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Introduction

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Algebraic Primitives for Sparse Data Structures in Python

algebrax treats Python's native dict as a first-class sparse algebraic object, unifying linear algebra, graph algorithms, formal language theory, signal transforms, and information metrics under a single polymorphic framework.

Quickstart

Install algebrax using your favorite package manager:

# Using uv (recommended)
uv add algebrax

# Using pip
pip install algebrax

Core Philosophy

  • ⚡ Zero Heavy Dependencies: Built entirely with pure Python, requiring no heavy external libraries or C++ compilation. Includes native conversion between sparse dict representations and dense multidimensional arrays.
  • 🔄 Polymorphic Semiring Computing: By changing the algebraic semiring \((\oplus, \otimes)\), the exact same matrix algorithms compute standard linear algebra, tropical shortest path latencies, or symbolic rule provenance.
  • 🌌 Sparse Multidimensional Tensors: Arbitrary nested mappings behave as infinite-dimensional sparse tensors, tries, and lattices with custom key operators.

Overview

Below is a 10-line demonstration showing how swapping the semiring parameter in matrix.dot changes matrix multiplication from Standard Linear Algebra to Tropical Shortest Path and Symbolic Provenance Tracking:

import algebrax as ax

# 1. Define a Sparse Graph Adjacency / Distance Matrix
graph = {
    0: {1: 2.0, 2: 10.0},
    1: {2: 3.0},
}

# Standard Linear Matrix Multiplication (+, *)
linear_mult = ax.matrix.dot(graph, graph, semiring=ax.semiring.StandardSemiring())
print("Linear Multiplication (0->2):", linear_mult[0][2])
# Output: 30.0 (path combination weight)

# Tropical Shortest Path (min, +)
shortest_path = ax.matrix.dot(graph, graph, semiring=ax.semiring.TropicalSemiring())
print("Shortest Path Cost (0->1->2):", shortest_path[0][2])
# Output: 5.0 (min(2 + 3, 10 + inf))

# Symbolic Provenance Rule Tracking
provenance_graph = {
    0: {1: {("rule_A",): 1}, 2: {("rule_C",): 1}},
    1: {2: {("rule_B",): 1}},
}
provenance_mult = ax.matrix.dot(provenance_graph, provenance_graph, semiring=ax.semiring.ProvenanceSemiring())
print("Symbolic Derivation Polynomial:", provenance_mult[0][2])
# Output: {('rule_A', 'rule_B'): 1}

Next Steps

Explore the documentation sections:

  • 💡 Core Concepts: Learn the mathematical foundations of Monoids, Groups, Semirings, and Lattices.
  • ⚖️ Library Comparison: Feature matrix and trade-off analysis comparing AlgebraX vs SciPy, NumPy, Pandas, NetworkX, and SymPy.
  • 📖 Tutorials: Step-by-step guides for 24 semirings, matrix operations, decompositions, tries, signal transforms, and graph algorithms.
  • 🍳 Use Cases & Recipes: Executable real-world scripts, Jupyter notebooks, and the interactive DearPyGui laboratory.
  • 📜 Enhancement Proposals (EPs): Architectural designs, roadmap, and technical specs.