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Introducing fuzzy logic

What if we could have a continuous evaluation of a property? What if our degree of limitation increased continuously as slope increased? Fuzzy logic makes this possible.

The fact that something is true does not exclude the possibility that it is also false. Fuzzy logic (or approximate reasoning) is built upon this precept. With fuzzy logic, you can get a complete gradation of how true (or false) an interpretive statement is. Fuzzy logic allows you to translate the ranges of properties into a uniform basis. The uniform basis is a value from 0 to 1 where 1 means a statement is absolutely true and 0 means a statement is absolutely not true. For example, currently the percentage slope for picnic areas is rated as:

< 8      slight
8-15      moderate
> 15      severe

Minimum depth to water table is rated as:

> 100      slight
20-99      moderate
< 20      severe

With fuzzy logic, we can show a value in the middle or anywhere along a continuum. The easiest way to see this is to set up a graph. Notice that in the figures below, the values for slope and minimum depth to water table are translated into some measure of truthfulness about the statement of being too wet or too steep. (Of course, you might choose a different type of curve; but for this simple example, we will use a linear curve.)

           Percent Slope Along a Continuum

As shown above, with fuzzy logic, you can show a value in the middle. It is partly true that 10% slope is too steep. Its also partly not true.

Below you can see that 0.4 is more not wet than wet. Values greater than 100 are clearly not too wet.

           Minimum Depth to Water Table Along a Continuum

Compare the graphs above to the crisp limits table. The difference is, instead of crisp limits, you can now have gradational limits. To see how this really helps to improve the development of interpretive criteria, you will need to understand some concepts of fuzzy math.

Although in this demonstration the numerical values for too steep and too wet seem arbitrarily determined, they would actually be based on opinions and judgments of experts like yourself. Once you have numerical values for too steep and too wet, the possibilities of dealing with interactions and relative weights become real.

Understanding fuzzy math concepts

Applying fuzzy math opens up soil interpretations to the realm of handling interactions. For example, the interaction of slope and water table, where, as slope increases, water decreases. Fuzzy logic also allows the handling relative weights, such as when slope may have more importance to the interpretation than depth to water table. Before we discuss fuzzy math, lets examine our conventional way of thinking.

As stated previously, the fact that something is true does not exclude the possibility that it is also false, although our conventional bias is to believe that true excludes false. In the conventional way of thinking, a condition of A OR B is true under the first three conditions in the figure below. The condition of A OR B is false under the last condition:

Now lets turn to fuzzy math. The equations below are two fuzzy math rules relevant to our discussion.

Fuzzy Math Rules for OR and AND Operators

A OR B Max [A, B]
A AND B Min [A, B]

The next table shows a truth table for the Boolean OR operator. Using fuzzy math, the true values are equal to 1 and the false values are equal to 0. By inserting the fuzzy values of 0 to 1 and then applying the fuzzy math rule of A OR B ~ Max [A, B], the conditions are expressed for the OR statement.

The table demonstrates with true=1 and false=0 that OR is equivalent to Max.

The next table shows a truth table for the Boolean AND operator. Using fuzzy math, the true values are equal to 1 and the false values are equal to 0. By inserting the fuzzy values of 0 to 1 and then applying the fuzzy math rule of A AND B ~ Min [A, B], the conditions are expressed for the AND statement.

The table demonstrates with true=1 and false=0 that AND is equivalent to Min.

This demonstration of fuzzy math is not meant as a proof but simply as a demonstration of how the math works. Returning to the picnic area example, next you will insert into the equation the fuzzy values shown in the following graphs:

     A

           Fuzzy Logic Applied to Percent Slope

     B

           Fuzzy Logic Applied to Minimum Depth to Water Table

Remembering our interpretive statement and applying the fuzzy values from the graphs above, refer to the statement below for a picture of how it fits together.

"A soil has limitations for picnic areas if it is too steep or too wet" where too steep equals .6 and too wet equals .4, and OR means MAX.

Finally, lets compute the interpretive result given that we are dealing with the OR operator:

     A      OR B            Then

     T.6       T.4            T.6
 

A site has limitations for picnic areas if it is 0.6 too steep or 0.4 too wet. The statement has an OR condition so the fuzzy rule of A OR B ~ Max [A, B] was applied to produce the maximum value of 0.6. With fuzzy logic, you would say that there is a 0.6 truthfulness that the site has limitations for picnic areas and that the primary limitation is related to slope.

What if you had chosen to construct your statement of limitations this way: "A site has limitations for picnic areas if it is too wet and too steep?" You would use the math for AND statements and the result would be a 0.4 truthfulness that the site has limitations for picnic areas.

     A      AND B            Then
     T.6       T.4            T.4

You may wonder, is it good or bad that there is a 0.4 truthfulness that the site has limitations for picnic areas and that the limitation relates to the interaction of slope and wetness? Furthermore, once you derive a numerical value, what does it mean? How does it relate to the interpretive statement for picnic areas?

As always, it depends on the opinion and judgment of an expert or team of experts. Fuzzy logic gives you the ability to handle interactions and relative weights to interpret a soil interpretation, but you still need to use expert opinion and judgments when assigning meaning to the fuzzy numbers. As the expert, you decide what the values mean in the context of the land use.