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Fuzzy-Set Convexity

  • Status: Design contract for Task #77
  • Decision: ADR-0009
  • Scope: one-dimensional scalar fuzzy sets
  • Public API: not implemented

Meaning of convexity

For a fuzzy set A over a real interval X, FuzzyRoutines uses Zadeh's convexity condition:

mu_A(lambda*x + (1 - lambda)*y) >= min(mu_A(x), mu_A(y))

for every x, y in X and every lambda in [0, 1].

The graph of mu_A does not need to be a convex function. The condition says that mu_A is quasiconcave: along an interval it can rise, possibly remain level, and fall, but it cannot contain a valley between two higher grades. Monotone membership functions are therefore fuzzy-convex too.

Fuzzy convexity does not imply normality. A subnormal hill, a monotone shoulder, and the empty fuzzy set can all be convex.

Alpha-cut equivalence

The weak alpha-cut is

A_alpha = {x in X | mu_A(x) >= alpha}, 0 < alpha <= 1.

The following statements are equivalent on a continuous scalar universe:

  1. A satisfies the membership inequality for every pair and every convex combination.
  2. Every positive weak alpha-cut is a convex crisp subset of X.

The proof is short and defines the implementation contract. If two coordinates are in A_alpha, both grades are at least alpha; the membership inequality puts every coordinate between them in the same cut. Conversely, set alpha = min(mu_A(x), mu_A(y)). A positive alpha invokes cut convexity, and alpha = 0 is already covered by the membership range.

This equivalence uses all positive levels, not a convenient finite selection. For a finite discrete universe it is sufficient to inspect its distinct positive membership grades because the cuts change only at those levels. For an arbitrary continuous callable, checking finitely many levels is not a proof.

Continuous representation

ContinuousUniverse is always a real interval, so the affine coordinate in the defining inequality remains inside the universe. Only coordinates that belong to the declared interval participate. An excluded endpoint, infinity, or a point outside the universe is never evaluated by implication.

The accepted analytical MFunction families are exactly certifiable from their formulas:

Family Shape proving quasiconcavity
Hyperbolic constant then decreasing
Bell increasing then constant then decreasing
Parabolic increasing shoulder
Triangle increasing then decreasing
Trapezium increasing then constant then decreasing
Exponential increasing then decreasing around its centre
Sigmoidal monotone
Desirability monotone

Parameter validation is part of each proof obligation. Registry aliases do not create new mathematics; they use the canonical family's certificate. Clipping a certified family to a bounded, open, closed, or half-bounded interval preserves convexity.

An arbitrary Python membership callable is different. Any algorithm making a finite number of evaluations can be given a second callable that returns the same grades at those coordinates but has a narrow lower valley elsewhere. Consequently, finite evaluation can find a counterexample but cannot certify global convexity. Such a representation needs an analytical certificate, symbolic structure, or a proved construction rule before an exact positive result is possible.

Discrete representation

Let a discrete universe declare

x_0 < x_1 < ... < x_(n-1).

Its exact relative-order condition is

mu_A(x_j) >= min(mu_A(x_i), mu_A(x_k)) for every i <= j <= k.

Every weak alpha-cut must therefore be one contiguous block of declared coordinates, or be empty. This is order-convexity. It deliberately does not claim that a finite set of coordinates contains every real point on a line segment.

For example, with X = {0, 2, 10}:

  • a cut containing {0, 2} is order-convex;
  • a cut containing {0, 10} but not 2 is not order-convex;
  • the gaps (0, 2) and (2, 10) have no membership semantics unless an explicit interpolation policy constructs a new continuous fuzzy set.

The check is exact because all declared coordinates are evaluated. A production implementation may use the equivalent unimodal-sequence rule: grades are nondecreasing through a maximum region and nonincreasing after it. Tests should retain the direct triple inequality as an independent oracle.

Sampled continuous evidence

Sampling a continuous membership function produces evidence with a declared domain and resolution. It does not change the universe or the representation.

A report may say:

COUNTEREXAMPLE
    sampled coordinates exhibit a strict valley

NO_SAMPLED_VIOLATION
    no tested coordinate triple exhibits a strict valley

It must not translate the second outcome into convex, approximately convex, a confidence percentage, or a global alpha-cut result. Without an additional regularity bound, a denser grid only adds observations; it does not close the universal proof obligation.

When sampled membership results are the authoritative runtime values, a strict valley is a counterexample to that evaluated callable. If those values are floating-point approximations to a separate ideal formula, the formula-level claim is limited by the declared numerical error policy. No implicit epsilon is part of exact convexity.

Cut and endpoint conventions

  • Alpha-cuts are weak: grade equality includes the coordinate.
  • Positive levels are 0 < alpha <= 1; the level 1 cut is the core.
  • A weak zero cut is the complete universe and is redundant for convexity.
  • Empty and singleton cuts are convex.
  • Cut endpoints may be open because the universe excludes an endpoint or the membership function is discontinuous. Convexity does not imply closedness or upper semicontinuity.
  • Positive support uses the separate strict condition mu_A(x) > 0.
  • An integration domain is an operational numerical interval and never limits a global convexity claim.

Future executable test matrix

Representation Exact positive result Negative result Required guard
Analytical audited family certificate explicit analytical witness unknown families remain unknown
Discrete exhaustive declared-coordinate evaluation violating declared triple no interpolation claim
Sampled never from samples alone sampled violating triple pass means only NO_SAMPLED_VIOLATION
Callable only a separate certificate or closure rule explicit evaluated counterexample finite queries never return PROVEN_CONVEX

The future implementation suite should:

  1. cover every canonical analytical family and alias at parameter boundaries;
  2. compare the optimized discrete algorithm with exhaustive triples for all short grade sequences over a small exact grade alphabet;
  3. compare discrete results with cuts at every distinct positive grade;
  4. verify deterministic witness coordinates and grades;
  5. exercise open, closed, bounded, and unbounded continuous universes;
  6. include a narrow off-grid valley that defeats a coarse and a refined grid;
  7. verify that sampled provenance survives serialization and display;
  8. reject any conversion from NO_SAMPLED_VIOLATION to a proven Boolean.

Non-goals

This contract does not define multidimensional universes, interpolation, strict fuzzy convexity, convexity degree, approximate/tolerance convexity, or closure laws for every t-norm and s-norm. Each changes the mathematical or evidence contract and requires separate design work.

References