Dagon
Dagon [...] is an ancient Mesopotamian AssyroBabylonian 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.
Overview
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" %% "dagoncore" % "0.3.0",
compilerPlugin("org.spiremath" %% "kindprojector" % "0.9.4"))
// use this snippet for JS, or crossbuilding
libraryDependencies ++= List(
"com.stripe" %%% "dagoncore" % "0.3.0",
compilerPlugin("org.spiremath" %% "kindprojector" % "0.9.4"))
We strongly encourage you to use kindprojector with Dagon. Otherwise, working with types like FunctionK
will be signficantly more painful.
Example
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
andSimplifyAddition
)
Dagon allows you to write very terse, natural rules that use partial functions (similar to pattternmatching) 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 everchanging 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)
: aleaf
node of your ASTLiteral.Unary(node, f)
: a childnode
and a unary functionf
Literal.Binary(lhs, rhs, g)
: two nodes (lhs
,rhs
) and a binary functiong
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 thirdparty use.

Costbased 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)
intox * z + y * z
. 
Benchmarking and performance optimization. While this code performs adequately for most realworld use cases it's likely quadratic or superquadratic in the worstcase. 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 20132017 Twitter.