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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.

© 2016 The Authors.Publishedby ElsevierB.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 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

* Corresponding author. Tel.: 0099412 5984509; fax: 0099412 5984509. E-mail address:raliev@asoa.edu.az

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

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 structures7^8 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 X, and X2 be two independent continuous random variables with pdfs pj and p2. A pdf pj2 of Xj2 = Xj * X2, where * is a two-place operation, is referred to as a convolution of pj and p2 (pdf of a random variable Xj2 obtained as a result of a two-place operation over Xj and X2 ) and is defined as follows.

pj2(*) = }}pj(X)p2(x2)dx,dx2,Q. = {(x,,x2)|x = x, * x2} .

Let Xj and X2 be two independent discrete random variables with the corresponding outcome spaces Xj = {x1j,...,xji,...,xj } and X2 = {x2j,...,x2i,...,x2n } and the corresponding discrete probability distributions pj and p2. The probability distribution of Xj * X2, *e {+, -, •,/} , comes as the convolution pj2 = pj o p2 of pj and p2 which is defined for any x e {xj * x2 |xj e Xj,x2 e X2} , xj e Xj , x2 e X2 as follows:

pj2(x) = S pj(xJ)P2(x2).

x=I, • 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) = (x)p(x)dx .

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

P(A) = £ Ma (xi)p(xi) = Ma (x,)p(x,) + Ma (x2 )p(x2 ) + ... + Ma (xn )p(xn ) .

3. U-numbers

Let X be a random variable and A be a fuzzy number playing a role of fuzzy constraint on values that the random variable may take: X is A. The definition of a usual value of X may be expressed in terms of the probability distribution ofX as follows5. If p(xi) is the probability ofXtaking xt as its value, then

USUally (X is A) = Vusualiy (X tP(xi *)Va (x,-)) (!)

usually (X is A) = ^ost (S ¡P(Xi )^A (Xi )) (2)

A usual number describing, "usually, professor's income is medium" is shown fig.1.

Fig.1. An example of U-number

Formula (2) indicates that the probability that the event A occurs as the value for the variable X, is "most'. As it was mentioned above, in5 Zadeh provided an outline for the theory of usuality, however this topic requires further investigation. It is needed a more general approach for other usuality quantifiers. In this paper "usuality" will be a composite term characterized by fuzzy quantities as always, usually, frequently / often, occasionally, seldom, almost never/rarely, never. The codebook for "usuality" is shown in fig.2.

Fig.2. The codebook of the fuzzy quantifiers of usuality

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 Xj2 _ Xj + X2 , and the

usual values of Xj and X 2 are given, what will be the usual value of Xj2 ?

- 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 Xj and X2 are dependent or independent. This will influence how a usuality quantifier related to the result Xj2 = X, * X2 should be determined on the basis of the usuality quantifiers related to X, and X2. 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 x,, 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 ^ Prob {X is Aj is usually ^ ¡J.mualiy ¡J.A (x)p(x)dxj ,

Mc (y) = suPp(x) (MusuaUy MA (x)p(x)dx)) , subject to

y = L mB (x)P(x)dx .

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, = (A,,B,) and U2 = (A2,B2) be U-numbers ( B, and B2 are fuzzy terms of the usuality codebook) describing values of random variables X, and X2. Assume that it is needed to compute the result U,2 = (A,2, B,2) of a two-place operation *e {+, -, •, /} : U,2 = U, * U2 .Computation of one-place operations U = U,2 and U = ,JU\ 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 A, and A2 on the basis of fuzzy arithmetic. For example, for sum U,2 = U, + U2 we have to calculate A,2 = A, + A2.

The second stage involves step-by-step construction of B12 and is related to propagation of probabilistic restrictions. We realize that in U-numbers U1 - (A1,B1)and U2 - (A2,B2) , the 'true' probability distributions p1 and p2 are not exactly known. In contrast, the information available is represented by the fuzzy restrictions:

Z (x1k )P1 (x1k ) is B1 Z (X2k )P2 (X2k ) iS B2

which are represented in terms of the membership functions as

( n ^ ( »2 ^

Mp,( Pi) = ZM Xlk ) M Xk ) /"p2( P2) = ZM ) Pl( X2k )

^ k=1 y ^ k=1

Given these fuzzy restrictions, extract probability distributions Pj, j = 1,2 by solvingthe following goal linear programming problem:

C1v! + C2V2 + ... + CnV'n ^ bjl

subject to

v1 + v2 + ... + v[ = 1 V, v2,..., vn > 0

where ct = jj.a (xjt) andvk = Pj(xjk), k = 1,..,nj, k = 1,..,nj .As a result, pJl(xjk),k = 1,..,» is found and, therefore, distribution pjl is obtained. Thus, to construct thedistributions pjl, we need to solve m simple problems (3)-(4).

Distributions p 1 (xjk),k = 1,..,nj naturally induce probabilistic uncertainty over the result X12 = X1 *X2. This is a critical point of computation of U-numbers, at which the issue of dependence between X1 and X2 should be considered. For simplicity, here we consider the case of independence between X1 and X2. This implies that given a pair p1i,p2lj , the convolutionp12s = p1l] o p2 ,s = 1,...,m2 is to be computed as on the basis of Definition 1.

For the case of dependence between X1 and X2, p12s should be computed as a joint probability distribution by taking into account dependence between random variables15,16.

Given p12s, the value of probability measure of A12can be computed: P(A12) = Z/°A (x12k)p12(x12k) . However,

the 'true' p12s is not exactly known as the 'true' pu , p2l are described by fuzzy restrictions. These fuzzy restrictions induce the fuzzy set of convolutions p12s,s = 1,...,m2 with the membership function defined as

MPn (P12.S) = max

Pu, = Pu, 0Pllj

[^p, (Put ) A ^ (P22 )]

subject to

MPj (Pjlj ) = Mbj (Xjk )PjlJ ( Xjk )

j = 1,2

where a is min operation.

As a result, fuzziness of information on p12s described by /u induces fuzziness of the value of probability measure P(A12) in a form ofa fuzzy number B12. The membership function of B12 is defined as

MB12(b12s ) = max(^p12( p12s )) (7)

subject to

b12s xk ) p12s (xk ) (8)

As a result, U12 - U1 * U2 is obtained as U12 = (A12, B12) .

4.2.2. Square of a U-number

Let us now consider construction ofU = U12. A = A2 is determined as follows:

A12 = U a[A12]a, (9)

ae[0,1]

[A?]a = {xf |x e 4°}. (10)

The probability distribution p is determined given p1 as17

p(X)=[)+ p1(~^*)], x^o. (11)

Next by noting that a 'true' p1 is not known, one has to consider fuzzy constraint /u to be constructed by solving a certain LP problem (3)-(4).The fuzzy set of probability distributions pu with membership function /u naturally induces the fuzzy set of probability distributions pj with the membership function /xp (pl) defined as Mp (pt) = Ap1( pu), 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 /j.p , we extend P(A) to a fuzzy set B by solving a problem analogous to (7)-(8). As a result, U2 is obtained on the basis of the extension principle for computation with U -numbers as U2 = (A, B).Let us mention that for X1 > 0 , it is not needed to compute ofB because it is the same as B117,18.Computation of U = U1", where n is any natural number, is carried out in an analogous fashion.

4.2.3. Square Root of a U-number

Let us consider computation of U = iJU\ based on the extension principle for computation with U -numbers.

A = is determined as follows:

-Ja = u «k/Aja

ae[0,1]

u/4f= Wx]x 6 Aa}. (13)

The probability distribution p is determined given p as17

p( x) = 2 xpi( x2). (14)

Then we compute ¡j. by solving problem (3)-(4) and recall that

Ap( pi ) = MPl( PU

where p is determined from pu on the basis of (14). Next we compute probability measure P(A) . Finally, given the membership function ¡j., we construct a fuzzy setB 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 ofB because it is the same as Bu 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 Xu is UX u = (AX] 1, BX u) and,..., and Xm is Ux_ u = (AX_ 1, BX_ u)then Y is UY = (Arl, BY1) If Xuis UX] 2 = (AX] 2,BX] 2) and,., and Xmis UX 2 = (AX 2,BX 2)then Yis UY = (AY2,BY2)

If Xiis Uxi,n = (Axi,n, Bxi,n) and,..., and Xm is Uxm,n = (Axm,n, BXm„ )then Y is UY = (AYn, BYn) and a current observation

Xi is UXx = (A'xu , B'xu ) and,., and Xm is U'x_ = (A'x_ , B'x_ ) , 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 UY is computed as

U'r = £ WjUYj ^Wj ( Ay, j , By, j ) , (15)

j=i j=i

whereUYj is the U -valued consequent of the j-th rule, wj =——— , j - 1,...,n; k - 1,...,n are coefficients of linear

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

Pi = mini.i.....m S(U'Xi,Ux,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, pj computes the similarity between a current input vector and the vector of the antecedents ofj-th rule.

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 U-number 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.

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