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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)

- A binary operation, *, is a t-norm if

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

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