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# Fuzzy Logic - PowerPoint PPT Presentation

Fuzzy Logic. Frank Costanzo – MAT 7670 Spring 2012. Introduction. Fuzzy logic began with the introduction of Fuzzy Set Theory by Lotfi Zadeh in 1965. Fuzzy Set Sets whose elements have degrees of membership .

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### Fuzzy Logic

Frank Costanzo – MAT 7670 Spring 2012

• 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}

• 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

• 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)

• 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}

• 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

• 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 *.

• 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.

• 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

• 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

• 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.

• Stanford Encyclopedia of Philosophy:

• http://plato.stanford.edu/entries/logic-fuzzy/

• Wikipedia:

• http://en.wikipedia.org/wiki/Fuzzy_logic