04 — Membership Functions

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A membership function defines a value μ(x) within [0,1]. Each shape's parameters are edited via named fields, so you don't need to memorize the order of an anonymous array.

Built-in Shapes

Shape	Parameters	Use
Triangle	Left, Peak, Right	A universal set with a single maximum. An edge can be made to coincide with the peak for a vertical side.
Trapezoid	LeftFoot, LeftShoulder, RightShoulder, RightFoot	A plateau of full membership; convenient for a "normal range" concept.
Gaussian	Center, Sigma	A smooth symmetric shape with no kinks. Sigma controls the width.
Bell	Center, Width, Slope	A generalized bell with a controllable flat top and edge steepness.
Sigmoid	Center, Slope	A smooth transition from 0 to 1. A negative Slope mirrors the shape.
Ramp	Foot, Shoulder	A linear rise or fall with saturation at the edges.
JShape	Threshold, Steepness	Zero up to the threshold, then a smooth rise starting from zero.
Singleton	Value, Tolerance	A narrow, impulse-like set for a discrete or near-discrete result.
Constant	Level	The same degree of membership across the whole range.

Inversion

Every shape has an Inverted flag. It replaces the degree with 1 - μ(x). This lets a single shape produce a falling J-profile, a notch in a Bell, or an opposite Ramp.

NOT in a rule also complements the degree. The difference is where it's authored: Inverted changes the set itself everywhere, while NOT affects only a specific rule statement.

How to Choose a Shape

• Start with Triangle and Trapezoid: they're easy to read and tune.
• Use Ramp or Sigmoid for "the more, the stronger" concepts.
• Choose Gaussian or Bell when the derivative needs to be smooth.
• Singleton pairs best with Weighted Average for fast, near-Sugeno-style systems.
• Constant is useful for a constant premise or a fixed alpha-cut.

Overlap and Coverage

Neighboring sets should generally overlap. An intersection near a degree of 0.5 often gives predictable, smooth blending. The exact value depends on the task.

If there's a gap between sets, no rule may fire. If all sets are too wide, the decision becomes insensitive. Check the shapes in the Variables and Inference tabs.

A Custom Shape in C++

Create a USTRUCT derived from FFuzzyMembershipFunction and implement EvaluateRaw:

USTRUCT(BlueprintType, DisplayName = "Cosine Lobe")
struct FMyMF_CosineLobe : public FFuzzyMembershipFunction
{
    GENERATED_BODY()

    UPROPERTY(EditAnywhere, BlueprintReadWrite, Category = "Fuzzy Logic")
    float Center = 0.0f;

    UPROPERTY(EditAnywhere, BlueprintReadWrite, Category = "Fuzzy Logic")
    float HalfWidth = 1.0f;

    virtual float EvaluateRaw(float X) const override
    {
        const float T = (X - Center) / FMath::Max(HalfWidth, UE_KINDA_SMALL_NUMBER);
        return FMath::Abs(T) >= 1.0f ? 0.0f
            : 0.5f * (1.0f + FMath::Cos(T * UE_PI));
    }
};

The editor, JSON serialization, the type registry, and drawing utilities discover the derived struct via reflection. Override GetSupport, GetRepresentativeValue, Validate, and ToDisplayString as needed.