Clarify distributive aggregates and correctness semantics

Author: cristianoc

reduce.pdf

@@ -187,7 +187,7 @@ \subsection{Reducers}
 \]
 \end{definition}
 
-In database terminology, a well-formed reducer defines an \emph{invertible distributive aggregate}: the fold can be computed over partitions independently (distributive), and individual values can be removed from the accumulated result (invertible).
+In database terminology, a well-formed reducer defines an \emph{invertible distributive aggregate} (see Section~\ref{subsec:distributive-aggregates}): the fold can be computed over partitions independently (distributive), and individual values can be removed from the accumulated result (invertible).
 
 \begin{remark}[Remove-Add Commutativity]
 For well-formed reducers where $\oplus$ and $\ominus$ arise from an abelian group action on $A$, the following property holds automatically:
@@ -198,6 +198,22 @@ \subsection{Reducers}
 All practical reducers (sum, count, product over commutative groups) satisfy this.
 \end{remark}
 
+\subsection{Distributive Aggregates}\label{subsec:distributive-aggregates}
+
+The database literature~\cite{viewmaintenance} classifies aggregates as \emph{distributive}, \emph{algebraic}, or \emph{holistic}.
+In our setting, a pair $(\iota, \oplus)$ defines a \emph{distributive aggregate} when folding over a union of multisets can be decomposed into folds over the parts.
+
+\begin{definition}[Distributive Aggregate]
+Let $\oplus : A \times V \to A$ be pairwise commutative and $\iota \in A$.
+We say that $(\iota, \oplus)$ is a \emph{distributive aggregate} if for all finite multisets $M, N \in \mathcal{M}(V)$:
+\[
+\mathsf{fold}_\oplus(\iota, M \uplus N) = \mathsf{fold}_\oplus\big(\mathsf{fold}_\oplus(\iota, M), N\big).
+\]
+\end{definition}
+
+By Lemma~1, any pair $(\iota, \oplus)$ with pairwise commutative $\oplus$ is a distributive aggregate in this sense.
+Moreover, a well-formed reducer $R = (\iota, \oplus, \ominus)$ (Definition~\ref{def:well-formed-reducer}) is precisely an \emph{invertible distributive aggregate}: the aggregate is distributive over partitions of the multiset, and individual contributions can be removed using $\ominus$.
+
 \subsection{Deltas}
 
 We model updates to collections as deltas.