Discopy

DisCoPy: String Diagrams in Python

"String diagrams are the syntax, functors are the semantics."

bmorphism Contributions

"all is bidirectional" — @bmorphism, Play/Coplay gist

Active Inference Implementation: DisCoPy provides the foundation for implementing Active Inference in String Diagrams (Tull, Kleiner, Smithe). The paper's core insight — that perception and action form a bidirectional loop — maps directly to DisCoPy's composition operators:

• Sequential (>>) → temporal flow (action → perception → action)
• Parallel (@) → concurrent sensory channels
• Dagger ([::-1]) → time reversal (perception as action's adjoint)

Categorical Cybernetics: DisCoPy's parametrised optics implement the cybernetic lens pattern from Towards Foundations of Categorical Cybernetics, enabling the Play/Coplay duality bmorphism references.

String Diagram Coloring: Gay.jl colors can be applied to DisCoPy diagrams for deterministic visualization — each wire type gets a consistent color from the splittable RNG.

Overview

DisCoPy is a Python library for computing with string diagrams - the graphical language of monoidal categories. It provides:

1. Categorical Framework: Ty, Ob, Box, Arrow, Diagram, Category
2. Operads: CFG as free operads, operad algebras, colored operads
3. Quantum Computing: Circuits, gates, ZX-calculus, pytket/qiskit integration
4. QNLP: Pregroup grammars, DisCoCat, ansätze for quantum NLP
5. Tensor Networks: NumPy/JAX/PyTorch backends, tensornetwork contraction
6. Visualization: Matplotlib/TikZ drawing with color customization

Core Architecture (DeepWiki 2025-12-22)

Class Hierarchy

Category
├── ob: Ob (objects/systems)
└── ar: Arrow (morphisms/processes)

Ob → Ty (monoidal: tuple of objects, tensor = concatenation)

Arrow → Diagram (monoidal: boxes + offsets, parallel composition)
     └── Box (atomic operations with dom/cod)

Composition Operators

from discopy.monoidal import Ty, Box, Diagram

x, y, z = Ty('x'), Ty('y'), Ty('z')
f = Box('f', x, y)
g = Box('g', y, z)

# Sequential: x → y → z
sequential = f >> g

# Parallel: x ⊗ y → y ⊗ z
parallel = f @ g

# Both: (f ⊗ id) >> (id ⊗ g)
mixed = f @ Diagram.id(y) >> Diagram.id(y) @ g

Operads (DeepWiki 2025-12-22)

DisCoPy implements operads via discopy.grammar.cfg:

Operad Structure

• Colors: Ty objects (types of operations)
• Operations: Tree objects (composable rules)
• Algebras: Functor from free operad to target operad

from discopy.grammar.cfg import Ty, Rule, Word, Tree

# Define colors (types)
n, v, s = Ty('N'), Ty('V'), Ty('S')
vp, np = Ty('VP'), Ty('NP')

# Define operations (rules)
Caesar = Word('Caesar', n)
crossed = Word('crossed', v)
VP = Rule(n @ v, vp)
NP = Rule(Ty('D') @ n, np)
S = Rule(vp @ np, s)

# Build tree (operadic composition)
sentence = S(VP(Caesar, crossed), NP(Word('the', Ty('D')), Word('Rubicon', n)))
# Axioms of multicategories (operads) hold on the nose

Colored Operads

Types (Ty) act as colors - operations can have different input/output colors:

x, y = Ty('x'), Ty('y')
f = Rule(x @ x, x, name='f')   # x ⊗ x → x
g = Rule(x @ y, x, name='g')   # x ⊗ y → x  
h = Rule(y @ x, x, name='h')   # y ⊗ x → x

# Operadic composition
assert f(g, h) == Tree(f, *[g, h])

Color Configuration (DeepWiki 2025-12-22)

Default Color Palette

from discopy.config import COLORS

COLORS = {
    "white": "#ffffff",
    "red": "#e8a5a5",
    "green": "#d8f8d8", 
    "blue": "#776ff3",
    "yellow": "#f7f700",
    "black": "#000000"
}

ZX-Calculus Colors

from discopy.quantum.zx import Z, X, Y

# Z spiders: GREEN
Z(1, 1, phase=0.5).color  # "green"

# X spiders: RED  
X(1, 1, phase=0.25).color  # "red"

# Y spiders: BLUE
Y(1, 1).color  # "blue"

Custom Box Colors

from discopy.monoidal import Ty, Box

x = Ty('x')
blue_box = Box('f', x, x, color="blue")
blue_box.draw()

# Spider with custom color
from discopy.frobenius import Spider
spider = Spider(2, 3, x)
spider.color = "red"
spider.draw_as_spider = True

Drawing API

diagram.draw(
    figsize=(8, 6),
    color="blue",           # Default box color
    draw_as_nodes=True,     # Draw boxes as nodes
    wire_labels=True,       # Show type labels on wires
    draw_box_labels=True,   # Show box names
    path="output.png",      # Save to file
    to_tikz=True           # Output TikZ code
)

Quantum Computing (DeepWiki 2025-12-22)

Quantum Circuits

from discopy.quantum import qubit, H, X, CX, Ket, Bra, Measure

# Bell state preparation
bell = Ket(0, 0) >> H @ qubit >> CX

# Evaluation
state_vector = bell.eval()  # Returns Tensor[complex]

# Measurement probability
experiment = Ket(0, 0) >> bell >> Bra(0, 0)
amplitude = experiment.eval().array
probability = abs(amplitude) ** 2

Gate Library

External Integration

# pytket integration
from discopy.quantum.tk import mockBackend

circuit = H @ qubit >> CX >> Measure() @ Measure()
tk_circuit = circuit.to_tk()  # Convert to pytket

# Run on backend
backend = mockBackend({(0, 1): 512, (1, 0): 512})
counts = circuit.eval(backend=backend, n_shots=1024)

# PennyLane integration
pennylane_qnode = circuit.to_pennylane()

ZX-Calculus

from discopy.quantum.zx import Z, X, circuit2zx

# Convert circuit to ZX diagram
zx_diagram = circuit2zx(circuit)

# Z spider (green) with phase
z_spider = Z(2, 1, phase=0.5)

# X spider (red) with phase  
x_spider = X(1, 2, phase=0.25)

# PyZX integration for optimization
pyzx_graph = zx_diagram.to_pyzx()
# Apply PyZX simplification algorithms
simplified = pyzx_graph.simplify()

QNLP (DeepWiki 2025-12-22)

Pregroup Grammar

from discopy.grammar.pregroup import Ty, Word, Cup, Diagram

s, n = Ty('s'), Ty('n')

Alice = Word('Alice', n)
loves = Word('loves', n.r @ s @ n.l)
Bob = Word('Bob', n)

# Parse sentence
sentence = Alice @ loves @ Bob >> Cup(n, n.r) @ s @ Cup(n.l, n)
sentence.draw()

DisCoCat: Diagrams to Quantum Circuits

from discopy.quantum import circuit, qubit, Ket, H, CX
from discopy.cat import Category

# Define semantic functor
F = circuit.Functor(
    ob={s: qubit ** 0, n: qubit ** 1},  # Type → Qubits
    ar={
        Alice: Ket(0),
        loves: sqrt(2) @ Ket(0, 0) >> H @ X >> CX,
        Bob: Ket(1)
    }
)
F.dom = Category(Ty, Diagram)

# Apply functor to get quantum circuit
quantum_sentence = F(sentence)
quantum_sentence.draw()

Ansätze

Parameterized quantum circuits for word meanings:

from discopy.quantum.ansatze import IQPAnsatz, Sim14Ansatz

# IQP ansatz for nouns
noun_ansatz = IQPAnsatz(n_qubits=2, n_layers=3)

# Sim14 ansatz for verbs
verb_ansatz = Sim14Ansatz(n_qubits=4, n_layers=2)

Tensor Networks (DeepWiki 2025-12-22)

Multi-Backend Support

from discopy.tensor import Tensor, backend
import jax.numpy
import torch

# Default: NumPy
assert isinstance(Tensor.id().array, np.ndarray)

# JAX backend
with backend('jax'):
    assert isinstance(Tensor.id().array, jax.numpy.ndarray)

# PyTorch backend
with backend('pytorch'):
    assert isinstance(Tensor.id().array, torch.Tensor)

Functor to Tensors

from discopy.tensor import Dim, Tensor, Functor
from discopy.cat import Category

# Define tensor functor
F = Functor(
    ob={n: Dim(2), s: Dim(1)},
    ar={
        Alice: [1, 0],
        loves: [[0, 1], [1, 0]],
        Bob: [0, 1]
    },
    cod=Category(Dim, Tensor)
)

result = F(sentence)

Contraction with tensornetwork

import tensornetwork as tn
from discopy.tensor import Box, Dim

vector = Box('v', Dim(1), Dim(2), [0, 1])
contracted = (vector >> vector[::-1]).eval(contractor=tn.contractors.auto)

Advanced Categories (DeepWiki 2025-12-22)

Hypergraph Categories

from discopy.hypergraph import Hypergraph

# Canonical form for diagram equality
hyp = diagram.to_hypergraph()
# Composition via pushouts

Frobenius Algebras and Spiders

from discopy.frobenius import Spider, Diagram

x = Ty('x')

# Special commutative Frobenius algebra
spider = Spider(n_legs_in=2, n_legs_out=3, typ=x)

# Unfuse to canonical primitives
primitives = diagram.unfuse()

Markov Categories

from discopy.markov import Copy, Merge, Discard, Diagram

x = Ty('x')

# Copy wire n times
copy = Diagram.copy(x, n=2)

# Merge n wires to one
merge = Diagram.merge(x, n=2)

# Discard wire
discard = Diagram.copy(x, n=0)  # Copy with n=0

Traced Categories

from discopy.traced import Diagram

# Feedback loop: output fed back to input
traced = diagram.trace()

GF(3) Triad Integration

DisCoPy as ERGODIC coordinator (trit 0):

three-match (-1) ⊗ discopy (0) ⊗ gay-mcp (+1) = 0 ✓  [Diagram Coloring]
sheaf-cohomology (-1) ⊗ discopy (0) ⊗ operad-compose (+1) = 0 ✓  [Operadic]
proofgeneral-narya (-1) ⊗ discopy (0) ⊗ rubato-composer (+1) = 0 ✓  [Music]
persistent-homology (-1) ⊗ discopy (0) ⊗ gay-mcp (+1) = 0 ✓  [Quantum]

Color ↔ Gay.jl Integration

from discopy.monoidal import Box, Ty

# Use Gay.jl deterministic colors for boxes
def gay_colored_box(name, dom, cod, seed, index):
    """Create box with deterministic color from Gay.jl"""
    # Gay.jl: golden angle dispersion
    hue = ((seed * 0x9E3779B97F4A7C15 + index) >> 16) % 360
    hex_color = hsl_to_hex(hue, 0.7, 0.55)
    return Box(name, dom, cod, color=hex_color)

# Example: colored diagram
x = Ty('x')
boxes = [gay_colored_box(f'f{i}', x, x, 0x42D, i) for i in range(5)]

DeepWiki Integration Notes

DeepWiki URL: https://deepwiki.com/discopy/discopy

Version: 2.0.0 Trit: 0 (ERGODIC - coordinates categorical computation) GF(3): Substitutes for other ERGODIC skills in triads Qualified: 2025-12-22 (8 DeepWiki interactions)

End-of-Skill Interface

Commands

just discopy-demo           # Run DisCoPy demonstration
just discopy-quantum        # Quantum circuit examples
just discopy-qnlp          # QNLP parsing examples
just discopy-zx            # ZX-calculus optimization
just discopy-tensor        # Tensor network contraction
just discopy-operad        # Operad composition examples

Related Skills

• operad-compose: Operadic composition patterns (+1)
• gay-mcp: Deterministic colors for diagram elements (+1)
• proofgeneral-narya: Type-theoretic diagram verification (-1)
• rubato-composer: Musical diagram applications (+1)
• three-match: 3-coloring for diagram validation (-1)

r2con Speaker Resources

Color and diagram repositories from r2con speakers relevant to DisCoPy: