Lecture notes Fuzzy logic

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Introduction to Fuzzy Sets and Fuzzy Logic Introduction to Fuzzy Sets and Fuzzy Logic Luca Spada Department of Mathematics and Computer Science University of Salerno www.logica.dmi.unisa.it/lucaspada REASONPARK. Foligno, 17 - 19 September 2009 1/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Contents of the Course PART I Introduction Fuzzy sets Operations with fuzzy sets t-norms A theorem about continuous t-norm 2/ 144Introduction to Fuzzy Sets and Fuzzy Logic Contents of part I Introduction What Fuzzy logic is? Fuzzy logic in broad sense Fuzzy logic in the narrow sense Fuzzy sets Operations with fuzzy sets Union Intersection Complement t-norms A theorem about continuous t-norm 7/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Introduction There are no whole truths; all truths are half- truths. It is trying to treat them as whole truths that plays the devil. - Alfred North Whitehead 8/ 144Introduction to Fuzzy Sets and Fuzzy Logic Introduction What Fuzzy logic is? What Fuzzy logic is? Fuzzy logic studies reasoning systems in which the notions of truth and falsehood are considered in a graded fashion, in contrast with classical mathematics where only absolutly true statements are considered. From the Stanford Encyclopedia of Philosophy: The study of fuzzy logic can be considered in two di erent points of view: in a narrow and in a broad sense. 9/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Introduction Fuzzy logic in broad sense Fuzzy logic in broad sense Fuzzy logic in broad sense serves mainly as apparatus for fuzzy control, analysis of vagueness in natural language and several other application domains. It is one of the techniques of soft-computing, i.e. computational methods tolerant to suboptimality and impreciseness (vagueness) and giving quick, simple and suciently good solutions. Klir, G.J. and Yuan, B. Fuzzy sets and fuzzy logic: theory and applications. Prentice-Hall (1994) Nguyen, H.T. and Walker, E. A rst course in fuzzy logic. CRC Press (2006) Novak, V. and Novbak, V. Fuzzy sets and their applications. Hilger (1989) Zimmermann, H.J. Fuzzy set theoryand its applications. Kluwer Academic Pub (2001) 10/ 144Introduction to Fuzzy Sets and Fuzzy Logic Introduction Fuzzy logic in the narrow sense Fuzzy logic in the narrow sense Fuzzy logic in the narrow sense is symbolic logic with a comparative notion of truth developed fully in the spirit of classical logic (syntax, semantics, axiomatization, truth-preserving deduction, completeness, etc.; both propositional and predicate logic). It is a branch of many-valued logic based on the paradigm of inference under vagueness. Cignoli, R. and D'Ottaviano, I.M.L. and Mundici, D. Algebraic foundations of many-valued reasoning. Kluwer Academic Pub (2000) Gottwald, S. A treatise on many-valued logics. Research Studies Press (2001) Hajek, P. Metamathematics of fuzzy logic. Kluwer Academic Pub (2001) Turunen, E. Mathematics behind fuzzy logic. Physica-Verlag Heidelberg (1999) 11/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Fuzzy sets Fuzzy sets and crisp sets In classical mathematics one deals with collections of objects called (crisp) sets. Sometimes it is convenient to x some universe U in which every set is assumed to be included. It is also useful to think of a set A as a function from U which takes value 1 on objects which belong to A and 0 on all the rest. Such functions is called the characteristic function of A,  : A ( 1 if x2A  (x) = A def 0 if x2= A So there exists a bijective correspondence between characteristic functions and sets. 12/ 144Introduction to Fuzzy Sets and Fuzzy Logic Fuzzy sets Crisp sets Example Let X be the set of all real numbers between 0 and 10 and let A = 5; 9 be the subset of X of real numbers between 5 and 9. This results in the following gure: 13/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Fuzzy sets Fuzzy sets Fuzzy sets generalise this de nition, allowing elements to belong to a given set with a certain degree. Instead of considering characteristic functions with value in f0; 1g we consider now functions valued in 0; 1. A fuzzy subset F of a set X is a function  (x) assigning to F every element x of X the degree of membership of x to F : x2X7 (x)2 0; 1: F 14/ 144Introduction to Fuzzy Sets and Fuzzy Logic Fuzzy sets Fuzzy set Example (Cont.d) Let, as above, X be the set of real numbers between 1 and 10. A description of the fuzzy set of real numbers close to 7 could be given by the following gure: 15/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Operations with fuzzy sets Operations between sets In classical set theory there are some basic operations de ned over sets. Let X be a set andP(X) be the set of all subsets of X or, equivalently, the set of all functions between X andf0; 1g. The operation of union, intersection and complement are de ned in the following ways: AB =fxjx2A or x2Bg i.e.  (x) = maxf (x); (x)g AB A B A\B =fxjx2A and x2Bg i.e.  (x) = minf (x); (x)g A\B A B 0 A =fxjx2= Ag i.e.  0(x) = 1 (x) A A 16/ 144Introduction to Fuzzy Sets and Fuzzy Logic Operations with fuzzy sets Union Operations between fuzzy sets: union The law  (x) = maxf (x); (x)g: AB A B gives us an obvious way to generalise union to fuzzy sets. Let F and S be fuzzy subsets of X given by membership functions  and  : F S 17/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Operations with fuzzy sets Union Operations between fuzzy sets: union We set  (x) = maxf (x); (x)g FS F S 18/ 144Introduction to Fuzzy Sets and Fuzzy Logic Operations with fuzzy sets Intersection Operations between fuzzy sets: intersection Analogously for intersection:  (x) = minf (x); (x)g: A\B A B We set  (x) = minf (x); (x)g F\S F S 19/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Operations with fuzzy sets Complement Operations between fuzzy sets: complement Finally the complement for characteristic functions is de ned by,  0(x) = 1 (x): A A We set  0(x) = 1 (x): F F 20/ 144Introduction to Fuzzy Sets and Fuzzy Logic t-norms Operations between fuzzy sets 2 Let's go back for a while to operations between sets and focus on intersection. We de ned operations between sets inspired by the operations on characteristic functions. Since characteristic functions take values overf0; 1g we had to choose an extension to the full set 0; 1. It should be noted, though, that also the product would do the job, since onf0; 1g they coincide:  (x) = minf (x); (x)g = (x) (x): A\B A B A B 21/ 144 Introduction to Fuzzy Sets and Fuzzy Logic t-norms Operations between fuzzy sets 2 So our choice for the interpretation of the intersection between fuzzy sets was a little illegitimate. Further we have  (x) = minf (x); (x)g = maxf0; (x) + (x) 1g A\B A B A B It turns out that there is an in nity of functions which have the same values as the minimum on the setf0; 1g. This leads to isolate some basic property that the our functions must enjoy in order to be good candidate to interpret the intersection between fuzzy sets. 22/ 144Introduction to Fuzzy Sets and Fuzzy Logic t-norms t-norms In order to single out these properties we look again back at the crisp case: It is quite reasonable for instance to require the fuzzy intersection to be commutative, i.e.  (x)\ (x) = (x)\ (x); F S S F or associative:  (x)\  (x)\ (x) =  (x)\ (x)\ (x): F S T F S T Finally it is natural to ask that if we take a set  bigger than F  than the intersection  \ should be bigger or equal than S F T  \ : S T If for all x2X (x) (x) then  (x)\ (x) (x)\ (x) F S F T S T 23/ 144 . Introduction to Fuzzy Sets and Fuzzy Logic t-norms t-norms Summing up the few basic requirements that we make on a function that candidates to interpret intersection are: I To extend thef0; 1g case, i.e. for all x2 0; 1. 1x =x and 0x = 0 I Commutativity, i.e., for all x;y;z2 0; 1, xy =yx I Associativity, i.e., for all x;y;z2 0; 1, (xy)z =x (yz); I To be non-decreasing, i.e., for all x ;x ;y2 0; 1, 1 2 x x implies x yx y: 1 2 1 2 24/ 144Introduction to Fuzzy Sets and Fuzzy Logic t-norms t-norms Objects with such properties are already known in mathematics and are called t-norms. Example (i) Luk asiewicz t-norm: x y = max(0;x +y 1). (ii) Product t-norm: xy usual product between real numbers. (iii) G odel t-norm: xy = min(x;y).  2 0 if (x;y)2 0; 1 (iv) Drastic t-norm: x y = D min(x;y) otherwise. (v) The family of Frank t-norms is given by: 8 x y if  = 0 xy if  = 1  x y = F min(x;y) if  =1 x y : ( 1)( 1) 25/ 144 log (1 + ) otherwise.  1 Introduction to Fuzzy Sets and Fuzzy Logic t-norms Examples 1 1 1 1 1 1 y y y z z z 00 00 00 xx xx xx 11 1 1 Lu kasiewicz Product Minimum 26/ 144Introduction to Fuzzy Sets and Fuzzy Logic A theorem about continuous t-norm Mostert and Shields' Theorem An element x2 0; 1 is idempotent with respect to a t-norm , if xx =x. For each continuous t-norm, the set E of all idempotents is a closed subset of 0; 1 and hence its complement is a union of a setI (E) of countably many non-overlapping open intervals. open Let a;b2I(E) if and only if (a;b)2I (E). For I2I(E) open 2 letjI the restriction of to I . Theorem (Mostert and Shields, '57) If;E;I(E) are as above, then (i) for each I2I(E),jI is isomorphic either to the Product t-norm or to Lukasiewicz t-norm. (ii) If x;y2 0; 1 are such that there is no I2I(E) with x;y2I, then xy = min(x;y). 27/ 144 Introduction to Fuzzy Sets and Fuzzy Logic A theorem about continuous t-norm Examples 1 1 0.75 0.75 1 1 0.5 0.5 0.8 0.8 0.25 0.25 0 0.6 0 0.6 00 00 0.2 0.2 0.4 0.2 0.2 0.4 0.4 0.4 0.4 0.4 0.2 0.2 0.6 0.6 0.6 0.6 0.8 0.8 0.8 0.8 0 0 1 1 Two copies of Luk asiewicz Luk asiewicz plus Product 28/ 144Introduction to Fuzzy Sets and Fuzzy Logic A theorem about continuous t-norm Summing up We have seen that it is possible to generalise the classic crisp sets to objects which naturally admits a notion of graded membership. Also the fundamental operations between sets can be generalised to act on those new objects. ...but there is not just one of such generalisations. A few natural requirements drove us to isolate the concept of t-norm as a good candidate for intersection. There is a plenty of t-norms to choose from, but all of them can be reduced to a combination of three basic t-norms. next aim: we have fuzzy properties and we can combine them, let us try to reason about them. 29/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Part II Mathematical logic 30/ 144Introduction to Fuzzy Sets and Fuzzy Logic Contents of part II What is a logic? Propositional logic Syntax The axioms Deductions Semantics Truth tables Multiple truth values Again t-norms The other connectives 31/ 144 Introduction to Fuzzy Sets and Fuzzy Logic What is a logic? What is a logic? In mathematics a logic is a formal system which describes some set of rules for building new objects form existing ones. Example I Given the two words ab and bc is it possible to build new ones by substituting any b with ac or by substituting any c with a. So the words aac;aaa;acc;aca;:: are deducible from the two given ones. I The rules of chess allow to build new con gurations of the pieces on the board starting from the initial one. I The positions that we occupy in the space are governed by the law of physics. I ... 32/ 144Introduction to Fuzzy Sets and Fuzzy Logic Propositional logic Propositional logic Propositional logic studies the way new sentences are derived from a set of given sentences (usually called axioms). Example If there is no fuel the car does not start. There is no fuel in this car. This car will not start. If you own a boat you can travel in the see. If you can travel in the see you can reach Elba island. If you own a boat you can reach Elba island. 33/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Syntax Propositional logic De nition The objects in propositional logic are sentences, built from an alphabet. The language of propositional logic is given by: I A set V of propositional variables (the alphabet): fX ;:::;X ;:::g 1 n I Connectives:_,,:, (conjunction, disjunction, negation and implication). I Parenthesis ( and ). 34/ 144Introduction to Fuzzy Sets and Fuzzy Logic Syntax Sentences of propositional calculus De nition Sentences (or formulas) of propositional logic are de ned in the following way. i) Every variable is a formula. ii) If P and Q are formulas then (P_Q), (PQ), (:P ), (PQ) are formulas. iii) All formulas are constructed only using i) and ii). Parenthesis are used in order to avoid confusion. They can be omitted whenever there is no risk of misunderstandings. 35/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Syntax The axioms The axioms of propositional logic The axioms of propositional logic are : 1. (A (BA)) 2. ((A (BC)) ((AB) (AC))) 3. ((:A:B) (BA)) plus modus ponens: if AB is true and A is true, then B is true. A deduction is a sequence of instances of the above axioms and use of the rule modus ponens. The other connectives are de ned as I A_B = :AB def I AB = :(:A_:B) =:(::A:B) def 36/ 144Introduction to Fuzzy Sets and Fuzzy Logic Syntax Deductions A deduction in propositional logic Example An instance of 1 gives :X (X :X ) 1 2 1 and an instance of 2 gives :X (X :X ) ((:X X ) (:X :X )); 1 2 1 1 2 1 1 the use of modus ponens leads (:X X ) (:X :X ) 1 2 1 1 which, by de nition, can be written as (:X X ) (X _:X ): 1 2 1 1 37/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Semantics The semantics of a calculus Just as happens in mathematics, where one makes calculations with numbers and those numbers represent, e.g: physical quantities, or amount of money, or points in a space, one can associate to a logic one (or several) interpretation, called the semantics of the logic. 38/ 144Introduction to Fuzzy Sets and Fuzzy Logic Semantics Evaluations De nition An evaluation of propositional variables is a function v :V f0; 1g mapping every variable in either the value 0 (False) or 1 (True). In order to extend evaluations to formulas we need to interpret connectives as operations overf0; 1g. In this way we establish a homomorphism between the algebra of formulas (with the operation given by connectives) and the Boolean algebra onf0; 1g: v :Formf0; 1g 39/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Semantics The semantics of connectives The evaluation v can be extended to a function v total on Form by using induction: I Variables: v(X ) =v(X );:::; v(X ) =v(X ). 1 1 n n I v(PQ) = 1 if both v(P ) = 1 and v(Q) = 1. v(PQ) = 0 otherwise. I v(P_Q) = 1 if either v(P ) = 1 or v(Q) = 1. v(P_Q) = 0 otherwise. I v(PQ) = 0 if v(P ) = 1 and v(Q) = 0. v(PQ) = 1 otherwise. I v(:P ) = 1 if v(P ) = 0, and vice-versa. A formula is a tautology if it only takes values 1. Tautologies are always true, for every valuation of variables. 40/ 144Introduction to Fuzzy Sets and Fuzzy Logic Semantics Truth tables Truth tables The above rules can be summarized by the following tables: A B AB A B A_B A B AB 1 1 1 1 1 1 1 1 1 A :A 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 1 Conjunction Disjunction Implication Negation 41/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Semantics Truth tables Truth tables Using the tables for basic connectives we can write tables for any formula: Example Let us consider the formula X (Y_:X): X Y :X Y_:X X (Y_:X) 1 1 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 1 1 1 42/ 144Introduction to Fuzzy Sets and Fuzzy Logic Multiple truth values Adding a third truth values It is easy now to gure out how to extend the previous logical apparatus with a third truth value, say 1=2. We keep the same syntactical structure of formulas: we just change the semantics. Evaluations are now functions from the set of variables into f0; 1=2; 1g. Accordingly to the de nitions of truth tables for connectives we have di erent three-valued logics. 43/ 144 Introduction to Fuzzy Sets and Fuzzy Logic Multiple truth values Kleene's logic Kleene strong three valued logic is de ned as A and B 0 1=2 1 A or B 0 1=2 1 0 0 0 0 0 0 1=2 1 1=2 0 1=2 1=2 1=2 1=2 1=2 1 1 0 1=2 1 1 1 1 1 Conjunction Disjunction A implies B 0 1/2 1 A not A 0 1 1 1 1 0 1/2 1/2 1/2 1 1=2 1=2 1 0 1/2 1 0 1 Implication Negation 44/ 144

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