Fuzzy Logic

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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

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