Fuzziness?

Just like a light switch the value is either "on" or "off", we all know the value for a Boolean is either a False (zero or "off") or a True (one or "on"). Fuzzy numbers fit in the infinite space between the Boolean zero and one and provide us with a "dimmer" switch through which to view the world. The applications of this "dimmer switch" approach are all around us in products such as anti-lock breaks, washing machines, thermostats, auto-focus cameras, and more.

Here is a simple example of what we are talking about: If you have someone who is 6ft (72in) tall, would they be considered tall or short? It's not that simple, right. They are kind of tall, or depending on the population they are being compared against, they could be considered not tall at all.

If you look at the graph below, you'll see a person of 72in is 0.63 tall, or has a 0.63 correlation to the "tall" concept. This person is more tall than not tall, but not 100% (1.00) tall. These are our "shades of grey" in respect to tall. The membership function can be defined differently or modified through what are called hedges (but that's a subject for a later article). At a certain point the fuzzy logic lines up with regular Boolean logic where the correlation to the "tall" concept is either 0.00 (absolutely false) or 1.00 (absolutely true). If we need to convert a fuzzy number (from 0.00 to 1.00) to a Boolean value, we just say it is true if the fuzzy value is greater than 0.50.

Fuzzy Controller

Usually, a fuzzy controller has multiple "area" definitions as in the graph below. Traditionally, they are labeled: L1, L0, M, H0, and H1 (L standing for Low and H standing for High).

If we apply this method to the concept of tall well be using the following ranges: "short", "somewhat short", "medium", "somewhat tall", and "tall".

And now if we look at 72in, we have a fuzzy relation to the somewhat tall and tall groups (with values 0.75 and 0.25 respectively). Now we can code against these fuzzy definitions and can think in terms of ranges of heights instead of specifics.

Just as a side note, defined fuzzy regions don't necessarily have to be exactly the same. They can also be triangles and the slopes on each section can vary instead of the symmetric plateau-shaped polygons I am using in the examples. It all depends on how you want to look a the data.

Boolean Operators

We can use basic Boolean operators on fuzzy numbers. The table below shows how they would be evaluated.

Operator Boolean Logic Fuzzy Implementation
| (OR) Sort OR Somewhat Short MAX(Short, Somewhat short)
& (And) Short AND Somewhat Short MIN(Short, Somewhat short)
! (NOT) NOT Short 1.0 - Short

Code Sample

The sample application consists of a bunch of points on a windows app. Each point has a momentum which is modified depending on how far from the center of the screen they are located. The momentum of each point is also modified by the relations with all other points. This gives us a chaotic system which is kind of fun to watch.

Here is our fuzzy number class which is pretty straightforward. Notice the overloaded Boolean operators:

public class FuzzyNumber
{
    public FuzzyNumber(double value)
    {
        m_value = value;
    }
    private double m_value;
    public double Value
    {
        get { return m_value; }
        set { m_value = value; }
    }
    public bool IsTrue()
    {
        return m_value > 0.5;
    }
    public static FuzzyNumber operator | (FuzzyNumber A, FuzzyNumber B)
    {
        return new FuzzyNumber(Math.Max(A.m_value, B.m_value));
    }
    public static FuzzyNumber operator & (FuzzyNumber A, FuzzyNumber B)
    {
        return new FuzzyNumber(Math.Min(A.m_value, B.m_value));
    }
    public static FuzzyNumber operator !(FuzzyNumber A)
    {
        return new FuzzyNumber(1.0 - A.Value);
    }
}

I have defined the distances and positions with enums as follows:

public enum HorizontalPosition : int
{
    FarLeft, Left, Center, Right, FarRight
}

public enum VerticalPosition : int
{
    ExtremeLow, Low, Middle, High, ExtremeHigh
}

public enum Distance : int
{
    VeryClose, Close, SlightlyClose, Normal, SlightlyFar, Far, VeryFar
}

And so can define how the velocities of each point will changed based on these enums:

m_horizAdjustments = new Dictionary<HorizontalPosition, float>();
m_horizAdjustments.Add(HorizontalPosition.FarLeft, -.075F);
m_horizAdjustments.Add(HorizontalPosition.Left, -0.25F);
m_horizAdjustments.Add(HorizontalPosition.Center, 0.0F);
m_horizAdjustments.Add(HorizontalPosition.Right, 0.25F);
m_horizAdjustments.Add(HorizontalPosition.FarRight, 0.75F);
m_vertAdjustment = new Dictionary<VerticalPosition, float>();
m_vertAdjustment.Add(VerticalPosition.ExtremeHigh, 0.75F);
m_vertAdjustment.Add(VerticalPosition.High, 0.25F);
m_vertAdjustment.Add(VerticalPosition.Middle, 0.0F);
m_vertAdjustment.Add(VerticalPosition.Low, -0.25F);
m_vertAdjustment.Add(VerticalPosition.ExtremeLow, -0.75F);
m_distAdjustment = new Dictionary<Distance, float>();
m_distAdjustment.Add(Distance.VeryFar, 100.0F);
m_distAdjustment.Add(Distance.Far, 10.0F);
m_distAdjustment.Add(Distance.SlightlyFar, 1.0F);
m_distAdjustment.Add(Distance.Normal, 1.0F);
m_distAdjustment.Add(Distance.SlightlyClose, 1.0F);
m_distAdjustment.Add(Distance.Close, 1.0F);
m_distAdjustment.Add(Distance.VeryClose, 1.0F);

In Conclusion

I'm not going to go through all the code for implementing the project because it would take too long, however it is pretty self-documenting and if you have any questions, let me know and I'll do my best to answer them.

Download the source and try adjusting the settings (above) to see how the points interact differently. If nothing else, it is a fun toy to play with.

In upcoming articles, I'll dig into other ways to implement fuzzy computing, but for now I hope you found this article interesting.

Until next time,

-Happy coding