Membership-Function Formula and Parameter Contracts

• Status: Contract reference for Task #50
• Related ADR: ADR-0003
• Related executable surface: fuzzyroutines.FuzzyRoutines.MFunction

Purpose

This table records the mathematics currently exposed through the historical factory identifiers. It distinguishes a protected historical identifier from a standard mathematical name. It does not silently redefine an existing parameter convention.

A membership function maps an input to a grade in [0, 1], following the fuzzy-set formulation introduced by Zadeh (1965). Task #51 enforces the exact parameter sets, finite real values, and geometric constraints below at construction and through the public parameter setter.

Contract table

Historical identifier	Canonical mathematical description	Legacy formula and parameter meaning	Non-degenerate domain	Alias policy
hyperbolic	Rational-power decreasing right shoulder; not a standard hyperbola name	1 for x <= c; otherwise 1 / (1 + (a * (x - c)) ** b). c is the left cutoff, a scale, b exponent.	Finite x, a, b, c; a > 0, b > 0.	Keep hyperbolic; do not add a false standard-name alias.
bell	Flat-top piecewise-quadratic bell; not the generalized bell of Jang (1993)	Rising quadratic shoulder from a to b, plateau from b to c, then a mirrored falling shoulder ending at c + b - a.	Finite a, b, c; a < b <= c.	Keep bell; no generalizedBell alias.
parabolic	Quadratic S-shaped increasing shoulder	Let t = (x - a) / (b - a). It is 0 for x <= a, 2t**2 for 0 < t <= 1/2, 1 - 2(1 - t)**2 for 1/2 < t < 1, and 1 for x >= b.	Finite a, b; a < b.	sShoulder is an exact additive alias for the same implementation and parameters.
triangle	Triangular membership function with legacy parameter order	a is left foot, c is apex, b is right foot. The geometric order is a < c <= b, unlike the historical argument order.	Finite a, b, c; a < c <= b. The c = b boundary preserves the historical default high term.	Preserve triangle(a, b, c); a conventional-order alias must use a different explicit name.
trapezium	Trapezoidal membership function with legacy parameter order	a is left foot, c plateau start, d plateau end, b right foot. The geometric order is a < c <= d < b.	Finite a, b, c, d; a < c <= d < b.	Preserve trapezium(a, b, c, d); a conventional-order alias must use a different explicit name.
exponential	Gaussian membership function	exp(-0.5 * ((x - a) / b)**2). a is the centre and b is the scale.	Finite x, a, b; canonical scale b > 0. The legacy formula is sign-invariant for non-zero b.	gaussian is an exact additive alias for the same implementation and parameters.
sigmoidal	Logistic sigmoid	1 / (1 + exp(-a * (x - b))). a is slope and b is midpoint. Positive a rises; negative a falls.	Finite x, a, b; a != 0.	logistic is an exact additive alias for the same implementation and parameters.
desirability	Harrington one-sided desirability / Gumbel CDF transform	exp(-exp(-y)). It accepts direct input y and stores no parameters.	Finite y.	harringtonDesirability is an exact additive alias for the same implementation.

Reference naming

The standard parameterized generalized bell has three parameters for width, slope, and centre. It is not the same formula as the legacy flat-top quadratic bell implementation. Jang's original ANFIS paper defines the three-parameter generalized bell and identifies the centre, half-width, and slope roles: Jang (1993). The legacy exponential formula is exactly Gaussian-shaped, while legacy parabolic is an increasing quadratic shoulder rather than a Gaussian.

Transitional compatibility registry names

The forward-looking spellings currently available through the historical MFunction factory are additional registry keys, not separate formulas:

Registry name	Historical identifier	Interim parameter contract
sShoulder	parabolic	identical a, b
gaussian	exponential	identical a, b
logistic	sigmoidal	identical a, b
harringtonDesirability	desirability	no parameters

Each alias resolves to the same bound method as its historical identifier, so validation and evaluation share one implementation. This does not make the historical method canonical. The focused modern module owns the formula implementations and conventional parameter contracts; the legacy factory identifiers are aliases or explicit compatibility adapters.

No aliases are added for bell, triangle, or trapezium: their historical shape or parameter ordering would make a conventional name ambiguous without a distinct modern API contract.

The desirability transform is attributed to E. C. Harrington, The Desirability Function, Industrial Quality Control 21(10), 494–498 (1965). That provenance is documented by the peer-reviewed journal Metrika in Trautmann and Weihs (2006).

Compatibility rule

Historical factory identifiers, keyword names, and parameter ordering remain part of the public compatibility surface. Invalid configurations are rejected when a membership function is constructed or its complete parameter mapping is reassigned rather than being converted to a plausible membership grade of zero.