Fuzzy logic
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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

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Fuzzy logic

Fuzzy Logic

Frank Costanzo – MAT 7670 Spring 2012


Introduction

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}


What is fuzzy logic

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


Fuzzy connectives

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)


Fuzzy connectives continued

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}


Basic fuzzy propositional logic

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


Basic fuzzy propositional logic cont

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 *.


Basic fuzzy propositional logic cont1

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.


Basic fuzzy predicate logic

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


Various types of fuzzy logic

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


Applications

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.


References

References

  • Stanford Encyclopedia of Philosophy:

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

  • Wikipedia:

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


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