Drawing wiring diagrams in Compose.jl

Catlab can draw wiring diagrams using the Julia package Compose.jl.

For best results, it is recommended to load the packages Convex.j and SCS.jl. When available they are used to optimize the layout of the outer ports.

using Catlab.WiringDiagrams, Catlab.Graphics

import Convex, SCS

Examples

Symmetric monoidal category

using Catlab.Theories

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

To start, here are a few very simple examples.

to_composejl(f)

to_composejl(f⋅g)

to_composejl(f⊗g)

Here is a more complex example, involving generators with compound domains and codomains.

h, k = Hom(:h, C, D),  Hom(:k, D, C)
m, n = Hom(:m, B⊗A, A⊗B), Hom(:n, D⊗C, C⊗D)
q = Hom(:l, A⊗B⊗C⊗D, D⊗C⊗B⊗A)

to_composejl((f⊗g⊗h⊗k)⋅(m⊗n)⋅q⋅(n⊗m)⋅(h⊗k⊗f⊗g))

Identities and braidings appear as wires.

to_composejl(id(A))

to_composejl(braid(A,B))

to_composejl(braid(A,B) ⋅ (g⊗f) ⋅ braid(A,B))

The isomorphism $A \otimes B \otimes C \to C \otimes B \otimes A$ induced by the permutation $(3\ 2\ 1)$ is a composite of braidings and identities.

σ = (braid(A,B) ⊗ id(C)) ⋅ (id(B) ⊗ braid(A,C) ⋅ (braid(B,C) ⊗ id(A)))

to_composejl(σ)

By default, anchor points are added along identity and braiding wires to reproduce the expression structure in the layout. The anchors can be disabled to get a more "unbiased" layout.

to_composejl(σ, anchor_wires=false)

Biproduct category

A, B, C = Ob(FreeBiproductCategory, :A, :B, :C)
f = Hom(:f, A, B)

to_composejl(mcopy(A))

to_composejl(delete(A))

to_composejl(mcopy(A)⋅(f⊗f)⋅mmerge(B))

to_composejl(mcopy(A⊗B), orientation=TopToBottom)

to_composejl(mcopy(A⊗B⊗C), orientation=TopToBottom)

Compact closed category

The unit and co-unit of a compact closed category appear as caps and cups.

A, B = Ob(FreeCompactClosedCategory, :A, :B)

to_composejl(dunit(A))

to_composejl(dcounit(A))

In a self-dual compact closed category, such as a bicategory of relations, every morphism $f: A \to B$ has a transpose $f^\dagger: B \to A$ given by bending wires:

A, B = Ob(FreeBicategoryRelations, :A, :B)
f = Hom(:f, A, B)

to_composejl((dunit(A) ⊗ id(B)) ⋅ (id(A) ⊗ f ⊗ id(B)) ⋅ (id(A) ⊗ dcounit(B)))

Abelian bicategory of relations

In an abelian bicategory of relations, such as the category of linear relations, the duplication morphisms $\Delta_X: X \to X \oplus X$ and addition morphisms $\blacktriangledown_X: X \oplus X \to X$ belong to a bimonoid. Among other things, this means that the following two morphisms are equal.

X = Ob(FreeAbelianBicategoryRelations, :X)

to_composejl(plus(X) ⋅ mcopy(X))

to_composejl((mcopy(X)⊕mcopy(X)) ⋅ (id(X)⊕swap(X,X)⊕id(X)) ⋅ (plus(X)⊕plus(X)))

Custom styles

The visual appearance of wiring diagrams can be customized by passing Compose properties.

using Compose: fill, stroke

A, B, = Ob(FreeSymmetricMonoidalCategory, :A, :B)
f, g = Hom(:f, A, B), Hom(:g, B, A)

to_composejl(f⋅g, props=Dict(
  :box => [fill("lavender"), stroke("black")],
))

X = Ob(FreeAbelianBicategoryRelations, :X)

to_composejl(plus(X) ⋅ mcopy(X), props=Dict(
  :junction => [fill("red"), stroke("black")],
  :variant_junction => [fill("blue"), stroke("black")],
))

The background color can also be changed.

to_composejl(f⋅g, background_color="lightgray", props=Dict(
  :box => [fill("white"), stroke("black")],
))

By default, the boxes are rectangular (:rectangle). Other available shapes include circles (:circle) and ellipses (:ellipse).

to_composejl(f⋅g, default_box_shape=:circle)

Output formats

The function to_composejl returns a ComposePicture object, which contains a Compose.jl context as well as a recommended width and height. When displayed interactively, this object is rendered using Compose's SVG backend.

Any backend can be used by calling Compose's draw function. The SVG and PGF (LaTeX) backends are always available. To use the PNG or PDF backends, the extra packages Cairo.jl and Fontconfig.jl must be installed.

For example, here is how to use the PGF backend.

using Compose: draw, PGF

pic = to_composejl(f⋅g, rounded_boxes=false)
pgf = sprint() do io
  pgf_backend = PGF(io, pic.width, pic.height,
    false, # emit_on_finish
    true,  # only_tikz
    texfonts=true)
  draw(pgf_backend, pic.context)
end
println(pgf)

\begin{tikzpicture}[x=1mm,y=-1mm]
\definecolor{mycolor000000}{rgb}{0,0,0}
\begin{scope}
\path [fill=mycolor000000,draw=mycolor000000] (32,8) .. controls (36,8) and (36,8) .. (40,8);
\end{scope}
\begin{scope}
\path [fill=mycolor000000,draw=mycolor000000] (16,8) .. controls (20,8) and (20,8) .. (24,8);
\end{scope}
\begin{scope}
\path [fill=mycolor000000,draw=mycolor000000] (0,8) .. controls (4,8) and (4,8) .. (8,8);
\end{scope}
\begin{scope}
\path [fill=mycolor000000,fill opacity=0,draw=mycolor000000] (24,4) rectangle +(8,8);
\end{scope}
\begin{scope}
\draw (28,8) node [text=mycolor000000,rotate around={-0: (0,0)},inner sep=0.0]{\fontsize{12mm}{14.4mm}\selectfont $\text{g}$};
\end{scope}
\begin{scope}
\path [fill=mycolor000000,fill opacity=0,draw=mycolor000000] (8,4) rectangle +(8,8);
\end{scope}
\begin{scope}
\draw (12,8) node [text=mycolor000000,rotate around={-0: (0,0)},inner sep=0.0]{\fontsize{12mm}{14.4mm}\selectfont $\text{f}$};
\end{scope}
\end{tikzpicture}