eml-cost 0.1.0a0

Pfaffian chain order and EML routing depth for symbolic expressions.

eml-cost

Pre-release. Patent pending. Not for redistribution.

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

Status

This package is pre-release. It is being developed alongside an in-flight patent application. Do not redistribute, fork, or publish.

When the patent files, the public release will move to PyPI under the same package name.

Quick start

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(θ) · Φ(φ)), 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.

Installation (local only, pre-release)

pip install -e /path/to/eml-cost-pkg

PyPI release follows patent filing.

License

PROPRIETARY-PRE-RELEASE. See LICENSE.

Citation

Citation form will be locked at public release.