ALGEBRAIC APROACH TO MEANING OF ALGEBRAIC APROACH TO MEANING OF LINGUISTIC TERMS, FUZZY LOGIC AND LINGUISTIC TERMS, FUZZY LOGIC AND APPROXIMATE REASONING APPROXIMATE REASONING Nguyen Cat Ho Nguyen Cat Ho Institute of Information Technology Institute of Information Technology (VAST) (VAST) E-mail: ncatho@hn.vnn.vn E-mail: ncatho@hn.vnn.vn July 1, 2014 2 I. Introduction I. Introduction  Well know that: Languages are human tools to cognize the reality → They sufficient to model facts, processes in reality in their own way. However, up to now one has still not discovered the mathematical structure of term-domains of linguistic variables, as Rinks wrote in “A heuristic approach to aggregate production scheduling using linguistic variables, Proc. of Inter. Congr. on Appl. Systems Research and Cybernetics, Vol. VI (1981)” that "verbal coding is a human way of repackaging material into a few chunks of rich information. Natural language is rather unique in this characteristic. Until recently, a unified theory for manipulating in a strict mathematical sense non-numerical- valued variables, such as linguistic terms, did not exist." A question is how can we find out a mathematical structure to model the meaning of linguistic terms ? July 1, 2014 3 I. Introduction I. Introduction  1965 L.A. Zadeh, in his paper “Fuzzy sets, Information and Control 8 (1965)”, Fuzzy sets were introduced to model the meaning of linguistic terms of natural languages. In natural language there are vague terms such as young, old, rather large, approximately 9, … 1 0.5 25 35 U July 1, 2014 4 I. Introduction I. Introduction  1965 L.A. Zadeh, in his paper “Fuzzy sets, Information and Control 8 (1965)”, Fuzzy sets were introduced to model the meaning of linguistic terms of natural languages. - Allow manipulate and process fuzzy, uncertain and inexact information; - Establish a computation approach (analytic one) to simulate human reasoning methods, based on fuzzy sets and fuzzy logics; - Open a new era of developing applications in uncertain environment of industry, sciences, social-economy, … - Now, we can find everywhere applications of fuzzy sets like in cars, washing machines, air conditioners, …. The achievements of fuzzy sets theory are very great and incontrovertible. However, it inherits still some problems. July 1, 2014 5 I. Introduction I. Introduction Some problems: Representation of linguistic terms by fuzzy sets means that one should establish an embedding: ϕ : Dom(X) → F(U,[0,1]). Since the image of Dom(X) under ϕ has no mathematical structure, operations on fuzzy sets are defined on the whole space F(U,[0,1]). It seems to be unreasonable and, moreover, not correct, because - Dom(X) is finite, but F(U,[0,1]) is infinite; - We observe that Dom(X) has a semantics-based ordering relation ≤ , but ϕ does not preserve this relation. + Maybe, by these reasons the efficiency of fuzzy sets- based methods is limited. + Moreover, these methods are complicated, in general. July 1, 2014 6 1 Introduction 1 Introduction Term-domain Dom(X) as a rich enough algebra: The question is that can we discover an algebraic structure of Dom(X) sufficient for studying fuzzy logic and approximate reasoning methods. + The answer is affirmative + Since late 1980s: an algebraic approach to the structure of term-domains of linguistic variables has been introduced: - Dom(X) = AX = (X,C,H, ≤ ); - A wide class of HAs: Lattices; - Symmetrical HAs: algebraic foundation of non-classical logics; July 1, 2014 7 1 Introduction 1 Introduction Application of HAs: + Linguistic reasoning: Inference rules allow handle linguistic terms directly; + Relational Databases with Linguistic data: Unified way for manipulating data types; + New approximate reasoning method: Interpolative reasoning based on SQMs + Application in Fuzzy Control: the results are much better. July 1, 2014 8 1 Introduction 1 Introduction We shall talk about  An overview of fuzzy sets theory and its computation mechanism  Hedge algebras – a semantics- based structure of terms-domains  Quantification method of hedge algebras: Fuzziness Measure of linguistic terms, hedges and Semantically Quantifying Mappings  Applicability of hedge algebras  Some Conclusions July 1, 2014 9 2 Fuzy Sets: An Overview 2 Fuzy Sets: An Overview Definition of Fuzzy Sets and Their Operations Example: Representation of the meaning of young 1 0.5 25 35 U + Definition: Fuzzy set is an (ordinary) set A of pairs {(u, µ A (u)) : u ∈ U, µ A (u) ∈ [0,1]}. µ A (u) is membership degree of u and µ A is a membp function. Denote the set of all fuzzy sets on U by F(U,[0,1]). July 1, 2014 10 2 Fuzy Sets: An Overview 2 Fuzy Sets: An Overview + Fuzzy Sets Operations Given fuzzy sets A and B Union: A ∪ B µ A ∪ B (u) = µ A (u) ∨ µ A (u) 1 Intersection: A ∩ B 0.5 µ A ∩ B (u) = µ A (u) ∧ µ A (u) 25 35 U Complement: CA µ CA (u) = 1 - µ A (u). µ A (u) is membership degree of u and µ A is a membp function. Denote the set of all fuzzy sets on U by F(U,[0,1]). [...]... µ A(u) + Fuzzy Sets representing logical connectives Given fuzzy sets A and B - AND, OR, NEGATION by Union, Intersection and Complement - If X is A then Y is B : A ⇒ B µ A⇒ B (u,v) = µ A(u) → * µ A(v) , where → * is a logical implication RB/A denotes fuzzy relation with µ RB/A(u,v) = µ A⇒ B (u,v) + Approximate Reasoning July 1, 2014 11 2 Fuzy Sets: An Overview + Approximate Reasoning E.g in fuzzy control... and Y linguistically: If X1 = A11 and and Xm = A1m then Y = B1 If X1 = An1 and and Xm = Anm then Y = Bn It is called a fuzzy model representing expert knowledge Approximate reasoning problem: FMCR problem A method which solves this problem is called FMCR method July 1, 2014 12 2 Fuzy Sets: An Overview + Main Steps of FMCR methods: 1) To construct appropriate fuzzy sets (membership problem) 2) To. .. 4) Problem of determining an appropriate composition operator to compute the output fuzzy set 5) To define a suitable defuzzification method to transform an output fuzzy set into a real value July 1, 2014 13 2 Fuzy Sets: An Overview + Example: Aircraft Landing Problem (Ross T J., Fuzzy logic with engineering application, International Edition Mc Graw-Hill, Inc, 1997) Aim: Landing gently to avoid h... appropriate fuzzy sets (membership problem) 2) To define a fuzzy relation Ri = RBi/Ai on U1× × Un× V to represent the semantics of if-then sentence in the given fuzzy model, by choosing suitable implication operator 3) To define a fuzzy relation R on U1× × Un× V to represent the semantics of the fuzzy model by choosing an aggregation operator for aggregating fuzzy relations Ri defined above: R = Union {R1,... True and P True are incomparable !! X - a linguistic; X = Dom(X )- a set of terms X := (X, H, C, ≤) is at least a Poset (Partially Ordered Set) July 1, 2014 19 II Hedge algebras - algebraic models of linguistic domains based on ling meaning  An algebraic structure of X : AX = (Dom(X ),C,LH,≤ )  X = Dom(X) can be ordered based on meaning of terms: – X owns an ordering relation ≤ , induced by term meaning. .. + fi and hi+1 = hi + vi 3) Define initial conditions and conduct a simulation for k cycles: + Now, inputs for Cycle 1: h1 = 980, v1 = - 14.2 ft/s High h fires L at 0.96 and M at 0.64; Velocity v fires only DS at 0.58 and DL at 0.42 L (.96) AND DS (.58) ⇒ DS (.58); (.42) L (.96) AND DL (.42) ⇒ Z M (.64) AND DS (.58) ⇒ Z (.58) ; (.42) M (.64) AND DL (.42) ⇒ US Compute f0 = Centroid of the union of outputs... ≤ x' and/ or y ≥ y', (iv) x ⇒ y = 1 iff either x = 0 or y = 1, (v) 1 ⇒ x = x and x ⇒ 1 = 1; 0 ⇒ x = 1 and x ⇒ 0 = − x, (vi) x ⇒ y ≥ W iff either x ≤ W or y ≥ W, and x ⇒ y ≤ W iff x ≥ W and y ≤ W July 1, 2014 34 III Fuzziness measure and quantifying semantic mappings Granularity information H(x) + H(x) models fuzziness of x H(V true) + The “size” of H(x) : fuzziness measure of x H(true) But first of all,... if khx ≠ hx and k is positive w.r.t h; Ex.: VPA true (- + - + = +) ; LPAV false (+ + - + - = +) (!) July 1, 2014 23 II Hedge algebras Semantic inheritance of hedges: Because each hedge h modifies only the meaning of term, it preserves essential meaning of terms and we can formulate this property in term of ≤ : For ex L true ≤ A true ⇒ true ≤ L true ≤ PL true ≤ LA true ≤ A true Formulation of the property:... C\{W}, is a representation of x with respect to a An element y is said to be a contradictory element of x if it can be represented as hn h1a', with a'∈ C\{W} and a' ≠ a Exam Little Very Possibly true ; Very Very Possibly false  Definition: A hedge algebra AX = (X,C,LH,≤ ), where C is defined as above, is said to be symmetrical, provided every element x in X has a unique contradictory element in X, denoted... = Centroid of the union of outputs = - 0.5 lbs 0.96 0.64 1.0 L NZ S M 0.4 0 July 1, 2014 300 500 800 980 1000 18 II Algebraic approach: Hedge algebras - Models of terms- domains based on linguistic meaning  Why we can and need introduce algebraic approach ?  Consider a set of terms of the TRUTH variable: True, V true, M True, A True, P True, VA True, MA True, VP True, MP True, VV true, VM True False, . ALGEBRAIC APROACH TO MEANING OF ALGEBRAIC APROACH TO MEANING OF LINGUISTIC TERMS, FUZZY LOGIC AND LINGUISTIC TERMS, FUZZY LOGIC AND APPROXIMATE REASONING APPROXIMATE REASONING Nguyen. algebras - algebraic models of II. Hedge algebras - algebraic models of linguistic domains based on ling. meaning linguistic domains based on ling. meaning  An algebraic structure of X : AX. discover an algebraic structure of Dom(X) sufficient for studying fuzzy logic and approximate reasoning methods. + The answer is affirmative + Since late 1980s: an algebraic approach to the structure