20 Mar Thresholds in Fuzzy Logic

At what point does temperature change from cold to hot? Tell a person (or a computer) that the beach will be closed if the water is too cold, and their interpretation may differ from yours. This statement is subjective, as are many of the values applied to the thermometer at right. But if you tell a person to place the eggs in the water when it begins to boil, the threshold is easy to measure and your intent will be understood easily. Threshold logic is a useful variation of multi-valued logic. It lets us establish points on continuum where decisions are triggered.

If you have read Section 3 or studied biological or artificial neural networks, you will recall that threshold functions are probably the most important biological computing mechanisms active in the brain (and the core functions of neural networks). The importance of threshold logic in both the biological and cybernetic domains is well established. In this section, we will look at psychological aspects of threshold logic and explore how, as a variation of multivalued logic, threshold theory can help us develop a clear and correct formulation of logical rules on which to build a simulation or mechanical mind.

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Section 5 #22

Table of Context

MIPUS considers himself to be a sensitive robot. His sensitivity, however, has lead to trouble more than once. One of his duties is to communicate with the home environment control computer to regulate heating and air conditioning. After weeks of responding to conflicting requests from the mister and missus, he just froze the setting at 69°F.
The thresholds, some fuzzy, some firm, may act as decision, or trigger points.

Firm  Thresholds in Fuzzy Logic

A firm threshold can be used to implement a binary or multi-valued logic function, while fuzzy thresholds support only multi-valued logic. There are methods for implementing both firm and fuzzy (subjective) thresholds. Mathematical functions that yield square curves can be used to implement firm thresholds. The formula states that below the threshold, the trigger is not activated, but above the threshold, the trigger is activated. For example, the freeway entrance-ramp control signal may be set to flash yellow until 4:00 PM, at which time the signal will be set to a pre-established red/green cycle (see program 1 below).

Mathematical functions that yield sigmoidal curves can be used to implement fuzzy thresholds. For example, a different freeway entrance-ramp control signal (see program 2 below) may be set to flash yellow until the volume of freeway traffic reaches 20 vehicles per lane per minute, at which time it will be set to a red/green cycle that permits one vehicle every four seconds. Then, for each increment of four cars per lane per minute, the signal adjusts the red/green cycle down by one or two seconds until the volume is 40 vehicles per lane per minute. The final flow-control factor is one vehicle every 9 to14 seconds. The determination of whether to increment one or two seconds may be determined by the speed of traffic.

A square curve indicates functions that produce firm thresholds. A sigmoidal curve represents functions that produce fuzzy thresholds.

Not So Fuzzy Absolutes

If we accept the propositions of threshold logic, we can look at the threshold as a converter that enables us to transform fuzzy principles into absolute theories that can be expressed mathematically. Let’s look at random probability and the possibility threshold. Assume, for the sake of this argument, that such a threshold exists. The threshold may be dynamic in that it moves up or down depending on external constraints. If we look at the threshold, we see that it represents an absolute dividing line with absolutely possible things on one side and absolutely impossible things on the other side.

This condition of absoluteness adds stability to the proposed structure of information in that it will be more difficult to disrupt the hierarchy of cause and effect. If there is a threshold governing all causal interactions at a quantum level, then it would be theoretically possible to develop an artificial or mechanical system that contains a complete set of cause and effect associations.

If, however, random probability is asymptotic, then it would be impossible to reflect all possible interactions in any finite system. In other words, if anything is possible, the set of all cause and effect associations is limitless and impossible to encode. Absolute thresholds do not preclude fuzzy and multi-valued interpretations. We may accurately say that sunrise occurred at 5:17 AM and still acknowledge that dawn stretched out over many minutes.

Whether this discussion of thresholds reflects truth in the physical universe or not, it serves as a valuable model for developing automated processing systems that mimic the brain. Even if our computer system does not capture all possible causes and effects, the fact that we can organize them into causal chains and explicitly associate them enables us to model the knowledge that is stored in finite human brains.

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