Realizing applied category theory in Julia

An introduction to Catlab.jl, an experimental framework for ACT

Evan Patterson
Stanford University, Statistics Department

MIT (Applied) Categories Seminar
January 16, 2020

Applied category theory in 2020

Applications of category theory began in logic/type theory but have now spread to

  • Quantum mechanics
  • Chemistry (reaction networks, Petri nets)
  • Databases and knowledge representation
  • Control theory and robotics
  • Probability and statistics
  • ...

Why? CT offers a unifying and widely applicable toolbox for building mathematical models of compositional phenomena.

But where are the tools?

In applied math, the math is only half the story.

Applied mathematics must ultimately feed back into better scientific and computational methods.

Software is a big part of this, but software tools for ACT are early and fragmented.

Question: How do we usefully realize the mathematical models of ACT on a computer?

Two views on categories and software

  1. Designing with categories: CT informing the design, architecture, and implementation of computer programs, in any domain

    Slogan: "Design patterns, but better"

  2. Computing on categories: data structures and algorithms for computing with objects and morphisms in categories

    Slogan: "Computer algebra of categories" or "computational category theory"

Both are valuable, but they are not the same. In principle, they are orthogonal.

This talk is about computational category theory (mostly).

Catlab.jl: an experimental framework for ACT

We are developing Catlab.jl, a Julia package that aims to be:

  • programming library
  • computer algebra system
  • interactive computing environment (mainly through Jupyter)

Some specific features:

  • Work with symbolic expressions for objects and morphisms
  • Create, manipulate, and visualize wiring diagrams for morphisms
  • Generate efficient Julia programs for algorithms expressed categorically

Overview: representing morphisms in monoidal categories

As of today, one of Catlab's most powerful capabilities is transforming between different computational representations of morphisms in symmetric monoidal categories.

Why Julia?

Why use Julia for ACT? Julia is a modern programming language:

  • Multiple dispatch gives generic operations of composition, monoidal products, etc.
  • Unicode operators enable familiar mathematical notation
  • Macro system ala Lisp allows further customization of syntax

Yet Julia is also practical:

  • Designed to be fast, thanks to JIT (just-in-time) compiler
  • Growing force in conventional applied math community:
    • Numerical linear algebra
    • Differential equations
    • Optimization
    • Statistics and machine learning

Some other projects in computational category theory

  • Cateno by Jason Morton, in Julia
    • Prototype, unmaintained since 2015
    • Early inspiration for Catlab
    • In turn, draws inspiration from Haskell type classes
  • Categorical Query Language by Conexus AI (AGPL/proprietary)
    • Solves word problems in finitely presented categories
  • Statebox (AGPL/proprietary)
    • Compositional programming with formal verification
  • Rewriting and proof assistants for monoidal/higher categories:

Talk outline

  1. Major features
    • Signatures, instances, and symbolic expressions
    • Wiring diagrams
    • Generating and parsing Julia programs
  2. Applications
  3. Future directions

Signatures, instances, and symbolic expressions

In Catlab, you can define signatures of algebraic structures (@signature macro).

In [2]:
module Sandbox # Encapsulate in module to avoid name conflicts later
export dom, codom, id, compose

using Catlab

@signature Category(Ob,Hom) begin

    compose(f::Hom(A,B), g::Hom(B,C))::Hom(A,C) <= (A::Ob, B::Ob, C::Ob)


Note: Usually use Category from "standard library" (Catlab.Doctrines)

Signature interface

Signatures define generic functions in Julia.

In [3]:
methodswith(Sandbox.Category.Ob, Sandbox)
1-element Array{Method,1}:
  • id(A::Main.Sandbox.Category.Ob) in Main.Sandbox
In [4]:
methodswith(Sandbox.Category.Hom, Sandbox)
3-element Array{Method,1}:
  • codom(::Main.Sandbox.Category.Hom) in Main.Sandbox
  • compose(f::Main.Sandbox.Category.Hom, g::Main.Sandbox.Category.Hom) in Main.Sandbox
  • dom(::Main.Sandbox.Category.Hom) in Main.Sandbox


Two ways to instantiate a signature:

  1. As Julia functions on concrete data types (@instance macro or standard Julia)
  2. As typed systems of symbolic expressions (@syntax macro)

Instance: category of matrices

Instance of Category with (element type, dimension) pairs as objects and matrices as morphisms.

In [5]:
using Catlab, Catlab.Doctrines
import Catlab.Doctrines: Ob, Hom, dom, codom, id, compose, , 
using LinearAlgebra: I

struct MatrixDomain

@instance Category(MatrixDomain, Matrix) begin
    dom(M::Matrix) = MatrixDomain(eltype(M), size(M,1))
    codom(M::Matrix) = MatrixDomain(eltype(M), size(M,2))

    id(m::MatrixDomain) = Matrix{m.eltype}(I, m.dim, m.dim)
    compose(M::Matrix, N::Matrix) = M*N

Instance: Category of matrices

In [6]:
M = [0 1; -1 0]
2×2 Array{Int64,2}:
  0  1
 -1  0
In [7]:
2×2 Array{Int64,2}:
 -1   0
  0  -1
In [8]:
2×2 Array{Int64,2}:
 1  0
 0  1

Symbolic expressions

Automatically generate system of symbolic expressions for Category.

In [9]:
@syntax CategoryExprs(ObExpr, HomExpr) Category begin
    # Empty, so no simplification logic!

A, B, C, D = Ob(CategoryExprs, :A, :B, :C, :D)
f, g, h = Hom(:f, A, B), Hom(:g, B, C), Hom(:h, C, D)
$f : A \to B$
In [10]:
$f \cdot g : A \to C$
In [11]:
$\left(f \cdot g\right) \cdot h : A \to D$

Symbolic expressions

Expression types

In [12]:
[ expr => typeof(expr) for expr in [A, f, id(A), compose(f,g) ] ]
4-element Array{Pair{A,DataType} where A,1}:
            A => Main.CategoryExprs.Ob{:generator} 
            f => Main.CategoryExprs.Hom{:generator}
        id(A) => Main.CategoryExprs.Hom{:id}       
 compose(f,g) => Main.CategoryExprs.Hom{:compose}  

Unicode operators

In [13]:
f⋅g == compose(f,g)
In [14]:
g∘f == compose(f,g)

Simplification of symbolic expressions

By default, no simplification or normalization is performed.

In [15]:
(f⋅g)h == f⋅(g⋅h)

But we can define symbolic expressions that normalize at construction.

In [16]:
@syntax NormalizingCategoryExprs(ObExpr, HomExpr) Category begin
    compose(f::Hom, g::Hom) = associate_unit(new(f,g), id)

A, B, C, D = Ob(NormalizingCategoryExprs, :A, :B, :C, :D)
f, g, h = Hom(:f, A, B), Hom(:g, B, C), Hom(:h, C, D);

Note: Rewriting function associate_unit is built-in. You can also define your own.

Simplification of symbolic expressions

Associativity and unitality

In [17]:
(f⋅g)h == f⋅(g⋅h)
In [18]:
id(A)f == f && f⋅id(B) == f

Unbiased representation

In [19]:
3-element Array{Main.NormalizingCategoryExprs.Hom{:generator},1}:

Math interlude: generalized algebraic theories (GATs)

Signatures are based on GATs (Cartmell 1978; Cartmell 1986)

  • GATs = algebraic theories + simple dependent types
  • Same expressivity as esentially algebraic theories (Freyd 1972)
  • Minimal logical system for axiomatizing categorical doctrines

Caveat: Catlab only supports signature of theory; axioms for the future.

Wanted: Math and data structures for treating GATs, or doctrines, as data.

Wiring diagrams

Wiring diagrams, also known as

  • string diagrams in physics and CT
  • port graphs in computer science,

are a graphical syntax for morphisms in symmetric monoidal categories
(Spivak 2013; Rupel, Spivak 2013; Yau 2018).

$\left(f \otimes g\right) \cdot \sigma_{B,B} \cdot h \cdot \sigma_{C,C} : A \otimes A \to C \otimes C$
h g f

Two kinds of wiring diagrams

  1. Combinatorial: Wiring diagrams as data structure, no geometry (Catlab.WiringDiagrams)
  2. Topological: Wiring diagrams as visualization, embedded in the plane (Catlab.Graphics)

By default, "wiring diagram" has the first meaning.

Another distinction: Catlab has extensive support for directed wiring diagrams.
Undirected wiring diagrams are planned for the future.

Wiring diagrams as a data structure

Wiring diagrams are built on top of digraphs from LightGraphs.jl:

  • Each box corresponds to a vertex
  • Two special vertices for inputs/outputs of outer box
  • Arbitrary metadata on boxes, ports, and wires

Two interfaces for wiring diagrams:

  1. Imperative: Mutate wiring diagram by adding/removing boxes and wires
  2. Categorical: Construct new diagrams from old by composing, tensoring, etc

Imperative interface for wiring diagrams

Manually constructing the wiring diagram visualized above:

In [22]:
using Catlab.WiringDiagrams

d = WiringDiagram([:A,:A], [:C,:C])
fv = add_box!(d, Box(:f, [:A], [:B]))
gv = add_box!(d, Box(:g, [:A], [:B]))
hv = add_box!(d, Box(:h, [:B,:B], [:C,:C]))
add_wires!(d, [ (input_id(d),1) => (fv,1), (input_id(d),2) => (gv,1),
                (fv,1) => (hv,2), (gv,1) => (hv,1),
                (hv,1) => (output_id(d),2), (hv,2) => (output_id(d),1) ])

WiringDiagram([:A,:A], [:C,:C], 
[ 1 => {inputs},
  2 => {outputs},
  3 => Box(:f, [:A], [:B]),
  4 => Box(:g, [:A], [:B]),
  5 => Box(:h, [:B,:B], [:C,:C]) ],
[ Wire((1,1) => (3,1)),
  Wire((1,2) => (4,1)),
  Wire((3,1) => (5,2)),
  Wire((4,1) => (5,1)),
  Wire((5,1) => (2,2)),
  Wire((5,2) => (2,1)) ])

Wiring diagrams as a monoidal category

Making the same diagram, compositionally:

In [23]:
A, B, C = Ports([:A]), Ports([:B]), Ports([:C])
f = singleton_diagram(Box(:f, A, B))
g = singleton_diagram(Box(:g, A, B))
h = singleton_diagram(Box(:h, B⊗B, C⊗C))

(f⊗g)braid(B,B)h⋅braid(C,C) == d
  • All wiring diagrams are an instance of SymmetricMonoidalCategory
  • Extra structure by parametrizing the base type, e.g., WiringDiagram{CartesianCategory.Hom}

Wiring diagrams as an operad

The fundamental operation is substitution of diagrams for boxes (substitute), aka operadic composition (ocompose).

Example: Implementing compose for wiring diagrams

function compose(f::WiringDiagram, g::WiringDiagram)
    d = WiringDiagram(dom(f), codom(g))
    fv, gv = add_box!(d, f), add_box!(d, g)
    add_wires!(d, ((input_id(d),i) => (fv,i) for i in eachindex(dom(f))))
    add_wires!(d, ((fv,i) => (gv,i) for i in eachindex(codom(f))))
    add_wires!(d, ((gv,i) => (output_id(d),i) for i in eachindex(codom(g))))
    substitute(d, [fv, gv])

Inverse of substitution is encapsulation of subdiagrams (encapsulate).

Wiring diagrams and morphism expressions

Morphism expressions $\to$ wiring diagrams:

  • Easy
  • Uniquely defined
  • One-liner using generic functor from expressions to another instance

Wiring diagrams $\to$ morphism expressions:

Caveat: Need to make math out of this algorithm!

Normalization by round-tripping?

Decomposition algorithm maximizes parallelism (prefers $\circ$ over $\otimes$).

In [24]:
roundtrip(expr) = to_hom_expr(FreeCartesianCategory, to_wiring_diagram(expr))

A, B, C = Ob(FreeCartesianCategory, :A, :B, :C)
f, g, h, k = Hom(:f, A, B), Hom(:g, B, C), Hom(:h, A, B), Hom(:k, B, C)

expr = compose(otimes(f,h), otimes(g,k))
$\left(f \otimes h\right) \cdot \left(g \otimes k\right) : A \otimes A \to C \otimes C$
In [25]:
expr2 = roundtrip(expr)
$\left(f \cdot g\right) \otimes \left(h \cdot k\right) : A \otimes A \to C \otimes C$
In [26]:
roundtrip(expr2) == expr2

Normalization by round-tripping?

Implicitly uses coherence axioms for copying, deleting, etc.

In [27]:
$\Delta_{A \otimes B} : A \otimes B \to A \otimes B \otimes A \otimes B$
In [28]:
$\left(\Delta_{A} \otimes \Delta_{B}\right) \cdot \left(\mathrm{id}_{A} \otimes \sigma_{A,B} \otimes \mathrm{id}_{B}\right) : A \otimes B \to A \otimes B \otimes A \otimes B$
In [29]:
$\lozenge_{A \otimes B} : A \otimes B \to I$
In [30]:
$\lozenge_{A} \otimes \lozenge_{B} : A \otimes B \to I$

Visualizing wiring diagrams

Wiring diagrams in Graphviz

In [31]:
using Catlab.Graphics

to_graphviz(d, orientation=LeftToRight)
G n3 f n1p1:e->n3:w n4 g n1p2:e->n4:w n5 h n3:e->n5:w n4:e->n5:w n5:e->n2p1:w n5:e->n2p2:w
In [32]:
to_graphviz(d, orientation=TopToBottom, labels=true, label_attr=:taillabel)
G n3 f n1p1:s->n3:n A n4 g n1p2:s->n4:n A n5 h n3:s->n5:n B n4:s->n5:n B n5:s->n2p1:n C n5:s->n2p2:n C

Wiring diagrams in Compose.jl/TikZ

In [33]:
using Compose: fill, stroke

to_composejl(FreeSymmetricMonoidalCategory, d,
h g f
In [34]:
to_tikz(FreeSymmetricMonoidalCategory, d,
        labels=true, labels_pos=0.25, arrowtip="Latex", arrowtip_pos=1.0)

Serializing wiring diagrams

Read and write wiring diagrams as:

Warning: No standards in this area, so interoperability is limited.

Generating and parsing Julia code

Catlab can transform Julia programs to/from other representations of morphisms.

Convenient to specify available functions by presenting a cartesian monoidal category.

In [35]:
@present TrigFunctions(FreeCartesianCategory) begin
    R::Ob  # Real numbers

generator(TrigFunctions, :+)
$+ : R \otimes R \to R$

Generating Julia programs from morphism expressions

Functorial compilation of morphism expression into Julia function:

In [36]:
gen(name::Symbol) = generator(TrigFunctions, name)

expr = compose(mcopy(gen(:R)), otimes(gen(:sin),gen(:cos)), gen(:*))
$\Delta_{R} \cdot \left(\mathrm{sin} \otimes \mathrm{cos}\right) \cdot * : R \to R$
In [37]:
using Catlab.Programs

compile_expr(expr, name=:my_trig_polynomial, args=[:x])
:(function my_trig_polynomial(x)
      v1 = sin(x)
      v2 = cos(x)
      v3 = v1 * v2
      return v3

Parsing wiring diagrams from Julia code

Conversely, parse wiring diagrams by evaluating code in fragment of Julia language.

In [38]:
parsed = @parse_wiring_diagram TrigFunctions (x::R) begin
    y1, y2 = sin(x), cos(x)
    return y1 * y2

to_graphviz(parsed, orientation=LeftToRight,
            node_attrs=Dict(:fontname => "Courier"))
G n3 sin n1p1:e->n3:w n4 cos n1p1:e->n4:w n5 * n3:e->n5:w n4:e->n5:w n5:e->n2p1:w
In [39]:
parsed == @parse_wiring_diagram(TrigFunctions, (x::R) -> sin(x) * cos(x))

Application: semantics of data science code

Project at IBM Research AI and Stanford (Patterson, Baldini, Mojsilovic, Varshney 2018):

$\xrightarrow{\text{analyze}}$ $\xrightarrow{\text{enrich}}$

Application: compositional assembly planning

Prototype project at Siemens (Master, Patterson, Yousfi, Canedo 2019):

  • Planning and scheduling for automated assembly
  • Toy domain: simulated LEGO assembly in Minecraft

Application: scientific model augmentation

Ongoing project at Georgia Tech Research Institute (Halter, Herlihy, Fairbanks 2019):

  • Model augmentation: transform existing scientific models to create new models
  • Current focus on epidemiology (SIR, SIRS, etc): agent-based models and ODEs
  • Code available as SemanticModels.jl

Future directions

Many! I'll just mention a few areas.

Computer algebra

  • Declarative, rule-based rewriting for
    • symbolic expressions
    • wiring diagrams
  • Solving word problems
  • Calculating hom-sets
  • GATs and beyond
    • Data structures for, and operations on, GATs
    • Extensions for unbiased operations, such as products and permutations

Future directions

"Standard library" for applied category theory



epatters/Catlab.jl on GitHub

#catlab on the Julia Slack

Getting involved

I'm looking for both users and contributors. There is much to do!
Please contact me if interested.