Re: [lojban] &Lang

[Jorge Llambías <jjllambias@gmail.com>]

On Fri, Aug 24, 2012 at 5:34 PM, And Rosta <and.rosta@gmail.com> wrote:
> Jorge Llambías, On 15/08/2012 23:13:
>> On Tue, Aug 14, 2012 at 10:55 PM, And Rosta<and.rosta@gmail.com>  wrote:
>>>
>>> E.g. F(a,b) and G(c,d) merge into a compound predicate H(a,b=c,d).
>>
>> I guess that means you need Bell number B(n) of inflections per
>> connective for predicates that together have n arguments.
>
> I don't think it is Bell number, since the Livagian system requires 1--1
> correspondences between members of the two sets, whereas Bell number seems
> to be the number of possible correspondence patterns among members of a set.

So you can't have F(a,b) and G(c,d) merge directly into H(a=b=c, d).

Then the nth Bell number is the number of unary mergers that you need
for predicates with n arguments, the binary mergers will be fewer
(assuming you can make use the unary mergers first). But presumably
you could also do the unary mergers in steps, with only two arguments
merging at each step, so for an n argument predicate you only need
n(n-1)/2 mergers. Which you could also reduce by combining them with
argument shuffling operators.

> Each member of each of the two sets may be in correspondence with either
> exactly one member of the other set or no member of the other set. For
> reasons I won't go into, there are four different sorts of correspondence.
> So, each member of each of the two sets is either in exactly one sort of
> correspondence with either exactly one member of the other set or is in no
> correspondence with any member of the other set.

OK, that's a smaller amount of correspondences than I was thinking of.

> That must be an easyish
> formula to find for somebody with more maths nous than me.

If my calculations (for a single sort of correspondence) are correct:

N(n,m) = Sum from i=1 to min(n,m) of n!m! / i!(n-i)!(m-i)!

N(n,1) = n
N(n,2) = n(n+1)
N(n,3) = n(n^2+2)

Probably this number has a name, but I don't know what it is.

For four sorts of correspondence, it gets more complicated.

>
> And yes, the inflections are compositional.

The nth Bell number is the number of different compositions you need
to cover all cases of n arguments. (With one kind of merge. With four
kinds of merge, I don't know.)

mu'o mi'e xorxes

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