A Fuzzy Story: Refined Time

...And so ... The Story Begins....

I considered that I could have a better idea of when the morning began if I could perceive smaller increments of time. At the moment the alarm had stopped I was aware that my perceptive recognition of time was approximately 300 milliseconds. Call this amount G(me) = .300 seconds, where G of (me) is the smallest distinguishable time slots. At different times of the day { or different moods } G(me) might change - and G(me) may not equal G(you) ... or:

Considering a finer (smaller) G(me) my definition of when the morning began would be more precise. At some point during my finer perception of the transitory event (the stopping of the alarm) I would find that the interface became "fuzzy". That is, no longer would a solid line of demarcation exist - but rather an area where there was a "shared" state that was both morning and pre-morning.

The morning, by definition, began when the alarm hammer stopped hitting the bell. Yet, if I increased G(me) greatly, I could discern the compression of the sound waves from the last hit of the hammer against the bell. Graphically:

With n number of rings, there is one last ring - as discrete values, the ending of the ringing is clear-cut and well defined.

By looking at one alarm event - there is still discrete definition.

Expanding out the time line of the last hammer-hit of the alarm shows less discrete boundaries.

Although this example is theoretically based on our observation of the world around us, clearly some events depend on our definition of belonging. Extending the limits of what constitutes some event belonging to a set of criteria may make the definition less precise.

Fuzzy numbers give a membership grade to what normally would be considered the probability that an attribute occurs. Instead of an event having a likely-hood of happening ( some fractional number between 0 and 1 ) fuzzy numbers allow events to share membership. Given a membership function of an Universe U, M is the membership function - with all cases being 0 or 1 defines a "crisp" set of data results - which is how normal probability defines events. We have seen from the above example that initially the alarm event was "crisp"; when the time line was expanded, the data became fuzzy - that is to say, maps to a non-crisp set. With fuzzy sets, "people can define new operations ... i.e. ( where triangle is the operation ) ... this is not an application for what is in "crisp" sets [4]." But these new operations are useful for manipulating fuzzy set definitions to coincide with "crisp" sets. One new use of an operation ( perhaps ) might be to consider the refinement of time recognition ( G(me) ) to be evenly distributed for certain physical systems: Rami Zwick writes, " ... if a vague concept is represented by a random set, then it is justifiable to use the law of comparative judgment to estimate the means of the membership function distributions ". The law of comparative judgment ( simplified ) assumes that normal distributions can represent events provided that each stimulus is independently compared to each other. The number of comparisons should be large [2]; given a large amount of small fluctuations of the ringing bell - this condition is met.

Conclusion:

In conclusion, there are seemingly discrete events which we use as demarcation points between one event and another. Upon examination to finer levels of definition ( in my example, time was expanded in order to find a higher tolerance of where the morning began and the ringing stopped ), the data points become fuzzy - that is, membership can be given a probability of belonging to more than one event. New operations can be used to link membership with the more inclusive standards of probability analysis. One technique that could be use to analyze physical events which are very close to each other is the law of comparative judgment.

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