eml-cost 0.7.0

Pfaffian chain order and EML routing depth for symbolic expressions.

eml-cost

Stable beta. Patent pending. Source-available; see LICENSE.

Pfaffian chain order and EML routing depth for symbolic expressions — a programmatic complexity measure on SymPy expression trees.

Installation

pip install eml-cost

For local development:

git clone https://github.com/almaguer1986/eml-cost
cd eml-cost
pip install -e ".[dev]"
pytest

Quick start

Three things you can do in under 10 lines each.

1. Get a complexity profile for any expression

from eml_cost import analyze

result = analyze("exp(exp(x)) + sin(x**2)")
print(result.pfaffian_r, result.max_path_r, result.predicted_depth)
# 5 5 7

2. Plug into SymPy's simplify as a cost function

import sympy as sp
from eml_cost import measure

x = sp.Symbol("x", real=True)
sp.simplify(sp.cos(x)**2 + sp.sin(x)**2, measure=measure)
# 1

3. Detect Pfaffian-but-not-EML expressions (Bessel, Airy, Lambert W)

import sympy as sp
from eml_cost import is_pfaffian_not_eml

is_pfaffian_not_eml(sp.besselj(0, sp.Symbol("x")))   # True
is_pfaffian_not_eml(sp.exp(sp.Symbol("x")))          # False

4. Canonicalize before profiling (eliminate form-fragility)

50% of textbook expressions yield different cost classes when written in algebraically equivalent forms. canonicalize() is a curated, content- preserving rewrite-rule sequence that drops drift to ~35% in our audit.

import sympy as sp
from eml_cost import PfaffianProfile

x = sp.Symbol("x")
forms = [
    1 / (1 + sp.exp(-x)),
    sp.exp(x) / (sp.exp(x) + 1),
    1 - 1 / (1 + sp.exp(x)),
]
for f in forms:
    p = PfaffianProfile.from_expression(f)  # canonicalize=True is default
    print(f"{f}  ->  {p.cost_class}")
# All three collapse to the same cost class.

5. Compare two expressions with a real distance metric

from eml_cost import PfaffianProfile

a = PfaffianProfile.from_expression("exp(x)")
b = PfaffianProfile.from_expression("sin(x)")
a.distance(b)        # weighted Euclidean in (r, d, w, c) space
a.compare(b)         # per-axis deltas + same_class flag
a.is_elementary()    # True (not Pfaffian-not-EML)

The metric satisfies identity, symmetry, and the triangle inequality (verified in tests/test_profile_metric.py). Default weights: r=4, d=1, w=2, c=1 — chain order dominates.

6. Run on the bundled 50-expression cross-domain corpus

import csv
from importlib.resources import files
from eml_cost import PfaffianProfile

corpus_path = files("eml_cost").joinpath("data/demo_corpus.csv")
with open(corpus_path) as f:
    rows = list(csv.DictReader(f))
profiles = [PfaffianProfile.from_expression(r["sympy_expr"]) for r in rows]

# 50 expressions, 9 domains (polynomial, exp_log, trig, pfaffian_not_eml,
# ml_activation, physics, biology, engineering, random_null), all with
# citations.

For an interactive walk-through with plots, see notebooks/quickstart.ipynb.

Result shape

from eml_cost import analyze

result = analyze("exp(exp(x)) + sin(x**2)")

result.pfaffian_r           # total Pfaffian chain order
result.max_path_r           # chain order along the deepest path
result.eml_depth            # EML routing tree depth
result.structural_overhead  # tree-structural depth
result.corrections          # Corrections(c_osc, c_composite, delta_fused)
result.predicted_depth      # max_path_r + corrections + structural
result.is_pfaffian_not_eml  # True for Bessel, Airy, Lambert W, ...

Drop-in measure for SymPy's simplify:

import sympy as sp
from eml_cost import measure

x = sp.Symbol("x", real=True)
sp.simplify(sp.cos(x)**2 + sp.sin(x)**2, measure=measure)
# 1

Public API

from eml_cost import (
    analyze,                # main entry point
    measure,                # SymPy simplify(..., measure=...) helper
    AnalyzeResult,          # frozen dataclass (result type)
    Corrections,            # frozen dataclass (correction terms)
    pfaffian_r,             # total chain order
    max_path_r,             # path-restricted chain order
    eml_depth,              # routing tree depth
    structural_overhead,    # Add/Mul/poly-Pow tree depth
    is_pfaffian_not_eml,    # True for Bessel/Airy/Lambert W/hyper
    PFAFFIAN_NOT_EML_R,     # registry: name -> chain order
)

What gets counted

Khovanskii r-counting throughout:

Operator	Chain contribution
exp(g)	1
log(g)	1
sin(g), cos(g) (pair)	2
tan(g)	1
tanh, atan, atanh, asinh, acosh	1 each
sinh(g), cosh(g) (pair)	2
sqrt(g), Pow(g, non-integer)	1
Pow(g, integer), Add, Mul	0
Bessel J/Y/I/K, Airy Ai/Bi, Lambert W, hyper	per registry

max_path_r differs from pfaffian_r only at Add and Mul nodes: pfaffian_r sums children, max_path_r takes the max. For independent- variable products like atomic orbital wavefunctions (R(r) * Y(theta) * Phi(phi)), the path-restricted count is dramatically smaller than the total — capturing the parallel-composition behavior.

EML routing depth

The eml_depth function models SuperBEST routing:

Operator	Depth contribution
exp, log	1
sin, cos	3 (Euler bypass)
tan	4
tanh, atan, sinh, cosh	1 (F-family primitive)
Pow, Add, Mul	1 + max over children

F-family fusion patterns are recognized:

• log(c + exp(g)) (LEAd / softplus shape) -> depth 1 + depth(g)
• 1/(1 + exp(-g)) (sigmoid shape) -> depth 1 + depth(g)

Pfaffian-but-not-EML class

Bessel J/Y/I/K, Hankel, Airy Ai/Bi, hypergeometric, and Lambert W are Pfaffian (admit polynomial-coefficient ODE chains) but lie outside the EML-elementary class. They are flagged by is_pfaffian_not_eml(expr) and contribute their registered chain order under pfaffian_r.

Links

• Project home: monogate.org
• Source: github.com/almaguer1986/eml-cost
• Package: pypi.org/project/eml-cost

License

PROPRIETARY-PRE-RELEASE. See LICENSE.

Citation

Citation form will be locked at public release.