EP-0131: Algebraic Law Verification Engine¶

Abstract¶

The library provides 23 semiring implementations, each claiming to satisfy specific algebraic axioms (associativity, distributivity, identity, annihilation). Currently, these axioms are tested implicitly through application-level tests. This proposal introduces an Algebraic Law Verification Engine that systematically and automatically tests every axiom for every semiring type.

Motivation¶

Popper (Falsifiability): "Assume the happy path is a lie." The library claims algebraic correctness but does not have a systematic mechanism to prove it across all semiring types with random inputs.

Specification¶

1. verify_semiring_laws(semiring, samples) Function¶

def verify_semiring_laws(
    semiring: Semiring[T],
    samples: list[T],
) -> dict[str, bool]:
    """
    Test all semiring axioms with the given sample elements.

    Returns a dict mapping axiom name to pass/fail:
      - 'add_associativity'
      - 'add_commutativity'
      - 'add_identity'
      - 'mul_associativity'
      - 'mul_identity'
      - 'left_distributivity'
      - 'right_distributivity'
      - 'left_annihilation'
      - 'right_annihilation'
    """

2. Parametrized Test Suite¶

A pytest parametrized test that iterates over all semiring types from Semiring.catalog() and verifies all axioms with type-appropriate random samples.

3. Runtime Auditor (Optional)¶

A python -m algebrax.verify CLI entry point that runs the law verification and prints a pass/fail report.

Backwards Compatibility¶

Purely additive. No existing behavior changes.

Change Log¶

• 2026-08-02: Initial Draft.
• 2026-08-02: Implemented verification.py, CLI runner verify.py, ModularSemiring fix, and parametrized test suite (274 tests passing). Status → Final.