Foundation

Functional Programming

• Typeclasses
• Curry-Howard Isomorphism
• Theorems for Free
• Referential Transparency
• Equational Reasoning
• Life After Monoids
• Algebird
• Kmett's mapping of abstract algebra to software patterns

Scala

Scala is the host of Axle for many reasons, principled and not.

It's support for functional programming is chief among them.

Axle makes extensive use of Scala 2's "Context Bounds".

When Axle moves to Scala 3, much of this will use the enw given / using syntax.

Typelevel and Cats

Axle makes use of several Typelevel libraries including

• Cats (Show, Eq)
• Cats Effect (IO, Async)
• Spire (Field, Ring)
• Monix
• ...

Architecture

Ideal

Laws are organized into a separate axle-laws jar for use in tests by code that builds upon these typeclasses. Many such witnesses are provided by Axle for native Scala collections.

Witnesses are also defined for other common jars from the Java and Scala ecosystems. Read more about "these third party libraries".

Remaining Design Issues

Please get in touch if you'd like to discuss these or other questions.

Algebra

The spire project is a dependency of Axle. spire.algebra defines typeclasses for Monoid, Group, Ring, Field, VectorSpace, etc, and witnesses for many common numeric types as well as those defined in spire.math

The axle.algebra package defines several categories of typeclasses:

• higher-kinded: Functor, Finite, Indexed, Aggregatable
• mathematical: LinearAlgebra, LengthSpace
• visualization: Tics, Plottable

Axioms are defined in the axle.algebra.laws package as ScalaCheck properties.

They are organized with Discipline.

Support for Third Party Libraries

Witnesses for 3rd party libraries.

Parallel Collections

"org.axle-lang" %% "axle-parallel" % "0.6.6"

For use with Scala Parallel Collections library ("org.scala-lang.modules" %% "scala-parallel-collections" % ...)

XML

"org.axle-lang" %% "axle-xml" % "0.6.6"

For use with Scala XML library ("org.scala-lang.modules" %% "scala-xml" % ...)

XML includes axle.web, where HTML and SVG visualizations reside.

JBLAS

"org.axle-lang" %% "axle-jblas" % "0.6.6"

Linear Algebra and other witnesses for JBLAS which itself is a wrapper for LAPACK. Includes Principal Component Analysis (PCA).

JODA

"org.axle-lang" %% "axle-joda" % "0.6.6"

Witnesses for the Joda time library.

JUNG

"org.axle-lang" %% "axle-jung" % "0.6.6"

Directed and Undirected Graph witnesses for the JUNG library.

AWT

"org.axle-lang" %% "axle-awt" % "0.6.6"

Witnesses for AWT

Future Work

Scala 3

• Scala 3
• convert to scalameta munit
• correct "Package Objects" doc

Bugs and adoption barriers

• Fix LogisticRegression and move LogisticRegression.md back

• Fix GeneticAlgorithmSpec

• Featurizing functions should return HLists or other typelevel sequences in order to avoid being told the number of features

• Redo Logic using Abstract Algebra

• Simple graph implementation so that axle-core can avoid including axle-jung

• svgJungDirectedGraphVisualization move to a axle-jung-xml jar?

• Will require externalizing the layout to its own.... typeclass?

• Layout of bayesian network is quite bad -- check ABE SVG

• axle-png to avoid Xvfb requirement during tests

• Chicklet borders / colors on site

• Factor axle.algebra.chain in terms of well-known combinators

Types and Axioms

• Replace Finite with Shapeless's version (eg Sized[Vector[_], nat.2])
• Delete Finite conversions for jung
• Replace with Cats: FoldLeft, Bijection, FunctionPair, Endofunctor
• Define laws for Scanner, Aggregator, Zipper, Indexed, Talliable, Finite?
• Sort out MapFrom, FromStream, FromSet
• Test axle.algebra.tuple2Field
• similarity syntax for SimilaritySpace (see axle.bio.*)
• Projections of jung graphs for Finite
• kittens or magnolia

• pattern match in FirstOrderPredicateLogic

• subtyping for Suit and Rank

• Machinist?

• Type-level matrix dimension using -Yliteral-types and singleton-ops in LinearAlgebra typeclass

• Make the Int abstract in KMeans{,Visualization}, LinearAlgebra, etc

• Eigenvectors

• Σ ⊣ Δ ⊣ Π means "sums are left adjoint to diagonals, which are left adjoint to products."

Compute Engines

• Bring back Spark spoke -- Solve the Spark ClassTag issue (see Frameless?)
• Performance benchmarking
• netlib-java Matrix
• GPU/CUDA support
• Algebird/Scalding for distributed matrices, HyperLogLog, etc
• Most MapRedicible witnesses are inefficient (eg calling toVector, toSeq, etc)

Hygiene

• Get rid of implicit arg passing to KMeans in ClusterIrises.md (and KMeansSpecification)
• Factor tics and tics-{joda,algebra,spire} into separate libs?
• remove unnecessary implicit Field, R{,i}ng, {Additive, Multiplicative}Monoid once spire/cats play well
• Fix "unreachable" default pattern match cases
• Review remaining usage of: asInstanceOf, ClassTag, and Manifest
• Review groupBy uses -- they use university equality. Replace with Eq
• axle.algorithms coverage > 80%
• axle.core coverage > 80%
• Rm throws from axle.jung
• Rm throws from axle.pgm.BayesianNetwork

Site

Near term / minor

• General

• Expand acronyms and include wikipedia links in "Future Work" sections

• Make dependencies clear in each section
• Introduction

• Smaller images for Gallery
• Foundation

• Architecture
• Math

• Intro section bullets not nesting
• Random, Uncertain

• Bayesian network rendering is missing tables
• Text

• Say more about Python Grammar

Later

• laikaIncludeAPI := true in build.sbt

• look at more of these options

• Meta tag with keywords: axle, scala, dsl, data, analysis, science, open-source, adam pingel

• update google analytics version

• test animation with monix 3.4.0

• Friend of Spire

• README: data sets from axle.data (Astronomy, Evolution, Federalist Papers, Irises)

Build

• ghpagesCleanSite leaving stale files?
• GitHub "Releases" in sidebar should show "latest"
• keep axle.g8 and axle versions in sync
• site publish (git push) via github action?
• make axle.g8 more axle-flavored (use cats.IO App as parent for HelloWorld)