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Monadic directed acyclic graph datastructures

Acyclicity provides a single data structure, Dag[T], representing a graph of nodes of type T, with monadic operations and several other utility methods, plus the means to generate DOT for input to GraphViz.


  • provides a simple immutable monadic implementation of a DAG
  • implements map, flatMap and filter for Dags
  • can deduce a partial order on a graph
  • generates Dot instances representing a DOT abstract syntax tree
  • serializes Dot instances to Strings, which can be rendered by GraphViz
  • can find the transitive closure, transitive reduction and inverse of a graph
  • methods for addition and subtraction of graph nodes

Availability Plan

Acyclicity has not yet been published. The medium-term plan is to build Acyclicity with Fury and to publish it as a source build on Vent. This will enable ordinary users to write and build software which depends on Acyclicity.

Subsequently, Acyclicity will also be made available as a binary in the Maven Central repository. This will enable users of other build tools to use it.

For the overeager, curious and impatient, see building.

Getting Started

Acyclicity provides a monadic representation of a directed, acyclic graph (DAG) called Dag, and support for generating DOT files which can be rendered with tools such as GraphViz.

All Acyclicity terms and types are defined in the acyclicity package.

import acyclicity.*

The Dag[T] type represents a mapping from nodes of type T to zero, one or many other nodes in the graph, and can be constructed by providing the mapping from each node to its Set of dependent nodes, or by calling,

val nodes: Set[Int] = Set(2, 3, 4, 5, 10, 15, 30)
def fn(n: Int): Set[Int] = (0 until n).filter(n%_ == 0).to(Set)
val dag = Dag(nodes)(fn)

where nodes is a Set of nodes, and fn is a function from each node to its dependencies.

For example,

val factors = Dag(
  30 -> Set(2, 3, 4, 5, 10, 15),
  15 -> Set(5, 3),
  10 -> Set(5, 2),
  5 -> Set(),
  3 -> Set(),
  2 -> Set()

Monadic Operations

A Dag[T] may be mapped to a Dag[S] with a function T => S, like so:

val dag2 =*10)

Care should be taken when more than one node in the domain maps to a single node in the range, but both incoming and outgoing edges will be merged in such cases.

It's also possible to flatMap with a function T => Dag[S]. This will replace every node of type T with a subgraph, Dag[S] with incoming edges attached to all source nodes of the subgraph, and pre-existing outgoing edges attached to all destination nodes of the subgraph.

A Dag[T] may also be filtered with a predicate, T => Boolean. The removal of a node during filtering will reattach every incoming edge to every outgoing edge of that node.

Other Operations

The method Dag#reduction will calculate the transitive reduction of the graph, removing any direct edge between two nodes when transitive edges exist between those nodes.

The dual of this operation is the transitive closure, which adds direct edges between each pair of nodes between which a transitive path exists. This is available with the Dag#closure method.

A list of nodes will be returned in topologically-sorted order by calling Dag#sorted.

DOT output

An extension method, dot, on Dags of Texts will produce a Dot instance, an AST of the DOT code necessary to render a graph. This can then be serialized to a Text with the serialize method.

Typical usage would be to first convert a Dag[T] to a Dag[Text], then produce the Dot instance and serialize it, for example:


def graph() = println(


This library is incomplete, inadequately tested and subject to further development, and is recommended to be used by developers who do not mind examining the source code to diagnose unexpected behavior.


Acyclicity is classified as fledgling. For reference, Scala One projects are categorized into one of the following five stability levels:

  • embryonic: for experimental or demonstrative purposes only, without any guarantees of longevity
  • fledgling: of proven utility, seeking contributions, but liable to significant redesigns
  • maturescent: major design decisions broady settled, seeking probatory adoption and refinement
  • dependable: production-ready, subject to controlled ongoing maintenance and enhancement; tagged as version 1.0.0 or later
  • adamantine: proven, reliable and production-ready, with no further breaking changes ever anticipated

Projects at any stability level, even embryonic projects, can still be used, as long as caution is taken to avoid a mismatch between the project's stability level and the required stability and maintainability of your own project.

Acyclicity is designed to be small. Its entire source code currently consists of 242 lines of code.


Acyclicity will ultimately be built by Fury, when it is published. In the meantime, two possibilities are offered, however they are acknowledged to be fragile, inadequately tested, and unsuitable for anything more than experimentation. They are provided only for the necessity of providing some answer to the question, "how can I try Acyclicity?".

  1. Copy the sources into your own project

    Read the fury file in the repository root to understand Acyclicity's build structure, dependencies and source location; the file format should be short and quite intuitive. Copy the sources into a source directory in your own project, then repeat (recursively) for each of the dependencies.

    The sources are compiled against the latest nightly release of Scala 3. There should be no problem to compile the project together with all of its dependencies in a single compilation.

  2. Build with Wrath

    Wrath is a bootstrapping script for building Acyclicity and other projects in the absence of a fully-featured build tool. It is designed to read the fury file in the project directory, and produce a collection of JAR files which can be added to a classpath, by compiling the project and all of its dependencies, including the Scala compiler itself.

    Download the latest version of wrath, make it executable, and add it to your path, for example by copying it to /usr/local/bin/.

    Clone this repository inside an empty directory, so that the build can safely make clones of repositories it depends on as peers of acyclicity. Run wrath -F in the repository root. This will download and compile the latest version of Scala, as well as all of Acyclicity's dependencies.

    If the build was successful, the compiled JAR files can be found in the .wrath/dist directory.


Contributors to Acyclicity are welcome and encouraged. New contributors may like to look for issues marked beginner.

We suggest that all contributors read the Contributing Guide to make the process of contributing to Acyclicity easier.

Please do not contact project maintainers privately with questions unless there is a good reason to keep them private. While it can be tempting to repsond to such questions, private answers cannot be shared with a wider audience, and it can result in duplication of effort.


Acyclicity was designed and developed by Jon Pretty, and commercial support and training on all aspects of Scala 3 is available from Propensive OÜ.


Acyclicity takes its name from the graphs it represents, which must not contain cycles.

In general, Scala One project names are always chosen with some rationale, however it is usually frivolous. Each name is chosen for more for its uniqueness and intrigue than its concision or catchiness, and there is no bias towards names with positive or "nice" meanings—since many of the libraries perform some quite unpleasant tasks.

Names should be English words, though many are obscure or archaic, and it should be noted how willingly English adopts foreign words. Names are generally of Greek or Latin origin, and have often arrived in English via a romance language.


The logo shows a single dot, alluding to the DOT language.


Acyclicity is copyright © 2024 Jon Pretty & Propensive OÜ, and is made available under the Apache 2.0 License.