Tools for rewriting and optimizing DAGs (directed-acyclic graphs) in Scala
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Dagon [...] is an ancient Mesopotamian Assyro-Babylonian and Levantine (Canaanite) deity. He appears to have been worshipped as a fertility god in Ebla, Assyria, Ugarit and among the Amorites. The Hebrew Bible mentions him as the national god of the Philistines with temples at Ashdod and elsewhere in Gaza.

-- Dagon Wikipedia entry


Dagon is a library for rewriting directed acyclic graphs (i.e. DAGs).

Quick Start

Dagon supports Scala 2.10, 2.11, and 2.12. It supports both the JVM and JS platforms.

To use Dagon in your own project, you can include this snippet in your build.sbt file:

// use this snippet for the JVM
libraryDependencies ++= List(
  "com.stripe" %% "dagon-core" % "0.3.0",
  compilerPlugin("org.spire-math" %% "kind-projector" % "0.9.4"))

// use this snippet for JS, or cross-building
libraryDependencies ++= List(
  "com.stripe" %%% "dagon-core" % "0.3.0",
  compilerPlugin("org.spire-math" %% "kind-projector" % "0.9.4"))

We strongly encourage you to use kind-projector with Dagon. Otherwise, working with types like FunctionK will be signficantly more painful.


To use Dagon you will need the following things:

  • a DAG or AST type (e.g. Eqn[T] below).
  • a transformation from your DAG to Dagon's literal types (e.g. toLiteral)
  • some rewrite rules (e.g. SimplifyNegation and SimplifyAddition)

Dagon allows you to write very terse, natural rules that use partial functions (similar to patttern-matching) to identify and transform some AST "shapes" while leaving others alone. These patterns will all be recursively applied until none of them match any part of the AST.

One consequence of this is that your rules should shrink the AST, or at least simplify it in some sense. If your rules do not converge on a final AST it's possible that the rewriter will not terminate (and will loop forever on an ever-changing AST).

Here's a complete, working example of using Dagon:

object Example {

  import com.stripe.dagon._
  // 1. set up an AST type

  sealed trait Eqn[T] {
    def unary_-(): Eqn[T] = Negate(this)
    def +(that: Eqn[T]): Eqn[T] = Add(this, that)
    def -(that: Eqn[T]): Eqn[T] = Add(this, Negate(that))

  case class Const[T](value: Int) extends Eqn[T]
  case class Var[T](name: String) extends Eqn[T]
  case class Negate[T](eqn: Eqn[T]) extends Eqn[T]
  case class Add[T](lhs: Eqn[T], rhs: Eqn[T]) extends Eqn[T]
  object Eqn {
    // these function constructors make the definition of
    // toLiteral a lot nicer.
    def negate[T]: Eqn[T] => Eqn[T] = Negate(_)
    def add[T]: (Eqn[T], Eqn[T]) => Eqn[T] = Add(_, _)
  // 2. set up a transfromation from AST to Literal

  val toLiteral: FunctionK[Eqn, Literal[Eqn, ?]] =
    Memoize.functionK[Eqn, Literal[Eqn, ?]](
      new Memoize.RecursiveK[Eqn, Literal[Eqn, ?]] {
        def toFunction[T] = {
          case (c @ Const(_), f) => Literal.Const(c)
          case (v @ Var(_), f) => Literal.Const(v)
          case (Negate(x), f) => Literal.Unary(f(x), Eqn.negate)
          case (Add(x, y), f) => Literal.Binary(f(x), f(y), Eqn.add)
  // 3. set up rewrite rules

  object SimplifyNegation extends PartialRule[Eqn] {
    def applyWhere[T](on: Dag[Eqn]) = {
      case Negate(Negate(e)) => e
      case Negate(Const(x)) => Const(-x)

  object SimplifyAddition extends PartialRule[Eqn] {
    def applyWhere[T](on: Dag[Eqn]) = {
      case Add(Const(x), Const(y)) => Const(x + y)
      case Add(Add(e, Const(x)), Const(y)) => Add(e, Const(x + y))
      case Add(Add(Const(x), e), Const(y)) => Add(e, Const(x + y))
      case Add(Const(x), Add(Const(y), e)) => Add(Const(x + y), e)
      case Add(Const(x), Add(e, Const(y))) => Add(Const(x + y), e)
  val rules = SimplifyNegation.orElse(SimplifyAddition)

  // 4. apply rewrite rules to a particular AST value

  val a:  Eqn[Unit] = Var("x") + Const(1)
  val b1: Eqn[Unit] = a + Const(2)
  val b2: Eqn[Unit] = a + Const(5) + Var("y")
  val c:  Eqn[Unit] = b1 - b2

  val simplified: Eqn[Unit] =
    Dag.applyRule(c, toLiteral, rules)

Dagon assumes your AST is paramterized on a T type. If yours is not, you can create a new type of the correct shape using a phantom type:

sealed trait Ast

object Ast {
  // T is a "phantom type" -- it's not actually used in the type alias.
  type Phantom[T] = Ast

val toLiteral: FunctionK[Ast.Phantom, Literal[Ast.Phantom, ?]] = ...

Implementing toLiteral

The function toLiteral has the type FunctionK[N, Literal[N, ?]]. This means that it can produce a N[T] => Literal[N, T]. The type N[_] is your AST type; in the example it was Eqn[_].

Dagon's Literal is sealed and has three subtypes:

  • Literal.Const(leaf): a leaf node of your AST
  • Literal.Unary(node, f): a child node and a unary function f
  • Literal.Binary(lhs, rhs, g): two nodes (lhs, rhs) and a binary function g

The functions f and g are mapping from inputs of type N[T1] to outputs of type N[T2] (where N[_] is your AST type). In the example above T1 and T2 are both Unit.

It's important that your toLiteral function is invertible. That means that the following should be true:

val node: Ast[T] = ...
toLiteral[T](node).evaluate == node

Future Work

Here are some directions possible future work could take:

  • Producing laws to generate and test your AST values against these rewrites. Many of the tests we use internally could be generalized and exported for third-party use.

  • Cost-based optimization: right now rules are applied until they don't match, which means that rules need to be conservative, and should not expand the size of the graph. Some rules could locally increase graph size but result in smaller graphs overall. One example of this would be arithmetic distribution, e.g. rewriting x * (y + z) into x * z + y * z.

  • Benchmarking and performance optimization. While this code performs adequately for most real-world use cases it's likely quadratic or super-quadratic in the worst-case. We could likely optimize some of the algorithms we are using as well as the actual code involved.

Copyright and License

Dagon is available to you under the Apache License, version 2.

Copyright 2017 Stripe.

Derived from Summingbird, which is copyright 2013-2017 Twitter.