Fuzzy Logic Frank Costanzo – MAT 7670 Spring 2012

Introduction Fuzzy logic began with the introduction of Fuzzy Set Theory by Lotfi Zadeh in 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 old person}

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 infimum of 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 Łukasiewicz t- 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: ▫ Wikipedia: ▫