Approximate Arithmetic Operations of U-numbers - ScienceDirect

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ScienceDirect Procedia Computer Science 102 (2016) 59 – 66

12th International Conference on Application of Fuzzy Systems and Soft Computing, ICAFS 2016, 29-30 August 2016, Vienna, Austria

Approximate arithmetic operations of U-numbers R.A. Aliev * Azerbaijan State University of Oil and Industry, Joint MBA Program, USA, Azerbaijan, 20 Azadlig Ave., AZ1010 Baku, Azerbaijan Department of Computer Engineering, Near East University, Lefkosa, North Cyprus

Abstract The theory of usuality suggested by L.A. Zadeh is widely used in many areas including decision analysis, system analysis, control and others where commonsense knowledge plays an important role. As a rule, this knowledge is imprecise, incomplete, and partially reliable. The concept of usuality is characterized by a combination of fuzzy and probabilistic information. Formally, it is handled by possibilistic-probabilistic constraint, where A is a fuzzy restriction on a value of a random variable X, and “usually” is a fuzzy restriction on a value of probability measure of A. Thus, usuality is a special case of a Z-number where second component is “usually”, and is referred to as U-number. Humans mainly use U-numbers in everyday reasoning. As usuality underlies human commonsense reasoning, arithmetic operations on U-numbers should be rather approximate than exact. In this study we develop a new approach to approximate arithmetic and algebraic operations on U-numbers. © Published by by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license ©2016 2016The TheAuthors. Authors. Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility ofthe Organizing Committee of ICAFS 2016. Peer-review under responsibility of the Organizing Committee of ICAFS 2016 Keywords:discrete fuzzy number, convolution, usuality, U-number, U-reasoning

1. Introduction The importance of the concept of usuality is dictated by the fact that it underlies commonsense knowledge-based human decision making and reasoning. Zadeh for the first time suggested the concept of “usuality” which plays central role in a theory of commonsense. In1,2,3Zadeh has suggested main principles of theory of usuality. In4 the

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1877-0509 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ICAFS 2016 doi:10.1016/j.procs.2016.09.370

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R.A. Aliev / Procedia Computer Science 102 (2016) 59 – 66

author shows that the concept of dispositionality is closely related to the notion of usuality. Theory of usuality is defined as a tool for computational framework for commonsense reasoning. In5 author outlines a theory of usuality based on a method of representing the meaning of usuality-qualified propositions. A system of inference for usuality-qualified propositions is developed.In6 Yager introduces a formal mechanism for representing and manipulating of usual values. This mechanism is based upon a combination of the linguistic variables and Shafer evidential structures7.In8 authors analyze the concepts usuality, regularity and dispositional reasoning from the point of view of approximate reasoning. Schwarts in9 discusses fuzzy quantifiers, fuzzy usuality modifiers and fuzzy likelihood modifiers. He analyzes these notions with unified semantics. Analyzing existing works on usuality concept we can conclude that in many types of commonsense knowledge it is used usual values of some variable. Almost always usual values are vague and imprecise and are represented by linguistic values. Usual information involves both a probabilistic and possibilistic granules. In existing studies, the meaning of usuality is defined in the terms of sequence of values of variable X. As Zadeh shows, the statement “Usually, X is A” indicates that the probability that the event A occurs as the value of X is “usually” (A occurs the most) and is represented as the possibility-probability granule. The main conclusion stemming from review of the mentioned above works is that arithmetic of U-numbers and reasoning under U-information should be rather approximate than exact. Indeed, for commonsense knowledge-based everyday reasoning, approximate and sufficient results are more effective than absolutely exact and time consuming results. Thus, a computational framework of operations of U-numbers should be based on a practically suitable tradeoff between accuracy and computational complexity. In this study we develop a new approach to approximate arithmetic operations on U-numbers. The rest of the paper is structured as follows. In Section 2 we present some prerequisite material including operations over random variables, probability measure of a fuzzy number etc. In Section 3 we present a general information on U-numbers. In Section 4 we give some arithmetic and algebraic operations on U-numbers. In Section 5 we consider approximate reasoning with usual information. Section 6 concludes. 2. Preliminaries Definition 1.Arithmetic operations over random variables10-12.Let X1 and X 2 be two independent continuous random variables with pdfs p1 and p2 . A pdf p12 of X 12 X 1 X 2 , where is a two-place operation, is referred to as a convolution of p1 and p2 (pdf of a random variable X 12 obtained as a result of a two-place operation over X1 and X 2 ) and is defined as follows.

³³ p ( x ) p

p12 ( x)

1

1

2

( x2 ) dx1 dx2 , : {( x1 , x2 ) x

x1 x2 } .

:

Let X1 and X 2 be two independent discrete random variables with the corresponding outcome spaces X1 { x11 ,..., x1i ,..., x1n1 } and X 2 { x21 ,..., x2i ,..., x2 n2 } and the corresponding discrete probability distributions p1 and p2 . The probability distribution of X 1 X 2 , {, , , /} , comes as the convolution p12

p1 $ p2 of p1 and p2

which is defined for any x { x1 x2 x1 X1 , x2 X 2 } , x1 X1 , x2 X 2 as follows:

¦

p12 ( x )

p1 ( x1 ) p2 ( x2 ) .

x x1 x2

Definition 2. Probability measure of a fuzzy number13,14. Let X be continuous random variable with pdf p . Let A be a continuous fuzzy number describing a possibilistic restriction on values of X . A probability measure of A denoted P( A) is defined as P ( A)

³P

A

( x) p ( x ) dx .

R

For a discrete fuzzy number and a discrete probability distribution, the probability measure is defined as n

P( A)

¦P i 1

A

( xi ) p( xi )

P A ( x1 ) p( x1 ) P A ( x2 ) p( x2 ) ... P A ( xn ) p( xn ) .

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3. U-n numbers Lett X be a random m variable and A be a fuzzy num mber playing a role r of fuzzy coonstraint on valu ues that the randdom variab ble may take: X is A. The definition of a usual value of X may be exprressed in terms of the probabbility distrib bution of X as fo ollows5. If p( xi ) is the probabiility of X taking g xi as its valuee, then usually X is A

Pusually ¦ i p( xi ) P A xi

(11)

usually X is A

Pmost ¦ i p ( xi ) P A xi

(22)

or

u number deescribing, “usuaally, professor’ss income is med dium” is shown fig.1. A usual

Fig.1. An A example of U-nu umber

obability that th he event A occu urs as the value for the variablee X, is “most”. A As it Forrmula (2) indicaates that the pro was mentioned m abovee, in5 Zadeh pro ovided an outlin ne for the theorry of usuality, hhowever this top pic requires furrther investigation. It is neeeded a more geeneral approach h for other usuaality quantifiers.. In this paper “usuality” “ will bbe a osite term charaacterized by fuzzzy quantities ass always, usually ly, frequently / ooften, occasiona ally, seldom, alm most compo never//rarely, never. The T codebook fo or “usuality” is shown in fig.2.

F Fig.2. The codebook k of the fuzzy quan ntifiers of usuality

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4. Operations on U-numbers 4.1. A general approach to computation with U-numbers In this paper we try to answer the questions raised by Zadeh5 concerning the concept of usuality. These basic questions are: - How can a usual value of a variable be computed? - How can the usual values of two or more variables be combined? More concretely, if X 12 X 1 X 2 , and the usual values of X1 and X 2 are given, what will be the usual value of X 12 ? - How can we construct an inference system for reasoning with usuality-qualified propositions? - How can decisions be made in usuality-qualified knowledge-based environment (i.e. when we know only usual values of probabilities, payoff’s etc)? The most critical question is that related to combination of U-numbers. It should be taken into account whether the variables X1 and X 2 are dependent or independent. This will influence how a usuality quantifier related to the result X 12 X 1 * X 2 should be determined on the basis of the usuality quantifiers related to X1 and X 2 . In this study the modality of a generalized constraint is considered as usuality: X is u A or Usuality r u , where X is the constrained variable, A is a constraining relation, and r identifies the semantics of the constraint. The usuality constraint presupposes that X is a random variable and the probability that X isu A is “usually”: Prob ^ X is A` is usually , where A is a usual value of X , for example A is “small”. Computation with U-numbers is related to usuality constraint propagation. Assume that X is a random variable taking values x1 , x2 ,... and p is probability distribution of X . The constraint propagation is as follows. X isu A , Prob ^ X is B` is C X isu A o Prob ^ X is A` is usually o Pusually

³ P R

A

( x ) p ( x ) dx ,

PC ( y ) sup p ( x ) ( Pusually ( ³ P A ( x) p( x)dx)) , R

subject to y ³ P B ( x) p ( x)dx . R

4.2. Operation on U-numbers We suggest an approach to computation with U-numbers according to basic two-place arithmetic operations , , , / and one-place algebraic operations as a square and a square root of U-numbers. Let U 1 ( A1 , B1 ) and U 2 ( A2 , B2 ) be U-numbers ( B1 and B2 are fuzzy terms of the usuality codebook) describing values of random variables X1 and X 2 . Assume that it is needed to compute the result U12 a two-place operation {, , , /} : U12

U1 U 2 .Computation of one-place operations U

2

( A12 , B12 ) of

U 1 and U

U1 is

treated analogously. 4.2.1. Arithmetic operations Consider the case of discretized version of components of usual numbers.The first stage is the computation of two-place operations of fuzzy numbers A1 and A2 on the basis of fuzzy arithmetic. For example, for sum U12 U1 U 2 we have to calculate A12 A1 A2 .

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The second stage involves step-by-step construction of B12 and is related to propagation of probabilistic restrictions. We realize that in U-numbers U 1 ( A1 , B1 ) and U 2 ( A2 , B2 ) , the ‘true’ probability distributions p1 and p2 are not exactly known. In contrast, the information available is represented by the fuzzy restrictions: n1

¦P

n2

A1

( x1k ) p1 ( x1k ) is B1

¦P

,k

k 1

A2

( x2 k ) p2 ( x2 k ) is B2

,

1

which are represented in terms of the membership functions as n1

§

·

P p ( p1 ) P B ¨ ¦ P A ( x1k ) p1 ( x1k ) ¸ 1

1

©k

1

¹ ,

1

§

n2

·

P p ( p2 ) P B ¨ ¦ P A ( x2 k ) p2 ( x2 k ) ¸ 2

2

©k

2

1

¹ .

Given these fuzzy restrictions, extract probability distributions p j , j 1, 2 by solvingthe following goal linear programming problem: c1 v1l c2 v2l ... cn vnl o b jl

(3)

subject to

v1l v2l ... vnl l 1

l 2

l n

v , v ,..., v t 0 where ck

1½° ¾. °¿

(4)

P A ( x jk ) and vk

p j ( x jk ), k 1,.., n j , k 1,.., n j .As a result, p jl ( x jk ), k

j

1,.., n j is found and, therefore,

distribution p jl is obtained. Thus, to construct thedistributions p jl , we need to solve m simple problems (3)-(4). Distributions p jl ( x jk ), k

1,.., n j naturally induce probabilistic uncertainty over the result X 12

X 1 X 2 . This is

a critical point of computation of U-numbers, at which the issue of dependence between X1 and X 2 should be considered. For simplicity, here we consider the case of independence between X1 and X 2 . This implies that given p1l1 $ p2 l2 , s

a pair p1l1 , p2 l2 , the convolution p12 s

1,..., m 2 is to be computed as on the basis of Definition 1.

For the case of dependence between X1 and X 2 , p12s should be computed as a joint probability distribution by taking into account dependence between random variables15,16. n

Given p12 s , the value of probability measure of A12 can be computed: P( A12 )

¦P

A12

( x12 k ) p12 ( x12 k ) . However,

k 1

the ‘true’ p12 s is not exactly known as the ‘true’ p1l1 , p2 l2 are described by fuzzy restrictions. These fuzzy restrictions induce the fuzzy set of convolutions p12 s , s 1,..., m 2 with the membership function defined as P p ( p12 s ) 12

max p12 s

p1 l1 $ p 2 l2

[ P p1 ( p1l1 ) P p2 ( p 2 l2 )]

(5)

subject to §

nj

·

P p ( p jl ) P B ¨¨ ¦ P A ( x jk ) p jl ( x jk ) ¸¸ , j 1, 2 k 1 j

j

j

©

j

j

¹

(6)

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where is min operation. As a result, fuzziness of information on p12 s described by P p induces fuzziness of the value of probability 12

measure P( A12 ) in a form ofa fuzzy number B12 . The membership function of B12 is defined as

P B (b12 s ) 12

max( P p12 ( p12 s ))

(7)

( xk ) p12 s ( xk )

(8)

subject to

b12 s

¦P

A12

k

As a result, U12

U1 U 2 is obtained as U12

4.2.2. Square of a U-number Let us now consider construction of U

A12 [ A12 ]D

2

( A12 , B12 ) .

U1 . A

A12 is determined as follows:

D [ A12 ]D ,

(9)

D [0,1]

{x12 x1 A1D } .

(10)

The probability distribution p is determined given p1 as17

p ( x)

1 ª p1 ( x ) p1 ( x ) º¼ , x t 0. 2 x¬

(11)

Next by noting that a ‘true’ p1 is not known, one has to consider fuzzy constraint P p1 to be constructed by solving a certain LP problem (3)-(4).The fuzzy set of probability distributions p1,l with membership function P p1 naturally induces the fuzzy set of probability distributions pl with the membership function P p ( pl ) defined as

P p ( pl )

P p ( p1l ) , l 1

1,..., m

where p is determined from p1 based on (11). The probability measure P ( A) given p is produced on basis of Definition 2. Finally, given a fuzzy restriction on p described by P p , we extend P ( A) to a fuzzy set B by solving a problem analogous to (7)-(8). As a result, U 2 is

obtained on the basis of the extension principle for computation with U -numbers as U 2 that for X1 t 0 , it is not needed to compute of B because it is the same as B1 is any natural number, is carried out in an analogous fashion.

17,18

( A, B ). Let us mention

.Computation of U

U1n , where n

4.2.3. Square Root of a U-number Let us consider computation of U

A

U1 based on the extension principle for computation with U -numbers.

A1 is determined as follows: A

D [ A1 ]D ,

D [0,1]

(12)

65


[ A1 ]D

{ x1 x1 A1D } .

(13)

The probability distribution p is determined given p1 as17 p( x)

2 xp1 ( x 2 ) .

(14)

Then we compute P p by solving problem (3)-(4) and recall that

P p ( pl )

P p ( p1l ) , 1

where p is determined from p1 on the basis of (14). Next we compute probability measure P ( A) . Finally, given the membership function P p , we construct a fuzzy set B by solving a problem analogous to (7)-(8).Let us mention that for the square root of a U-number, it is not needed to carry out computation of B because it is the same as B1 17,18. 5. Approximate reasoning with usual information

The approximate reasoning can be considered as a formal model of commonsense knowledge-based reasoning with imprecise and uncertain information19,20,21. Approximate reasoning is based on fuzzy logic22,23 and has found a lot of successful applications in various fields24,25. The problem of approximate reasoning with usual information is started as follows. Given the following U-rules: If X1 is U X1 ,1 ( AX1 ,1 , BX1 ,1 ) and,…, and X m is U X m ,1 ( AX m ,1 , B X m ,1 ) then Y is U Y ( AY ,1 , BY ,1 ) If X1 is U X1 ,2

( AX1 ,2 , B X1 ,2 ) and,…, and X m is U X m ,2

( AX m ,2 , BX m ,2 ) then Y is U Y

··· If X1 is U X1 , n ( AX1 , n , BX1 , n ) and,…, and X m is U X m , n ( AX m , n , BX m , n ) then Y is U Y and a current observation X1 is U X1 ( AXc 1 , BXc 1 ) and,…, and X m is U Xc m ( AXc m , BXc m ) ,

( AY ,2 , BY ,2 ) ( AY , n , BY , n )

find the U-value of Y . The idea underlying the suggested interpolation approach is that the resulting output should be computed as a convex combination of consequent parts. The coefficients of linear interpolation are determined on the basis of the similarity between a current input and the antecedent parts26. This implies for U-rules that the resulting output U Yc is computed as n

U Yc

¦w U j

n

Y,j

j 1

¦ w (A j

Y, j

, BY , j ) ,

(15)

j 1

where U Y , j is the U -valued consequent of the j-th rule, w j

Uj n

, j 1,..., n; k

1,..., n are coefficients of linear

¦ Uk k 1

interpolation, n is the number of U-rules. U j is defined as follows

Uj

min i

1,..., m

S (U Xc i , U X i , j ) ,

(16)

where S is the similarity between current i-th U-valued input and the i-th U-valued antecedent of the j-th rule. Thus, U j computes the similarity between a current input vector and the vector of the antecedents of j-th rule.

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6. Conclusion The concept of usuality underlines as usual all human decision making and reasoning on the basis of commonsense knowledge. L. Zadeh outlined a theory of usuality based on a method of representing the meaning of usuality-qualified propositions. However, this topic requires further investigation. It is needed more general approach for computation and approximate reasoning with usual information. Up to day, no systematic approach is suggested to solving such problems of usuality theory as computation of usual values of random variables, combination of usual values of two and more variables, reasoning with usuality-based IF-Then rules, decision making in usuality-qualified environment etc. We tried to provide in this study more effective approach to computation with U-numbers and commonsense reasoning on the basis of usual information. We considered a Unumber as a special case of a Z-number in which the second component “usually” may take one of the fuzzy quantifiers of usuality and developed a systematic framework for approximate arithmetic operations on U-numbers.

Acknowledgement

I would like to deeply thank Prof. L.A. Zadeh for his outstanding idea to develop a theory of approximate arithmetic operations on U-numbers and I am grateful to him for his valuable comments and suggestions. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Zadeh LA. Fuzzy sets and commonsense reasoning. Berkeley:Institute of Cognitive Studies report 21, University of California; 1984. Zadeh LA. Fuzzy sets as a basis for the management of uncertainty in expert systems. Fuzzy Sets and Systems 1983;11: 199-227. Zadeh LA. A computational theory of dispositions. In: Proc. of 1984 Int. Conference on Computation Linguistics. Stanford:1984. p.312-318. Zadeh LA. Syllogistic reasoning in fuzzy logic and its application to reasoning with dispositions. IEEE Trans. on Systems. Man and Cybernetics 1985; 15: 754-763. Zadeh LA. Outline of a theory of usuality based on fuzzy logic. Fuzzy sets, fuzzy logic, and fuzzy systems. USA: World Scientific Publishing Co., Inc. River Edge; 1996.p. 694-712. Yager RR. On Implementing Usual Values. UAI '86: Proceedings of the Second Annual Conference on Uncertainty in Artificial Intelligence. Philadelphia: University of Pennsylvania;1986. Shafer G. A Mathematical Theory of Evidence. Princeton: Princeton University Press; 1976. Whalen T, Usuality SB. Regularity, and fuzzy set logic. International Journal of Approximate Reasoning 1992 ; 64: 81-504. Schwartz DG. A Logic for Qualified Syllogisms. Advanced Techniques in Computing Sciences and Software Engineering 2009 ;45-50. Charles M, Grinstead J, Snell L. Introduction to Probability. American Mathematical Society; 1997. Springer MD. The Algebra of Random Variables. New York:John Wiley and Sons Inc; 1979. Williamson RC, Downs T. Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds. Int. J. Approx. Reason.1990; 4: 89–158. Pedrycz W, Gomide F. Fuzzy Systems Engineering. Toward Human-Centric Computing. New Jersey: John Wiley & Sons, Hoboken;2007. Zadeh LA. Probability measures of fuzzy events.J. Math. Anal. Appl.1968; 23: 421–427. Williamson RC. Probabilistic Arithmetic. Austrialia : University of Queensland;1989, http://theorem.anu.edu.au/~williams/papers/thesis300dpi.ps. Wise BP, Henrion M. A framework for comparing uncertain inference systems to probability. In: L.N. Kanal & J.F. Lemmer, editors. Uncertainty in artificial intelligence. Amsterdam: Elsevier Science Publishers; 1986 .p. 69-83. Aliev RA, Huseynov OH, Aliyev RR, Alizadeh AV. The Arithmetic of Z-numbers. Theory and Applications. Singapore:World Scientific; 2015 Aliev RA, Alizadeh AV, Huseynov OH. The arithmetic of discrete Z-numbers. Information Sciences 2015; 290: 134-155. Aliev RA, Mamedova GA, Aliev RR. Fuzzy Sets Theory and its application. Tabriz:Tabriz University; 1993. Aliev RA, Pedrycz W, Huseynov OH. Decision theory with imprecise probabilities. Int J Inf Tech Decis 2012; 11(02): 271-306.. Aliev RA, Alizadeh AV, Guirimov BG. Unprecisiated information-based approach to decision making with imperfect information. Proc. of the Ninth Inter. Conference on Application of Fuzzy Systems and Soft Computing, ICAFS-2010. Prague:2010. p. 387-397. Aliev RA, Tserkovny A. Systemic approach to fuzzy logic formalization for approximate reasoning. Information Sciences 2011;181:10451059. Aliev RA. Fuzzy expert systems. Soft computing Prentice-Hall Inc; 1994. p. 99-108. Aliev RA . Fundamentals of the fuzzy logic-based generalized theory of decisions. New York,Berlin:Springer; 2013. Aliev RA, Memmedova K. Application of Z-Number Based Modeling in Psychological Research. Computational Intelligence and Neuroscience 2015 ;Article ID 760403; 7 pages Kóczy LT, Hirota K. Rule Interpolation by Į-Level Sets in Fuzzy Approximate Reasoning. J. BUSEFAL 1991; 46 :115-123.