Fuzzy Logic

Fuzzy Logic Fuzzy logic can be viewed as an extension of multi-valued logic. Fuzzy logic deals with the approximate rather than precise models. Fuzzy logic is a matter of degree.

Basic Differences Between Two Logics In two-valued logic systems, a proposition p is either true or false. In fuzzy logic, the truth values are allowed to range over the fuzzy subsets of a finite or infinite truth value set T. The predicates in two-valued logic are constrained to be crisp. In fuzzy logic, the predicates may be crisp E.g., “mortal”, “even”, etc. They can also be more general E.g., “ill”, “tired”, “tall”, “very tall”, etc.

Basic Difference Between Two Logics (Cont’d) Two-valued logic allows only two quantifiers: “all” and “some”. In fuzzy logic, it allows, in addition, theuse of fuzzy quantifiers: “most”, “few”, “many”, “several”, “much of”, etc. In two-valued logical systems, a p may be quantified by associating with p Truth value, “true” or “false” A modal operator such as “possible” or “necessary” An intensional operator such as “know” or “believe”

Basic Difference Between Two Logics (Cont’d) Fuzzy logic has three principal modes of qualification: Truth-qualification, as in (Mary is young) is not quite true. Probability-qualification, as in (Mary is young) in unlikely. Possibility-qualification, s in (Mary is young) is almost impossible.

Meaning Representation and Inference

Canonical Form

Canonical Form (Example)

Inference Rules Categorical Dispositional Rules that do not contain fuzzy quantifiers Dispositional Rules in which one or more premises may contain, explicitly or implicitly, the fuzzy quantifier “usually”.

Inference Rules (Cont’d)

Inference Rules (Cont’d)

Inference Rules (Cont’d)

Linguistic Variable Definition: A linguistic variable is a variable whose values are words or sentences in a natural or synthetic language. For example: “Age” is a linguistic variable if its values are “young”, a”not young”, and so on. In general, the values of the linguistic variable can be generated from a primary term (for example, “young”), its antonym (“old”), a collection of modifiers (“not”, “very”, “quite”), and the connectives “and” and “or”. Furthermore, each value represents a possibility distribution:

Fuzzy Car by Sugeno

Control Rules

Fuzzy Controllers Fuzzy controllers are modeled according to the human behavior. Fuzzy controllers are simpler, since they have a smaller number of rules. Trade-off between imprecision and simplification.