Algebraic Rewriting

Algebraic Rewriting

Overview

AlgebraicRewriting.jl is a Julia library for performing category-theoretic rewrites over C-Sets and other Catlab.jl data structures.

Rewriting Approaches

Core Concepts

Rewrite Rules

A rewrite rule consists of:

• L (left) - Pattern to match
• K (interface) - What to preserve
• R (right) - Replacement pattern

using AlgebraicRewriting

# Define a rule: merge two vertices
L = @acset Graph begin V=2; E=1; src=[1]; tgt=[2] end
K = @acset Graph begin V=1 end
R = @acset Graph begin V=1 end

rule = Rule(L, K, R)

Apply Rewriting

G = @acset Graph begin
    V = 4
    E = 3
    src = [1, 2, 3]
    tgt = [2, 3, 4]
end

# Find matches and rewrite
matches = homomorphisms(L, G)
G′ = rewrite(rule, G, matches[1])

Double Pushout (DPO)

    L ←─ K ─→ R
    ↓    ↓    ↓
    G ←─ D ─→ H

The context D ensures no "dangling edges" after deletion.

Sesqui-Pushout (SqPO)

Supports cloning via the final pullback complement:

# Clone a vertex
L = @acset Graph begin V=1 end
K = @acset Graph begin V=1 end
R = @acset Graph begin V=2 end

clone_rule = Rule(L, K, R; type=:SqPO)

Gay.jl Integration

# sRGB boundary learning with rewriting seed
gay_seed!(0xabfca37b6b4bc699)

# Forward mode autodiff
∂params = Enzyme.gradient(Forward, loss, params, seed)

Documentation

• Full Documentation
• Brown 2022 - Theoretical foundation

Repository

• Source: plurigrid/​AlgebraicRewriting.jl (fork of AlgebraicJulia)
• Seed: 0xabfca37b6b4bc699
• Index: 496/​1055
• Color: #c25d0b

GF(3) Triad

algebraic-rewriting (-1) ⊗ acsets-hatchery (0) ⊗ gay-monte-carlo (+1) = 0 ✓

Related Skills

• acsets-hatchery - ACSet data structures
• topos-adhesive-rewriting - Adhesive categories
• dpo-rewriting - Graph transformation
• world-a - AlgebraicJulia ecosystem

SDF Interleaving

This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):

Primary Chapter: 3. Variations on an Arithmetic Theme

Concepts: generic arithmetic, coercion, symbolic, numeric

GF(3) Balanced Triad

algebraic-rewriting (+) + SDF.Ch3 (○) + [balancer] (−) = 0

Skill Trit: 1 (PLUS - generation)

Secondary Chapters

• Ch4: Pattern Matching
• Ch10: Adventure Game Example

Connection Pattern

Generic arithmetic crosses type boundaries. This skill handles heterogeneous data.