joseph-isaacs commented on issue #15524:
URL: https://github.com/apache/datafusion/issues/15524#issuecomment-4030636475
It seems like to me that aggregations AGG over SQL-integers
$\mathbb{Z}_\bot$ need to be examined on a case-by-case basis just like AGG
over integers $\mathbb{Z}$. For aggregations are defined over columns ($c_0,
..., c_n$) we see that
$\text{AGG}(c_0 \oplus^1 c_1 \oplus^2 ... \oplus^n c_n) = AGG(c_0 \oplus^1
c_1 \oplus^2 ... \oplus^n c_n\ \text{filter}\ m)$
where $m$ is the set of row where all elements are defined and all ops
$\oplus^1, .., \oplus^n$ are all null NULL-propagating/annihilating ($\forall
x. \bot \oplus^i x = \bot$). Furthermore RHS is all the contributing rows. This
means under the filter $m$ we can use integer reasoning since there are by
definition no nulls in any rows.
Also notice that is handle non-nullable columns easily.
Now lets think about specific examples:
## SUM
Consider a sum in sql `SUM(a+b+c)` we can write this as `SUM(a filter m) +
SUM(b filter m) + SUM(c filter m)` where `m = a and b and c is not null`. Then
we can use regular mathematical equivalencies over $\sum_i^n c^1_i \oplus ...
\oplus' c^n_i$.
## Count
I think the same idea holds for count `COUNT(a+b+c) = COUNT(m)`
## MAX/MIN
This is likely irreducible due to the [triangle
inequality](https://en.wikipedia.org/wiki/Triangle_inequality) $\forall a, b
\in \mathbb{Z}.\max(a + b) \leq \max(a) + \max(b)$.
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