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    Automated Reasoning in Uncertainty Using Fuzzy Logic Sets

    Automated reasoning in artificial intelligence tries to emulate human reasoning through automated deduction. The four color theorem is an example of this, and was the first major theorem to be proved using a computer (Appel & Haken, 1977). However, a large part of human reasoning involves decisions with uncertain, incomplete, ambiguous, or inaccurate input. A good primer for how fuzzy logic is used today in cancer research can be found here.

    We demand rigidly defined areas of doubt and uncertainty!

    Douglas Adams, The Hitchhiker’s Guide to the Galaxy

    Fuzzy set theory was introduced by Lotfi Zadeh (1965) which allowed for the use of classic set theory and their operations to a class of objects in sets that have varying degrees of membership.

    Fuzzy Logic Intro

    Using Aristotle logic in expert systems involving automated reasoning is impractical when dealing with uncertainty. Fuzzy logic (Zadeh, 1965) provided flexibility in reasoning for use in expert systems with uncertainty.

    In classical set theory, a set is a collection of crisp values. Crisp values are distinct and precise. These sets deal with strict boundaries. In a Boolean system truth values are also crisp, that is either 1 as true, or 0 as false. For example, we could have a set of geochemical samples with copper parts per million (Cu ppm) values over 100.

    In this example we have two groups, members and non-members. Members are any values over 100 and non-members are 100 or less. There is no such thing as partial membership. We can have a membership function defined as

    Fuzzy logic provides intermediate values between true and false and extends classic set theory. We can now have partial membership in the set , as seen in Figure 1.  

    Figure 1: Classical Set Theory versus Fuzzy Set Theory.

    In our fuzzy set we can represent our set , as a set of ordered pairs in the universe .

    In our case the universe  is finite and discrete, and our set can be described as

    Suppose we have a crisp set , with their respective partial membership . We visualize this in Figure 2, and can represent the fuzzy set in its discrete form.

    Figure 2: Fuzzy Set vs. Crisp Set example.

    In my next post, I will talk about how to do operations on these sets, and how to apply them to geological data.

    Citations

    Appel, K., & Haken, W. (1977). The Solution of the Four-Color-Map Problem. Scientific American, 237(4), 108–121. https://doi.org/10.1038/scientificamerican1077-108

    Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X

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