From 4f3045db1c62c17fa6641ffbae1fbcd5e2ae5975 Mon Sep 17 00:00:00 2001 From: Cristiano Calcagno Date: Thu, 27 Nov 2025 08:11:07 +0100 Subject: [PATCH 01/20] reduce combinator paper --- .gitignore | 2 + reduce.pdf | Bin 0 -> 318086 bytes reduce.tex | 528 +++++++++++++++++++++++++++++++++++++++++++++++++++++ 3 files changed, 530 insertions(+) create mode 100644 reduce.pdf create mode 100644 reduce.tex diff --git a/.gitignore b/.gitignore index 74db598..d476f1a 100644 --- a/.gitignore +++ b/.gitignore @@ -7,3 +7,5 @@ lib/ *.cmj *.cmi *.rei +reduce.aux +reduce.log diff --git a/reduce.pdf b/reduce.pdf new file mode 100644 index 0000000000000000000000000000000000000000..26537d3d776cba692920f561bb436b659b9062a2 GIT binary patch literal 318086 zcma&tQ+F=Fwl(V5w(S|)wv8Fvwv!p>jcwbuZQHi(d~2U}?)JLabydHh+S5n%L9Qq& zPRB&g217o#G`t4GLc~a9Z)63-%L~IGYieiiVnM{r!pTPT{~Z_xaZ4K)Qzs$@aT`My zQ&Cf6dlOR_etsBd7bjCgTNsaxWKEga!#3oe+1fig<~j8tj%Fc*Ofd{e1TBMwJ_5Jb 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mode 100644 index 0000000..a153ed5 --- /dev/null +++ b/reduce.tex @@ -0,0 +1,528 @@ +\documentclass{article} +\usepackage{amsmath, amssymb, amsthm} +\usepackage{mathpartir} +\usepackage{stmaryrd} +\usepackage{url} + +% Define big versions of oplus/ominus for folding notation +\newcommand{\bigominus}{\mathop{\text{\Large$\ominus$}}} + +\newtheorem{definition}{Definition} +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}{Lemma} +\newtheorem{remark}{Remark} + +\title{A Formal Semantics for Skip's Reactive \texttt{reduce} Combinator} +\author{Cristiano Calcagno} +\date{November 2025} + +\begin{document} +\maketitle + +\begin{abstract} +We present a formal semantics for the \texttt{reduce} combinator in Skip~\cite{skip}, a reactive programming framework that maintains aggregated views of collections with automatic incremental updates. +Skip exposes reducers as user-facing combinators: a reducer $R = (\iota, \oplus, \ominus)$ specifies an initial value, an add operation, and a remove operation, subject to an informal correctness condition. +This paper formalizes that condition and proves that incremental correctness---where updates produce the same result as recomputation---holds \emph{if and only if} the reducer is well-formed (i.e., $\ominus$ is the inverse of $\oplus$). +Our contribution combines the \textbf{algebraic foundations of incremental computation} (from differential dataflow and databases) with Skip's \textbf{user-facing combinator design} to provide a \textbf{formally characterized correctness contract}---giving Skip users a precise specification for writing correct custom reducers. +\end{abstract} + +\section{Introduction} + +Skip~\cite{skip} is a reactive programming framework that maintains derived views of collections with automatic incremental updates. +When the underlying data changes, Skip efficiently propagates updates to all dependent computations without manual intervention. +A central operation in Skip is the \texttt{reduce} combinator, which computes a summary (or \emph{view}) for each key in a collection by folding over its associated values---for example, computing the sum, count, or minimum. + +The key to efficient updates is that Skip's \texttt{reduce} supports \emph{retractions}: when values are removed from a collection, the reducer can incrementally update the accumulated result rather than recomputing from scratch. +Skip exposes this capability through a \emph{reducer} abstraction $R = (\iota, \oplus, \ominus)$, where $\iota$ is an initial value, $\oplus$ is an add operation, and $\ominus$ is a remove operation. +This allows $O(1)$ updates per change, rather than $O(n)$ recomputation. + +\paragraph{Skip's user-facing combinators.} +A distinguishing feature of Skip is that reducers are exposed as \textbf{user-facing combinators}---first-class programming constructs that developers use directly to build custom reactive services. +Rather than being internal implementation details hidden from users, Skip allows developers to define their own reducers for domain-specific aggregations. +Skip's documentation specifies an informal correctness condition: the result of applying the runtime's sequence of remove/add calls must equal recomputing from scratch. +This design enables extensibility (custom aggregations), composability (reducers combine with other Skip combinators), and---when the condition is satisfied---correctness guarantees. + +\paragraph{This paper.} +We formalize Skip's reducer abstraction and its correctness condition. +Our main result (Theorem~\ref{thm:equivalence}) proves that incremental correctness holds \emph{if and only if} the reducer is well-formed---that is, $\ominus$ is the inverse of $\oplus$. +This provides Skip users with a precise specification: satisfy the well-formedness condition, and your custom reducer is guaranteed to work correctly with Skip's incremental update mechanism. + +\paragraph{Synthesis of ideas.} +Our formalization builds on Skip's design and synthesizes ideas from multiple domains: +\begin{itemize} + \item From \textbf{incremental databases and streaming systems}, we take the insight that invertibility is key to efficient updates---aggregations with inverse operations can be maintained in $O(1)$ time per change. + \item From \textbf{Skip's reactive runtime}, we take the user-facing combinator design: a first-class, composable operator that developers use directly, rather than callbacks tied to internal machinery. + \item From \textbf{formal methods}, we provide a complete characterization: not merely sufficient conditions for correctness, but a precise \emph{if-and-only-if} theorem that fully characterizes when Skip's incremental updates are correct. +\end{itemize} + +\paragraph{Contributions.} +This paper formalizes Skip's \texttt{reduce} combinator. We provide: +\begin{itemize} + \item A denotational semantics for \texttt{reduce} as a derived view (Section~\ref{sec:reduce}) + \item A formal model of deltas and Skip's incremental update procedure (Section~\ref{sec:incremental}) + \item A precise well-formedness condition and proof that it is both necessary and sufficient for correctness (Section~\ref{sec:correctness}) + \item Concrete examples including sum, count, and min reducers (Section~\ref{sec:examples}) +\end{itemize} +We also position Skip's design relative to other streaming and reactive systems (Section~\ref{sec:related}) and analyze the complexity benefits of incremental updates. + +\section{Preliminaries}\label{sec:preliminaries} + +Let $K$ be a set of keys, $V$ a set of values, and $A$ a set of accumulator values. +For a set $V$, we write $\mathcal{M}(V)$ for the set of finite multisets over $V$; we use $\uplus$ and $\setminus$ for multiset union and multiset difference, respectively. + +\begin{definition}[Collection] +A \emph{collection} is a function $C : K \to \mathcal{M}(V)$. We write $C(k)$ for the multiset of values associated with key $k$. +\end{definition} + +\subsection{Commutative Operations} + +\begin{definition}[Pairwise Commutative Operation]\label{def:pairwise-commutative} +Let $\star : A \times V \to A$ be an update operation. +We say that $\star$ is \emph{pairwise commutative} if +\[ +\forall a \in A, v_1, v_2 \in V.\; (a \star v_1) \star v_2 = (a \star v_2) \star v_1. +\] +\end{definition} + +\subsection{Folds} + +\begin{definition}[Fold over Sequence for an Operation] +Let $\star : A \times V \to A$ be an update operation and let $s = [v_1,\dots,v_n]$ be a finite sequence of elements of $V$. +For any $a \in A$ we define: +\[ +\mathsf{fold}^{\mathsf{seq}}_\star(a, []) = a +\quad\text{and}\quad +\mathsf{fold}^{\mathsf{seq}}_\star(a, v_1 :: s') = \mathsf{fold}^{\mathsf{seq}}_\star(a \star v_1, s'). +\] +When a distinguished initial element $\iota \in A$ is understood from context, we write $\mathsf{fold}^{\mathsf{seq}}_\star(s)$ for $\mathsf{fold}^{\mathsf{seq}}_\star(\iota, s)$. +\end{definition} + +\begin{theorem}[Characterisation of Multiset Fold] +Let $\star : A \times V \to A$ be an update operation. +The following are equivalent: +\begin{enumerate} + \item For all $a \in A$, $M \in \mathcal{M}(V)$ and any two finite sequences $s_1, s_2$ enumerating $M$ (with multiplicity), we have + \[ + \mathsf{fold}^{\mathsf{seq}}_\star(a, s_1) = \mathsf{fold}^{\mathsf{seq}}_\star(a, s_2). + \] + That is, folding depends only on the multiset of elements, not on their enumeration. + \item The operation $\star$ is pairwise commutative in the sense of Definition~\ref{def:pairwise-commutative}. +\end{enumerate} +\end{theorem} + +\begin{proof} +Sketch: +For (2 $\Rightarrow$ 1), one shows first that swapping two adjacent elements in a sequence does not change the fold, using pairwise commutativity of $\star$. +Since any permutation of a finite sequence can be written as a product of adjacent transpositions, it follows that the fold depends only on the underlying multiset. +For (1 $\Rightarrow$ 2), instantiate (1) with the two sequences $[v_1,v_2]$ and $[v_2,v_1]$ enumerating the same multiset $\{v_1,v_2\}$, and expand the definition of the fold to obtain $(a \star v_1) \star v_2 = (a \star v_2) \star v_1$. +\end{proof} + +\begin{definition}[Fold over Multiset for an Operation] +Let $\star : A \times V \to A$ be pairwise commutative and let $M \in \mathcal{M}(V)$ be finite. +For $a \in A$ and any sequence $s$ enumerating $M$ (with multiplicity), we set +\[ +\mathsf{fold}_\star(a, M) := \mathsf{fold}^{\mathsf{seq}}_\star(a, s), +\] +which is well-defined by the Characterisation of Multiset Fold. +If an initial element $\iota \in A$ is fixed, we abbreviate $\mathsf{fold}_\star(M) := \mathsf{fold}_\star(\iota, M)$. +\end{definition} + +\section{The Reduce Combinator}\label{sec:reduce} + +The \texttt{reduce} combinator produces a \emph{view} of a collection by summarizing the values for each key. + +\begin{definition}[Reduce Combinator] +Let $\oplus : A \times V \to A$ be a pairwise commutative operation and $\iota \in A$ an initial value. +Given a collection $C : K \to \mathcal{M}(V)$, the \emph{reduce} combinator produces a view: +\[ +\mathsf{reduce}_{\iota,\oplus}(C) : K \to A +\] +defined as: +\[ +\mathsf{reduce}_{\iota,\oplus}(C)(k) = \mathsf{fold}_\oplus(\iota, C(k)) +\] +That is, for each key $k$, we fold the operation $\oplus$ over all values in $C(k)$, starting from $\iota$. +\end{definition} + +The view $\mathsf{reduce}_{\iota,\oplus}(C)$ is a derived collection that depends on $C$. +When $C$ changes, the view must be updated to remain consistent. +The next section addresses how to perform these updates efficiently. + +\section{Incremental Updates}\label{sec:incremental} + +When a collection $C$ changes, the view $\mathsf{reduce}_{\iota,\oplus}(C)$ must be updated. +A na\"ive approach would recompute the fold from scratch for each affected key, requiring $O(n)$ time where $n$ is the size of the multiset. +To achieve $O(1)$ updates, we introduce a \emph{remove} operation $\ominus$ that can undo the effect of $\oplus$. + +\subsection{Reducers} + +\begin{definition}[Reducer] +A \emph{reducer} is a triple $R = (\iota, \oplus, \ominus)$ where $\iota \in A$ is an initial value, and +\[ +\oplus, \ominus : A \times V \to A +\] +are update operations such that both $\oplus$ and $\ominus$ are pairwise commutative in the sense of Definition~\ref{def:pairwise-commutative}. +We call $\oplus$ the \emph{add} operation and $\ominus$ the \emph{remove} operation. +\end{definition} + +For a reducer $R = (\iota, \oplus, \ominus)$, we write $\mathsf{reduce}_R$ for $\mathsf{reduce}_{\iota,\oplus}$. + +\begin{definition}[Well-Formed Reducer]\label{def:well-formed-reducer} +A reducer $R = (\iota, \oplus, \ominus)$ is \emph{well-formed} if $\ominus$ is the \textbf{inverse} of $\oplus$: +\[ +\forall a \in A, v \in V.\; (a \oplus v) \ominus v = a +\] +\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). + +\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: +\[ +\forall a \in A, v_1, v_2 \in V.\; (a \ominus v_1) \oplus v_2 = (a \oplus v_2) \ominus v_1 +\] +This ensures that the order of interleaved adds and removes does not affect the final result. +All practical reducers (sum, count, product over commutative groups) satisfy this. +\end{remark} + +\subsection{Deltas} + +We model updates to collections as deltas. + +\begin{definition}[Delta] +A \emph{delta} $\Delta$ for a collection $C$ is a pair $(\Delta^+, \Delta^-)$ where: +\begin{itemize} + \item $\Delta^+ : K \to \mathcal{M}(V)$ represents added values + \item $\Delta^- : K \to \mathcal{M}(V)$ represents removed values +\end{itemize} +We require that $\Delta^-(k) \subseteq C(k)$ for all $k$ (i.e., we can only remove values that exist in the collection). +\end{definition} + +\begin{definition}[Delta Application] +Given a collection $C$ and a delta $\Delta = (\Delta^+, \Delta^-)$ for $C$, the updated collection $C \bullet \Delta$ is defined pointwise by: +\[ +C \bullet \Delta = \lambda k.\, (C(k) \setminus \Delta^-(k)) \uplus \Delta^+(k) +\] +Intuitively, $C \bullet \Delta$ is the collection obtained by first removing all values in $\Delta^-$ from $C$ and then adding all values in $\Delta^+$. +\end{definition} + +\begin{remark}[Operational Construction of Deltas] +In the Skip runtime, for each key $k$ we compute an \emph{old} multiset $\mathit{old}(k)$ of contributing values and a \emph{new} multiset $\mathit{new}(k)$. The delta is then constructed as: +\[ +\Delta^+(k) = \mathit{new}(k) \setminus \mathit{old}(k) +\quad\text{and}\quad +\Delta^-(k) = \mathit{old}(k) \setminus \mathit{new}(k). +\] +Note that $\Delta^+$ and $\Delta^-$ are disjoint by construction. +If $C_{\mathit{old}}$ is the collection with $C_{\mathit{old}}(k) = \mathit{old}(k)$, then $\Delta^-(k) \subseteq C_{\mathit{old}}(k)$, so $\Delta$ is a valid delta for $C_{\mathit{old}}$. +Moreover, $C_{\mathit{old}} \bullet \Delta = C_{\mathit{new}}$ where $C_{\mathit{new}}(k) = \mathit{new}(k)$. +\end{remark} + +\subsection{Incremental Update} + +\begin{definition}[Incremental Reduce] +Given the current accumulator value $a_k$ for key $k$ and a delta $\Delta$, the new accumulator is computed as: +\[ +\mathsf{update}_R(a_k, \Delta, k) = + \mathsf{fold}_\oplus\big( + \mathsf{fold}_\ominus(a_k, \Delta^-(k)),\; + \Delta^+(k) + \big) +\] +That is, we first apply all removals $\Delta^-(k)$ to $a_k$ using $\ominus$, and then apply all additions $\Delta^+(k)$ using $\oplus$. +This is well-defined since $\oplus$ and $\ominus$ are pairwise commutative. +\end{definition} + +\section{Correctness}\label{sec:correctness} + +We now characterize exactly when incremental updates are correct. +At some moment we have an \emph{old} collection $C$ and, for each key $k$, an accumulator +\[ +a_k = \mathsf{reduce}_R(C)(k) +\] +that agrees with the denotational semantics. +A change to the collection is described abstractly by a delta $\Delta$, yielding the \emph{updated} collection +\[ +C' = C \bullet \Delta. +\] +For each key $k$ there are then two ways to obtain the \emph{new} accumulator value: +\begin{itemize} + \item \emph{Denotational recompute:} ignore $a_k$ and compute + \[ + a'_k = \mathsf{reduce}_R(C')(k), + \] + i.e.\ start from $\iota$ and fold $\oplus$ over the current multiset $C'(k)$. + \item \emph{Incremental update:} update the old accumulator $a_k$ using the delta by + \[ + a'_k = \mathsf{update}_R(a_k, \Delta, k), + \] + i.e.\ first remove $\Delta^-(k)$ using $\ominus$, then add $\Delta^+(k)$ using $\oplus$. +\end{itemize} + +\begin{definition}[Incremental Correctness Property] +A reducer $R$ satisfies the \emph{incremental correctness property} if for all collections $C$, all valid deltas $\Delta$ for $C$, and all keys $k$: +\[ +\mathsf{reduce}_R(C \bullet \Delta)(k) = \mathsf{update}_R(\mathsf{reduce}_R(C)(k), \Delta, k) +\] +\end{definition} + +The following theorem shows that the inverse property is both necessary and sufficient for incremental correctness. + +\begin{theorem}[Characterization of Incremental Correctness]\label{thm:equivalence} +Let $R = (\iota, \oplus, \ominus)$ be a reducer (with pairwise commutative $\oplus$ and $\ominus$). +The following are equivalent: +\begin{enumerate} + \item $R$ is well-formed (i.e., $(a \oplus v) \ominus v = a$ for all $a \in A$, $v \in V$). + \item $R$ satisfies the incremental correctness property. +\end{enumerate} +\end{theorem} + +\begin{proof} +We prove both directions. + +\medskip +\noindent\textbf{(1 $\Rightarrow$ 2): Well-formedness implies correctness.} + +Assume $R$ is well-formed. Let $C$ be a collection, $\Delta = (\Delta^+, \Delta^-)$ a valid delta for $C$, and $k$ a key. +Write $M = C(k)$ for the old multiset, $M' = C'(k) = (M \setminus \Delta^-(k)) \uplus \Delta^+(k)$ for the new multiset, and $a = \mathsf{fold}_\oplus(\iota, M)$ for the old accumulator. + +We must show $\mathsf{fold}_\oplus(\iota, M') = \mathsf{fold}_\oplus(\mathsf{fold}_\ominus(a, \Delta^-(k)), \Delta^+(k))$. + +Since $\Delta^-(k) \subseteq M$, we can write $M = M_0 \uplus \Delta^-(k)$ for some multiset $M_0$. +By pairwise commutativity of $\oplus$: +\[ +a = \mathsf{fold}_\oplus(\iota, M) = \mathsf{fold}_\oplus(\iota, M_0 \uplus \Delta^-(k)) = \mathsf{fold}_\oplus(\mathsf{fold}_\oplus(\iota, M_0), \Delta^-(k)) +\] + +Let $a_0 = \mathsf{fold}_\oplus(\iota, M_0)$. +Then $a = \mathsf{fold}_\oplus(a_0, \Delta^-(k))$. + +By the inverse property applied inductively (once for each element of $\Delta^-(k)$): +\[ +\mathsf{fold}_\ominus(a, \Delta^-(k)) = \mathsf{fold}_\ominus(\mathsf{fold}_\oplus(a_0, \Delta^-(k)), \Delta^-(k)) = a_0 +\] + +Therefore: +\[ +\mathsf{fold}_\oplus(\mathsf{fold}_\ominus(a, \Delta^-(k)), \Delta^+(k)) = \mathsf{fold}_\oplus(a_0, \Delta^+(k)) +\] + +Since $M' = M_0 \uplus \Delta^+(k)$: +\[ +\mathsf{fold}_\oplus(\iota, M') = \mathsf{fold}_\oplus(\mathsf{fold}_\oplus(\iota, M_0), \Delta^+(k)) = \mathsf{fold}_\oplus(a_0, \Delta^+(k)) +\] + +Thus both sides are equal. + +\medskip +\noindent\textbf{(2 $\Rightarrow$ 1): Correctness implies well-formedness.} + +Assume $R$ satisfies the incremental correctness property. +We must show $(a \oplus v) \ominus v = a$ for all $a \in A$ and $v \in V$. + +Fix $a \in A$ and $v \in V$. +We first establish the property for $a$ of the form $a = \mathsf{fold}_\oplus(\iota, M)$ for some multiset $M$. + +Define a collection $C$ with a single key $k$ where $C(k) = M \uplus \{v\}$. +Then: +\[ +\mathsf{reduce}_R(C)(k) = \mathsf{fold}_\oplus(\iota, M \uplus \{v\}) = \mathsf{fold}_\oplus(\mathsf{fold}_\oplus(\iota, M), \{v\}) = a \oplus v +\] + +Define delta $\Delta$ with $\Delta^-(k) = \{v\}$ and $\Delta^+(k) = \emptyset$. +This is valid since $v \in C(k)$. +The updated collection is $C'$ with $C'(k) = M$. + +By the incremental correctness property: +\[ +\mathsf{reduce}_R(C')(k) = \mathsf{update}_R(\mathsf{reduce}_R(C)(k), \Delta, k) +\] + +The left side is: +\[ +\mathsf{reduce}_R(C')(k) = \mathsf{fold}_\oplus(\iota, M) = a +\] + +The right side is: +\[ +\mathsf{update}_R(a \oplus v, \Delta, k) = \mathsf{fold}_\oplus(\mathsf{fold}_\ominus(a \oplus v, \{v\}), \emptyset) = (a \oplus v) \ominus v +\] + +Therefore $(a \oplus v) \ominus v = a$. + +This establishes the inverse property for all $a$ in the image of $\mathsf{fold}_\oplus(\iota, -)$. +Since every accumulator value that arises during execution of $\mathsf{reduce}$ is of this form, this suffices for correctness. +\end{proof} + +\begin{remark} +The proof of (2 $\Rightarrow$ 1) establishes the inverse property for \emph{reachable} accumulator values---those expressible as $\mathsf{fold}_\oplus(\iota, M)$ for some finite multiset $M$. +If the accumulator type $A$ contains unreachable values, the inverse property need not hold for them. +In practice, for reducers like sum over integers or product over rationals, all values are reachable, so the distinction is immaterial. +\end{remark} + +\section{Examples}\label{sec:examples} + +\subsection{Sum Reducer} + +\[ +R_{\mathsf{sum}} = (0, \lambda(a,v).\, a + v, \lambda(a,v).\, a - v) +\] + +This reducer is well-formed: addition is commutative, and subtraction is the inverse of addition. + +\paragraph{Worked example.} +Suppose for key $k$ we have $C(k) = \{3, 5, 7\}$, so the current view is $a_k = 0 + 3 + 5 + 7 = 15$. +Now suppose value $5$ is removed and value $2$ is added, giving delta $\Delta^-(k) = \{5\}$ and $\Delta^+(k) = \{2\}$. +The incremental update computes: +\[ +a'_k = (15 - 5) + 2 = 12 +\] +This matches a full recompute: $0 + 3 + 7 + 2 = 12$. + +\subsection{Count Reducer} + +\[ +R_{\mathsf{count}} = (0, \lambda(a,v).\, a + 1, \lambda(a,v).\, a - 1) +\] + +This reducer counts the number of values, ignoring their content. +It is well-formed since $(a+1)-1 = a$. + +\paragraph{Worked example.} +For $C(k) = \{x, y, z\}$, we have $a_k = 3$. +If $y$ is removed ($\Delta^- = \{y\}$) and $w$ is added ($\Delta^+ = \{w\}$), then: +\[ +a'_k = (3 - 1) + 1 = 3 +\] + +\subsection{Min Reducer (Partial)} + +\[ +R_{\mathsf{min}} = (+\infty, \lambda(a,v).\, \min(a,v), \bot) +\] + +The min reducer does not have a well-defined remove operation $\ominus$ in general. +Consider $C(k) = \{3, 5\}$ with $a_k = 3$. +If we remove $5$, we need $a'_k = 3$, which is correct. +But if we remove $3$, we need $a'_k = 5$---yet from $a_k = 3$ alone, we cannot recover that $5$ was the second-smallest value. + +The Skip runtime handles such \emph{partial reducers} by either: +\begin{itemize} + \item Recomputing from scratch when values are removed, or + \item Maintaining additional state (e.g., a sorted structure of all values). +\end{itemize} + +\section{Complexity} + +\begin{theorem}[Time Complexity] +For a well-formed reducer with $O(1)$ add and remove operations: +\begin{itemize} + \item Adding a value: $O(1)$ to update the accumulator + \item Removing a value: $O(1)$ to update the accumulator +\end{itemize} +\end{theorem} + +This is in contrast to a na\"ive re-fold which would require $O(n)$ time where $n$ is the size of the multiset. + +\section{Related Work}\label{sec:related} + +The problem of efficiently maintaining aggregations over changing data has been studied extensively. +We position Skip's design relative to streaming systems, reactive programming, and incremental computation, and describe our formal contribution. + +\paragraph{Skip's design: combinators vs.\ callbacks.} +Skip's reducer $R = (\iota, \oplus, \ominus)$ is a \emph{first-class combinator}---a reusable, composable value that can be applied to any reactive collection via $\mathsf{reduce}_R$. +This differs from systems that allow users to supply add/remove \emph{callbacks} tied to specific engine internals. +Callbacks in other systems are bound to particular contexts (keyed tables, windows, SQL queries); Skip's combinator is context-agnostic and portable across different reactive pipelines. + +\paragraph{Streaming systems with add/remove callbacks.} +Several streaming systems allow users to supply both add and remove logic, but not as a first-class combinator: + +\emph{Apache Flink} (Table API) allows user-defined aggregate functions with \texttt{accumulate} and \texttt{retract} methods. +These are lifecycle methods on a stateful object that Flink's query planner plugs into its retraction-message algebra---not a standalone combinator. + +\emph{Apache Kafka Streams} provides \texttt{Aggregator} (add) and \texttt{Subtractor} (remove) interfaces for KTable aggregations. +These are callbacks passed to the \texttt{aggregate()} method, tightly bound to KTable's update mechanism. + +\emph{Esper CEP} supports \texttt{enter}/\texttt{leave} callbacks for windowed aggregation, but only within sliding-window contexts. + +In all cases, the add/remove logic is configuration for a specific operator, not a reusable combinator. +Moreover, none of these systems provide formal correctness conditions---they document informally what the remove function should do, but correctness is the user's responsibility. + +\paragraph{Systems without user-defined remove.} +\emph{Apache Beam} and \emph{Spark Streaming} do not expose user-defined inverse operations; they handle retractions internally via recomputation or diffing. +\emph{Functional reactive programming} libraries (Fran, Yampa, Reactive Banana~\cite{frp}) provide append-only fold combinators (\texttt{foldp}, \texttt{accumE}) with no built-in remove. +\emph{Incremental computation} frameworks (Adapton, Jane Street's Incremental) support removal internally via dependency tracking, but do not expose a user-facing inverse operation. + +\paragraph{Differential Dataflow and DBSP.} +Differential dataflow~\cite{differentialdataflow} and DBSP~\cite{dbsp} provide rigorous foundations for incremental computation using Z-sets (multisets with integer multiplicities) and abelian groups. +These frameworks \emph{internally} ensure that all built-in aggregations have inverses, but the user typically works at a higher level (SQL, dataflow graphs) without explicitly defining $\oplus$ and $\ominus$. +Skip's design surfaces this algebraic structure as a user-facing combinator; this paper provides the formal analysis. + +\paragraph{Database view maintenance.} +The database literature classifies aggregates as \emph{distributive}, \emph{algebraic}, or \emph{holistic}~\cite{viewmaintenance}. +Skip's well-formed reducers correspond to \emph{invertible distributive aggregates}. +Tangwongsan et al.~\cite{slidingwindow} note that ``prior work often relies on aggregation functions to be invertible'' for efficient sliding-window maintenance. +Yin et al.~\cite{incgraph} require inverse functions for incremental graph aggregation. +These works focus on algorithmic techniques; our contribution is to formalize correctness for Skip's user-facing abstraction. + +\paragraph{Summary: Skip's design and our contribution.} +Skip provides a first-class reduce combinator $R = (\iota, \oplus, \ominus)$ with user-defined add and remove operations, applicable across arbitrary reactive contexts. +Skip's documentation specifies an informal correctness condition for user-defined reducers. +Our contribution is to: +\begin{enumerate} + \item Formalize this correctness condition as a well-formedness property + \item Prove that correctness holds \emph{if and only if} the property is satisfied (Theorem~\ref{thm:equivalence}) + \item Connect Skip's design to the theory of invertible distributive aggregates +\end{enumerate} +This gives Skip users a precise specification for writing correct custom reducers, backed by a complete formal characterization. + +\begin{thebibliography}{9} + +\bibitem{skip} +Skip Team. +\textit{Skip: A Reactive Programming Framework}. +\url{https://github.com/SkipLabs/skip}, 2024. + +\bibitem{differentialdataflow} +Frank McSherry, Derek G.\ Murray, Rebecca Isaacs, and Michael Isard. +Differential dataflow. +In \textit{Proceedings of CIDR}, 2013. + +\bibitem{dbsp} +Mihai Budiu, Tej Chajed, Frank McSherry, Leonid Ryzhyk, and Val Tannen. +DBSP: Automatic incremental view maintenance. +In \textit{Proceedings of VLDB}, 16(7):1601--1614, 2023. + +\bibitem{rx} +Erik Meijer. +Your mouse is a database. +\textit{Communications of the ACM}, 55(5):66--73, 2012. + +\bibitem{selfadjusting} +Umut A.\ Acar, Guy E.\ Blelloch, and Robert Harper. +Adaptive functional programming. +In \textit{Proceedings of POPL}, pages 247--259, 2002. + +\bibitem{viewmaintenance} +Ashish Gupta and Inderpal Singh Mumick. +Maintenance of materialized views: Problems, techniques, and applications. +\textit{IEEE Data Engineering Bulletin}, 18(2):3--18, 1995. + +\bibitem{slidingwindow} +Kanat Tangwongsan, Martin Hirzel, Scott Schneider, and Kun-Lung Wu. +General incremental sliding-window aggregation. +\textit{Proceedings of the VLDB Endowment}, 8(7):702--713, 2015. + +\bibitem{frp} +Conal Elliott and Paul Hudak. +Functional reactive animation. +In \textit{Proceedings of ICFP}, pages 263--273, 1997. + +\bibitem{incgraph} +Shufeng Yin, Huanchen Zhang, Zhengyi Yang, Wentao Han, Wenguang Chen, and Yingxia Shao. +GraphBolt: Dependency-driven synchronous processing of streaming graphs. +In \textit{Proceedings of ICDE}, 2022. + +\end{thebibliography} + +\end{document} From 2bfd40cc5b43308132e9c8add7f3bc02da335eab Mon Sep 17 00:00:00 2001 From: Cristiano Calcagno Date: Thu, 27 Nov 2025 08:20:52 +0100 Subject: [PATCH 02/20] Formalize reduce semantics and incremental correctness --- reduce.pdf | Bin 318086 -> 320019 bytes reduce.tex | 33 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zX>WEDJNs6Rt(Kh?FL_YI&gW(56|>yKkEo<|bvpUsxJB#*M%w#iv*|X+{j)|JWruln za`8+&k2N=uq%J2CvZnS9Vjkl^vwh!C@CZ$Bq}Ht@{-u| ze72RrKLiqriw`JAMVY*!=>9ikxXm7N57~PkL=7Gw}+ z5JXn$y1OI80*f*TFz90QtGd~_BN6z&NRh!uQ6Fj7&|EPj8h=>cqi4H*AFmrXyw=_e)PY-g;^;lSAbz}7s{0Sqb zW?!Ax>T)qDy#Rw}mB<~bn!(P(U~<*UFAH-fHnS!a*^Z5r;9uT_3u$kcaT_T&C;zni z{d{=HZEW0>c+SN1B$Y%;7=nCR%Nj_9#<3k6;rMaJtr+QkOGh z*&rVdX8BB5Vr+GTZTLu<3}Ye=zr0)(Y*sK+CTIel-(}5fGO+*sJ0RQA z=R`hbX7h!*h58|gwI_wzptvUr!{GgQwC(EaK4+Ka6^v86r3%~)hvNAl-#nZu{GPis zS8u|gANrNhn*wMh;eE4PCHBptbyeTl=D4AL-&2;GJ{tL|y*Fl!%Sr`RP20u~0SlAe zhpP>isP Date: Thu, 27 Nov 2025 09:08:19 +0100 Subject: [PATCH 04/20] Improve exposition of partial reducers and algebraic aggregates - Move 'Partial Reducers' remark to after Theorem 4 (was prematurely placed) - Add proper definition of algebraic aggregates in Section 3.2 - Remove incoherent 'partial average reducer' paragraph - you cannot maintain average incrementally with only the average value (need count too) - Clarify min reducer as canonical example of non-invertible aggregate - Explain Skip's strategies for handling partial reducers --- reduce.pdf | Bin 321872 -> 336915 bytes reduce.tex | 51 ++++++++++++++++++++++++++++++++++++++++++++------- 2 files changed, 44 insertions(+), 7 deletions(-) diff --git a/reduce.pdf b/reduce.pdf index 983cac8843f58e6d5fb8fc3bfccf5235e1b5a5bf..5f587e5061fe885543407d5c01a40e4f272b983f 100644 GIT binary patch delta 142749 zcmV)CK*GP!(i4-&6tK89135T2mob6_D1Y5rNt4^g6~6mdFgLIywtHXX^1-pAl#+5% zWl!RRtpkR{jF2G-1^{O?|30r5EDbgnjb`K|7lHs9-EV*2d)Ov#Srqd_fh@R(YFW=Z(7)*}80tI&b%Nc$=@ADvT?B!+(Tt zi)Q}*>MvivRcgF~5J)IJ!+!33eOp}Db3wAaT>9Y=L7X6hc|ZF0wk~e-7G`z}{VLzT znJ+jIlQ$x3Kjf{QZnmI=AUoLtu0{lwmiczRV3cRqFkEMazgbqB-M%eW-79$W=>Ko> za%*o}xR%>n|1kF#%j?JPBR6Wlv46i|>e|1kZv1ssHf>#A@7wZzKCC<@-`#Ilj(QxE zy>nAmO}p6DmHWRen#L34m@dlN|E-H=zit~eUkHV>w{y+1D!eK_=9}HRcnx=20uwL` zkfMOn{I>8z%En*4cC2RLRyvMg+%n%cg@01EexQ|Sg5E*8hK!|aaCTEy8-M@2{SXEu zL<3JF(ZGGP58qiCxv7@berp+M{q233ck|tsO<8S02$HDmT}-0%B>sF??wlwL5aRg; 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\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 $\ominus$. +The database literature also defines \emph{algebraic} aggregates~\cite{viewmaintenance}: aggregates that can be computed by maintaining a fixed number of distributive aggregates and post-processing their results. +For example, average is algebraic because it can be computed from the distributive aggregates sum and count via division. +In Skip, this corresponds to using a well-formed reducer with richer accumulator state (e.g., $(sum, count)$ pairs) followed by a pointwise mapper to extract the final value. +We illustrate this pattern in Section~\ref{sec:examples}. + \subsection{Deltas} We model updates to collections as deltas. @@ -386,6 +391,14 @@ \section{Correctness}\label{sec:correctness} In many practical reducers (for example sum over integers or product over rationals), every accumulator value is reachable as $\mathsf{fold}_\oplus(\iota, M)$ for some multiset $M$, so the definition of well-formedness above coincides with the simpler global inverse law $(a \oplus v) \ominus v = a$ for all $a \in A$, $v \in V$. \end{remark} +\begin{remark}[Partial Reducers in Skip] +Theorem~\ref{thm:equivalence} characterizes when incremental updates are correct for well-formed reducers. +In Skip's concrete API, the remove operation is allowed to be \emph{partial}: a reducer's remove function can signal ``recompute from scratch'' (by returning \texttt{None} in the ReScript bindings) instead of producing an updated accumulator. +Such partial reducers handle cases where $\ominus$ cannot efficiently invert $\oplus$---for example, computing the minimum without maintaining auxiliary state. +The runtime responds by recomputing the fold from scratch for that key. +Examples of partial reducers appear in Section~\ref{sec:examples}. +\end{remark} + \section{Examples}\label{sec:examples} \subsection{Sum Reducer} @@ -421,21 +434,45 @@ \subsection{Count Reducer} a'_k = (3 - 1) + 1 = 3 \] +\subsection{Average Reducer (Algebraic)} + +The average is a classic example of an \emph{algebraic} aggregate~\cite{viewmaintenance}: it can be expressed as a post-processing of a distributive aggregate over richer state. +A standard encoding uses accumulator state $A = \mathbb{R} \times \mathbb{N}$ to track sum and count. + +Define: +\[ +R_{\mathsf{avgState}} = \big((0,0),\ \lambda((s,c),v).\, (s+v, c+1),\ \lambda((s,c),v).\, (s-v, c-1)\big) +\] +with accumulator state $A = \mathbb{R} \times \mathbb{N}$ tracking (sum, count). +On reachable states with $c > 0$, the corresponding average is $\mathsf{avg} = s / c$. +This reducer is well-formed: addition and subtraction on sum and increment/decrement on count satisfy the inverse law on all reachable accumulator states, so Theorem~\ref{thm:equivalence} applies. +In Skip, the average view can be implemented by first using $\mathsf{reduce}_{R_{\mathsf{avgState}}}$ and then applying a pointwise mapper that divides sum by count for each key. + +Note that maintaining \emph{only} the average (without count) is insufficient: to update the average when adding a value, one needs to know how many values contributed to the current average. +Thus, average is genuinely an algebraic aggregate requiring auxiliary state, unlike sum or count which can be maintained with a single accumulator value. + \subsection{Min Reducer (Partial)} +The min reducer demonstrates why invertibility is essential for incremental updates. +With accumulator $A = \mathbb{R} \cup \{+\infty\}$, the add operation is: \[ -R_{\mathsf{min}} = (+\infty, \lambda(a,v).\, \min(a,v), \bot) +\iota = +\infty, \quad \oplus = \lambda(a,v).\, \min(a,v) \] -The min reducer does not have a well-defined remove operation $\ominus$ in general, so $R_{\mathsf{min}}$ is not a reducer in the strict sense of Definition~\ref{def:well-formed-reducer}. +However, there is no inverse operation $\ominus$ that works in general. Consider $C(k) = \{3, 5\}$ with $a_k = 3$. -If we remove $5$, we need $a'_k = 3$, which is correct. -But if we remove $3$, we need $a'_k = 5$---yet from $a_k = 3$ alone, we cannot recover that $5$ was the second-smallest value. +\begin{itemize} + \item If we remove $5$: we need $a'_k = 3$. We could define $(3 \ominus 5) = 3$ (removing a non-minimum has no effect). + \item If we remove $3$: we need $a'_k = 5$. But from $a_k = 3$ alone, we cannot know that $5$ was the second-smallest value! +\end{itemize} + +This shows min is \emph{not} an invertible distributive aggregate: knowing only the accumulated minimum is insufficient to update the result when the minimum itself is removed. -The Skip runtime handles such \emph{partial reducers} by either: +In Skip's implementation, min is handled as a \emph{partial reducer}: \begin{itemize} - \item Recomputing from scratch when values are removed, or - \item Maintaining additional state (e.g., a sorted structure of all values). + \item The remove function signals ``cannot update incrementally'' (e.g., returns \texttt{None}) + \item The runtime responds by recomputing from scratch: $\mathsf{fold}_{\min}(\iota, C'(k))$ + \item Alternatively, one can maintain richer state (e.g., a sorted multiset of all values), making the remove operation invertible on that richer state---but this is no longer a constant-space reducer \end{itemize} \section{Complexity} From a89264d889f8605909f742592c158cdfe0f07463 Mon Sep 17 00:00:00 2001 From: Cristiano Calcagno Date: Thu, 27 Nov 2025 10:11:56 +0100 Subject: [PATCH 05/20] aggregates and more examples --- reduce.pdf | Bin 336915 -> 348226 bytes reduce.tex | 63 +++++++++++++++++++++++++++++++++++++++++++++++++++-- 2 files changed, 61 insertions(+), 2 deletions(-) diff --git a/reduce.pdf b/reduce.pdf index 5f587e5061fe885543407d5c01a40e4f272b983f..7786a300db3d0a2ddd59b6198fc0f7c7e6e58c98 100644 GIT binary patch delta 120804 zcmV)8K*qn5$P~h$6|mwemm!=4DSypcOLN;e629kGcyClygu(lftvO_7HdVVdnM2l< zIn3lhOSH{eNpy*nUHeMuT*X`?v&4LwGcu}w^iSVLk{fnRS6iBK289y$T+%kXm%xX*$$<4?hF8&^8 z$;m7O@{{8vmRrfdd4_Z67QvnDWD%Aj9T%C*>1C8~&Jd}Fjh^m-yDHBq4zik(%|(VP z4Fd0{@XUWAZpzE|8FE0msWL}`+~L%#rQnTK`H~tD?dtY}^9Cm;DuA+o7lY09|1_jsR41fvrM9{Px1VE&d zr+Mgz1y3@WbldDRn=(cCXyjkB>6F)EGF1^VD%HsA6iN{Tcx|#6$>1lDI`|H1qOPeN zR+EeuCXdxLOR(DSsH4?pvjk<8psd?!jegP<=Rq8!!C#(5ae__!2H;0lM3bp_0gHdf zdzFD-8r(g$l0CPUt0qkaG 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