E-Graph Geometric Algebra (EGGA)

Symbolic Geometric Algebra with E-Graphs

Things you can do with this library

• Simplify expressions
• Prove equalities
• Solve for variables

Things that are supported

• Any signature
• Arbitrary number of basis vectors
• Symplectic Geometric Algebra (aka Weyl Algebras)
• Derivatives
• Add your own expression types and rules (with egglog)

Based on the Python bindings for egglog

Setup

Supports Python 3.8 and higher.

pip install egga

Usage

The first step is to create a GeometricAlgebra object with a given signature. You can then use its basis vectors as well as functions exposed by it. Use the utility methods provided to do things like simplification and equation solving. In some cases you might need to interface with egglog directly. Below are some examples for common use-cases.

Simplification

from egga.geometric_algebra import GeometricAlgebra
from egga.utils import simplify

ga = GeometricAlgebra(signature=[1.0, 1.0])
e_0, e_1 = ga.basis_vectors
e_01 = e_0 * e_1

# Build an expression to simplify
expr = e_01 * e_0 * ~e_01

# Prints Simplified: -e("0")
print("Simplified:", simplify(ga, expr))

Equation solving

from egglog import union

from egga.geometric_algebra import GeometricAlgebra
from egga.utils import simplify

# Pass eq_solve=True to enable the equation solving rules.
# Add a cost to variable to it gets rewritten to something else.
ga = GeometricAlgebra(signature=[1.0, 1.0], eq_solve=True, costs={"variable": 1_000})

e_0, e_1 = ga.basis_vectors
e_01 = e_0 * e_1

# Solve e_01 * x * ~e_01 = e_0 for x
x = ga.expr_cls.variable("x")
lhs = e_01 * x * ~e_01
rhs = -e_0

# Make LHS equal to RHS
ga.egraph.register(union(lhs).with_(rhs))

assert str(simplify(ga, x)) == str(ga.expr_cls.e("0"))

Equality check

from egga.geometric_algebra import GeometricAlgebra
from egga.utils import check_equality

ga = GeometricAlgebra(signature=[1.0, 1.0])
e_0, e_1 = ga.basis_vectors
e_01 = e_0 * e_1

# Build an lhs to check for equality to an rhs
lhs = e_01 * e_01
rhs = ga.expr_cls.scalar_literal(-1.0)

assert check_equality(ga, lhs, rhs)

The /examples as well as the /tests directories contain more examples.

List of expressions

Operators

Functions

Unsupported but exists, might or might not work

Caveats

• Egraphs are bad with associativity (combined with commutativity?) so things can blow up
• Most operations aren't "fully" implemented (eg. pow only supports powers of two right now)

Contributing

Code contributions as well as suggestions and comments about things that don't work yet are appreciated. You can reach me by email at tora@warlock.ai or in the Bivector Discord.