Discopy Operads

DiscoPy Operads Skill

Repo Color: #64e3ec | Seed: 0x128b6ef4564e3a00 | Index: 224/1055

DisCoPy: Python toolkit for computing with string diagrams, monoidal categories, and operads.

Quick Reference

from discopy.monoidal import Ty, Box, Id, Diagram
from discopy.grammar.cfg import Tree, Rule, Word, Operad, Algebra
from discopy import symmetric, braided, compact, frobenius, hypergraph

String Diagram Syntax

# Types are objects in monoidal categories
x, y, z = Ty('x'), Ty('y'), Ty('z')
unit = Ty()  # monoidal unit

# Tensor product (horizontal composition)
xy = x @ y  # x ⊗ y

# Boxes are morphisms
f = Box('f', x, y)           # f: x → y
g = Box('g', y, z)           # g: y → z

# Sequential composition (vertical)
fg = f >> g                   # g ∘ f: x → z

# Parallel composition (tensor of morphisms)
f_par_g = f @ g              # f ⊗ g: x ⊗ y → y ⊗ z

# Identity morphisms
idx = Id(x)                  # id_x: x → x

# Interchange law
d = Id(x) @ g >> f @ Id(z)   # = (f @ g).interchange(0, 1)

Monoidal Category Operations

# Dagger (adjoint)
f_dag = f[::-1]              # f†: y → x

# Diagram slicing
d = f >> g >> h
first_two = d[:2]            # f >> g
last_box = d[2]              # h

# Interchange normalization
for step in (f @ g).normalize():
    print(step)              # yields normal form steps

normal = d.normal_form()     # boundary-connected normal form

Category Hierarchy

cat.Category
    └── monoidal.Category (planar diagrams)
        └── braided.Category (overcrossings)
            └── symmetric.Category (swaps)
                └── traced.Category (feedback loops)
                    └── compact.Category (cups/caps)
                        └── frobenius.Category (spiders)

Operad Composition

from discopy.grammar.cfg import Ty, Rule, Tree, Id, Operad

# Operads: multicategories with multi-input operations
x, y = Ty('x'), Ty('y')

# Rules are operad generators (atomic type codomain)
f = Rule(x @ x, x, name='f')  # f: x ⊗ x → x
g = Rule(x @ y, x, name='g')  # g: x ⊗ y → x
h = Rule(y @ x, x, name='h')  # h: y ⊗ x → x

# Tree construction via operadic composition
tree = f(g, h)               # plug g, h into f's inputs
assert tree == Tree(f, g, h)

# Axioms hold on the nose
assert Id(x)(f) == f == f(Id(x), Id(x))  # identity
left = f(Id(x), h)(g, Id(x), Id(x))
right = f(g, Id(x))(Id(x), Id(x), h)
assert f(g, h) == left == right          # associativity

# Nested diagram substitution (Patterson et al.)
diagram.substitute(i, other)  # replace box i with diagram other

Nested Operad Composition (Advanced)

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

# Multi-level nesting example
a, b, c = Ty('a'), Ty('b'), Ty('c')

# Arity-2 rules
add = Rule(a @ a, a, name='+')       # +: a ⊗ a → a
mul = Rule(a @ a, a, name='*')       # *: a ⊗ a → a
neg = Rule(a, a, name='-')           # -: a → a (unary)

# Deep composition: (a + b) * (-c)
# Corresponds to tree: mul(add(id_a, id_a), neg(id_a))
expr1 = mul(add, neg)                 # plug add and neg into mul's inputs
assert expr1.dom == a @ a @ a         # 3 leaves (a, a, a for left+, right*, and neg arg)

# Even deeper: ((a + b) * c) + (a * (b + c))
left_tree = mul(add, Id(a))           # (a + b) * c
right_tree = mul(Id(a), add)          # a * (b + c)  
full_expr = add(left_tree, right_tree)
print(f"Depth: {full_expr.depth}, Leaves: {len(list(full_expr.leaves))}")

# Operad morphism (algebra) evaluation
@Algebra.from_callable(a >> int)
def eval_int(rule: Rule, *args: int) -> int:
    if rule.name == '+': return args[0] + args[1]
    if rule.name == '*': return args[0] * args[1]
    if rule.name == '-': return -args[0]
    return args[0]

# Evaluate: (2 + 3) * (-4) = 5 * (-4) = -20
tree_with_leaves = mul(add(2, 3), neg(4))  # conceptual

CFG Example

n, d, v = Ty('N'), Ty('D'), Ty('V')
vp, np, s = Ty('VP'), Ty('NP'), Ty('S')

Caesar = Word('Caesar', n)
crossed = Word('crossed', v)
the, Rubicon = Word('the', d), Word('Rubicon', n)

VP = Rule(n @ v, vp)
NP = Rule(d @ n, np)
S = Rule(vp @ np, s)

sentence = S(VP(Caesar, crossed), NP(the, Rubicon))
# "Caesar crossed the Rubicon"

Hypergraph Categories

from discopy.hypergraph import Hypergraph, Spider

# Spiders: n-to-m operations with labeled nodes
spider = Spider(2, 3, label='x')  # 2 inputs, 3 outputs

# Wiring diagrams as cospans of hypergraphs
wires = (('a', 'b'), (('c',), ('d', 'e')), ('f',))

Color Integration via Gay.jl

# Initialize with repo seed
GAY_SEED = 0x128b6ef4564e3a00

def gay_color_box(box: Box, seed: int = GAY_SEED) -> str:
    """Generate deterministic color for diagram box."""
    h = hash((box.name, seed)) & 0xFFFFFFFF
    # SplitMix64 step
    h = ((h ^ (h >> 16)) * 0x85ebca6b) & 0xFFFFFFFF
    return f"#{h:06x}"[:7]

# Color diagram boxes
for box in diagram.boxes:
    color = gay_color_box(box)
    # Use in drawing.draw() with box_colors={box: color}

GF(3) Trit Conservation

def box_trit(box: Box) -> int:
    """Map box to balanced ternary trit."""
    return hash(box.name) % 3 - 1  # {-1, 0, +1}

def diagram_trit_sum(d: Diagram) -> int:
    """Diagrams conserve trit parity under composition."""
    return sum(box_trit(b) for b in d.boxes) % 3

GF(3) Integration with Verification

from discopy.monoidal import Box, Diagram, Ty
from typing import Dict, Tuple

# GF(3) = Z/3Z with balanced representation {-1, 0, +1}
class GF3Diagram:
    """Diagram with GF(3) trit annotations for conservation checking."""
    
    TRIT_NAMES = {-1: "MINUS", 0: "ZERO", 1: "PLUS"}
    
    def __init__(self, diagram: Diagram, trit_map: Dict[Box, int] = None):
        self.diagram = diagram
        self.trit_map = trit_map or {b: hash(b.name) % 3 - 1 for b in diagram.boxes}
    
    def total_trit(self) -> int:
        """Sum of trits mod 3, in balanced form."""
        s = sum(self.trit_map.values()) % 3
        return s if s <= 1 else s - 3
    
    def compose(self, other: 'GF3Diagram') -> 'GF3Diagram':
        """Compose diagrams, verify trit conservation."""
        new_diagram = self.diagram >> other.diagram
        new_trit_map = {**self.trit_map, **other.trit_map}
        result = GF3Diagram(new_diagram, new_trit_map)
        
        # Conservation check: sequential composition preserves total
        expected = (self.total_trit() + other.total_trit()) % 3
        expected = expected if expected <= 1 else expected - 3
        assert result.total_trit() == expected, "GF(3) conservation violated!"
        return result
    
    def verify_identity_neutral(self) -> bool:
        """Identity morphisms have trit 0 (neutral element)."""
        for box in self.diagram.boxes:
            if box.name.startswith('Id'):
                if self.trit_map.get(box, 0) != 0:
                    return False
        return True

# Usage
x, y = Ty('x'), Ty('y')
f = Box('f', x, y)  # trit = hash('f') % 3 - 1
g = Box('g', y, x)  # trit = hash('g') % 3 - 1

gf3_f = GF3Diagram(f)
gf3_g = GF3Diagram(g)
composed = gf3_f.compose(gf3_g)

print(f"f trit: {gf3_f.total_trit()}")
print(f"g trit: {gf3_g.total_trit()}")
print(f"f>>g trit: {composed.total_trit()}")
print(f"Conservation: {(gf3_f.total_trit() + gf3_g.total_trit()) % 3}")

Hyperlang Embedding

# Hyperlang: diagrams as executable specifications
from discopy.python import Function

# Interpret diagram as Python function
@Function.from_callable(x, y)
def f_impl(data):
    return transform(data)

# Functor maps syntax to semantics
F = Functor(
    ob={x: int, y: str},
    ar={f: f_impl},
    cod=Function
)

result = F(diagram)(input_data)

Quantum Circuit Embedding

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

# Quantum circuits as diagrams
circuit = Ket(0, 0) >> H @ qubit >> CX >> Measure() @ Measure()

# Interpret via tensor contraction
from discopy.quantum.circuit import Circuit
amplitude = circuit.eval()

Drawing Diagrams

from discopy.drawing import Equation

# Draw single diagram
diagram.draw(figsize=(8, 4))

# Draw equation
Equation(lhs, rhs).draw()

# Custom colors
diagram.draw(
    box_colors={f: '#64e3ec', g: '#150448'},
    wire_colors={x: '#ff0000'}
)

Recent Commits (Dec 2024)

• c456c37: Fix generic type handling in assert_isinstance
• 467c8c4: Operadic composition (#292) - nested diagram substitution
• 2cb5579: Fix Frobenius bubble

Links

• DiscoPy Docs
• GitHub
• Gay.jl Integration
• Patterson et al. - Wiring Diagrams (arXiv:2101.12046)

Patterson et al. "Wiring Diagrams as Morphisms of Operads" (2021)

Key concepts from the paper:

1. Wiring diagrams = morphisms in an operad of typed, directed wires
2. Operadic composition = nested substitution of boxes
3. Cospans of hypergraphs = the universal property for wiring

# Patterson's wiring diagram composition pattern
# (Box A with 2 outputs) composed with (Box B taking 2 inputs)

from discopy.hypergraph import Hypergraph, Spider

# Outer box: 2 inputs, 2 outputs
outer = Hypergraph(
    dom=['in1', 'in2'], 
    cod=['out1', 'out2'],
    boxes=[Spider(2, 2, 'process')]
)

# Inner box to substitute: 1 input, 1 output  
inner = Hypergraph(
    dom=['x'],
    cod=['y'],
    boxes=[Spider(1, 1, 'transform')]
)

# Operadic substitution: replace one port of outer with inner
# This is the key insight - wiring diagrams compose as operad morphisms

Citation:

@article{patterson2021wiring,
  title={Wiring diagrams as normal forms for computing in symmetric monoidal categories},
  author={Patterson, Evan and Baas, Amar and Hosgood, Timothy and Fairbanks, James},
  journal={arXiv:2101.12046},
  year={2021}
}

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