a set (packaged and unindexed),
a string (unpackaged and indexed), and
a list (packaged and indexed).

More terminology? Well, bunches and sets consist of elements, which
has the meaning we're familiar with from sets. Sets are sets; they
are a package of elements constructed with {} operator. , (comma)
is the set union operator. Unpackaging a set - interpolating the
contents of a set - yields a bunch, which also has a comma union
operator. So
a, b, c is a bunch
{a, b, c} is a set.

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 ?

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 ?

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 ?

RM has its own scheme: everything in RM is of type boolean, other
scalar, tuple, or relation.

It's just too lame, sorry. A trip to the library may fix the RM
misconceptions, though.

.

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