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1. Fuzzy Logic Frank Costanzo – MAT 7670 Spring 2012

2. Introduction • Fuzzy logic began with the introduction of Fuzzy Set Theory by LotfiZadeh in 1965. • Fuzzy Set • Sets whose elements have degrees of membership. • A fuzzy subset A of a set X is characterized by assigning to each element of x in X the degree of membership of x in A. • Example let X={x|x is a person} and A={x|x is an oldperson}

3. What is Fuzzy Logic? • In Propositional Logic, truth values are either True or False • Fuzzy logic is a type of Many-Valued Logic • There are more than two truth values • The interval [0,1] represents the possible truth values • 0 is absolute falsity • 1 is absolute truth

4. Fuzzy Connectives • t-norms (triangular norms) are truth functions of conjunction in Fuzzy Logic • A binary operation, *, is a t-norm if • It is Commutative • It is Associative • It is Non-Decreasing • 1 is the unit element • Example of a possible t-norm: x*y=min(x, y)

5. Fuzzy Connectives Continued • t-conorms are truth functions of disjunction • Example: max(x, y) • Negation – This function must be non-increasing and assign 0 to 1 and vice versa • 1-x • R-implication – The residuum of a t-norm; denoting the residuum as → and t-norm, * • x → y = max{z|x*z≤y}

6. Basic Fuzzy Propositional Logic • The logic of continuous t-norms (developed in Hajek 1998) • Formulas are built from proposition variables using the following connectives • Conjunction: & • Implication: → • Truth constant 0 denoting falsity • Negation ¬ φ is defined as φ → 0

7. Basic Fuzzy Propositional Logic cont…. • Given a continuous t-norm * (and hence its residuum →) each evaluation e of propositional variables by truth degrees for [0,1] extends uniquely to the evaluation e*(φ) of each formula φ using * and → as truth functions of & and → • A formula φ is a t-tautology or standard BL-tautology if e*(φ) = 1 for each evaluation e and each continuous t-norm *.

8. Basic Fuzzy Propositional Logic cont…. • The following t-tautologies are taken as axioms of the logic BL: • (A1) (φ → ψ) → ((ψ → χ) → (φ → χ)) • (A2) (φ & ψ) → φ • (A3) (φ & ψ) → (ψ & φ) • (A4) (φ & (φ → ψ)) → (ψ & (ψ → φ)) • (A5a) (φ → (ψ → χ)) → ((φ & ψ) → χ) • (A5b) ((φ & ψ) → χ) → (φ → (ψ → χ)) • (A6) ((φ → ψ) → χ) → (((ψ → φ) → χ) → χ) • (A7) 0 → φ • Modus ponens is the only deduction rule; this gives the usual notion of proof and provability of the logic BL.

9. Basic Fuzzy Predicate Logic: • Basic fuzzy predicate logic has the same formulas as classical predicate logic (they are built from predicates of arbitrary arity using object variables, connectives &, →, truth constant 0 and quantifiers ∀, ∃. • The truth degree of an universally quantified formula ∀xφ is defined as the infimumof truth degrees of instances of φ • Similarly ∃xφ has its truth degree defined by the supremum

10. Various types of Fuzzy Logic • Monoidal t-norm based propositional fuzzy logic • MTL is an axiomatization of logic where conjunction is defined by a left continuous t-norm • Łukasiewicz fuzzy logic • Extension of BL where the conjunction is the Łukasiewiczt-norm • Gödel fuzzy logic • the extension of basic fuzzy logic BL where conjunction is the Gödel t-norm: min(x, y) • Product fuzzy logic • the extension of basic fuzzy logic BL where conjunction is product t-norm

11. Applications • Fuzzy Control • Example: For instance, a temperature measurement for anti-lock breaks might have several separate membership functions defining particular temperature ranges needed to control the brakes properly. • Each function maps the same temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled.

12. References • Stanford Encyclopedia of Philosophy: • http://plato.stanford.edu/entries/logic-fuzzy/ • Wikipedia: • http://en.wikipedia.org/wiki/Fuzzy_logic