Why Unary and Binary Operations in Logic: General Result Motivated by Interval-Valued Logics Hung T

University of Texas at El Paso DigitalCommons@UTEP

Departmental Technical Reports (CS) Department of Computer Science

3-1-2001 Why Unary and Binary Operations in Logic: General Result Motivated by Interval-Valued Logics Hung T. Nguyen

Vladik Kreinovich University of Texas at El Paso, [email protected]

I. R. Goodman

Follow this and additional works at: http://digitalcommons.utep.edu/cs_techrep Part of the Computer Engineering Commons Comments: UTEP-CS-01-07. Published in the Proceedings of the Joint 9th World Congress of the International Fuzzy Systems Association and 20th International Conference of the North American Fuzzy Information Processing Society IFSA/NAFIPS 2001, Vancouver, Canada, July 25-28, 2001, pp. 1991-1996.

Recommended Citation Nguyen, Hung T.; Kreinovich, Vladik; and Goodman, I. R., "Why Unary and Binary Operations in Logic: General Result Motivated by Interval-Valued Logics" (2001). Departmental Technical Reports (CS). Paper 371. http://digitalcommons.utep.edu/cs_techrep/371

This Article is brought to you for free and open access by the Department of Computer Science at DigitalCommons@UTEP. It has been accepted for inclusion in Departmental Technical Reports (CS) by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. Why Unary and Binary Operations in Logic: General Result Motivated by Interval-Valued Logics

Hung T. Nguyen Vladik Kreinovich Mathem. Sciences, New Mexico State Univ. Comp. Science, Univ. of Texas at El Paso Las Cruces, NM 88003, USA El Paso, TX 79968, USA [email protected] [email protected] I.R. Goodman Code D42215, Space & Naval Warfare Systems Center San Diego, CA 92152, USA [email protected]

Abstract We are interested in explaining why unary and binary logical operations are the only basic ones. If we assume Traditionally, in logic, only unary and binary operations that the logic of human reasoning is the two-valued are used as basic ones – e.g., “not”, “and”, “or” – while (classical) logic, then the possibility to transform every the only ternary (and higher order) operations are the logical function into a DNF form explains this empirical operations which come from a combination of unary fact. and binary ones. For the classical logic, with the binary set of truth values ¢¡¤£¦¥¨§ , the possibility to express an However, classical logic is not a perfect description of arbitrary operation in terms of unary and binary ones is human reasoning: for example, it does not take into con- well known: it follows, e.g., from the well known pos- sideration fuzziness and uncertainty of human reason-

sibility to express an arbitrary operation in DNF form. ing. This uncertainty is taken into consideration in fuzzy ¡¤£¦¥

A similar representation result for © -based logic was logic [9, 25, 29]. In the traditional fuzzy logic, the set

¡¤£¥ ( proven in our previous paper. In this paper, we expand of truth values is the entire interval © . This this result to ﬁnite logics (more general than classical interval has a natural notion of continuity, so it is natu- logic) and to multi-D analogues of the fuzzy logic – both ral to restrict ourselves to continuous unary and binary motivated by interval-valued fuzzy logics. operations.

With this restriction in place, a natural question is: can

¡£¥ ¡¤£¥

#+% © an arbitrary continuous function )!*© be represented as a composition of continuous unary and 1. Introduction binary operations? The positive answer to this question was obtained in our papers [19, 22].

Traditionally, in logic, only unary and binary operations ¡£¥ In © -based fuzzy logic, an arbitrary logical operation are used as basic ones – e.g., “not”, “and”, “or” – while can be represented as a composition of unary and binary the only ternary (and higher order) operations are the ¡¤£¦¥ ones. However, the © -based fuzzy logic is, by itself, operations which come from a combination of unary only an approximation to the actual human reasoning and binary ones. about uncertainty. A natural question is: are such combinations sufﬁcient? Indeed, how can we describe the expert’s degree of con-

I.e., to be more precise, can an arbitrary logical op- / ﬁdence ,.-0/21 in a certain statement ? A natural way eration be represented as a combination of unary and to determine this degree is, e.g., to ask an expert to es-

binary ones? timate his degree of conﬁdence on a scale from 0 to 10.

¥¦¡ 6587

For the classical logic, with the binary set of truth values If he selects 8, then we take ,3-4/21 .

¨¨£ ¤¨§ ¡£¥§ (= ), the positive answer to this To get a more accurate result, we can then ask the same

question is well known. Indeed, it is known that an arbi- expert to estimate his degree of conﬁdence on a ﬁner $#&%'

trary logical operation "! can be represented, scale, e.g., from 0 to 100, etc. For example, if an expert

¥ ¥¦¡9¡ ¡¤: ¥

7 5 e.g., in DNF form and thus, it can indeed be represented 5 selects 81, we will take ,.-0/21 . If we as a combination of unary (“not”) and binary (“and” and want an even more accurate estimate, we can ask the “or”) operations.

expert to estimate his degree of conﬁdence on an even Uncertainty of expert estimates is only one reason why ¡¤£¥

ﬁner scale, etc. we may want to go beyond the traditional © -valued logic; there are also other reasons: The problem with this approach is that experts cannot

describe their degrees of too ﬁne scales. For example, an expert can point to 8 on a scale from 0 to 10, but this A 1-D value is a reqsonable way of describing the same expert will hardly be able to pinpoint a value on a uncertainty of a single expert. However, the con- ¯ scale from 0 to 100. ﬁdence strongly depends on the it consensus be- tween different experts. We may want to use ad-

So, to attain a more adequate description of human ditional dimensions to describe how many expert ¡£¥ reasoning, we must modify the traditional © -based share the original expert’s opinion, and to what de- fuzzy logic. Two types of modifications have been pro- gree; see, e.g., [13, 23]. posed. Different experts may strongly disagree. To de- One possibility is to take the finest (finite) scale which scribe the dgeree of this disagreement, we also an expert can still use, and take the values on this scale need additional numerical characteristics, which as the desired degrees of confidence. This approach make the resulting logic multi-D; see, e.g., [21]. leads to a finite-valued fuzzy logic, in which the set of truth values is finite. In all these cases, we need a multi-D logic to adequately This approach has been successfully used in practice; describe expert’s degree of confidence. see, e.g., [1, 5, 20, 26]. It is therefore desirable to check In this paper, we show that both for finite-valued logics whether in a finite logic, every operation can be repre- and for multi-D logics, every logical operation can be sented as a composition of unary and binary operations. represented as a composition of unary and binary op- The problem with finite-valued logics is that the set erations. Thus, we give a general explanation for the of resulting truth values depends on which scale we use. above empirical fact.

Instead of ﬁxing a ﬁnite set, we can describe the expert’s ¡¤£¦¥

degree of confidence by an interval from © . For example, if an expert estimates his degree of confidence 2. Finite-Valued Logics by a value 8 on a 0 to 10 scale, then the only thing that we know about the expert’s degree of confidence is that 2.1. What Was Known Before

it is closer to 0.8 (8/10) than to 0.7 or to 0.9, i.e., that it

¡¤: £¡¤:

¢¡ 5£¡

belongs to the interval © . In the Introduction, we have already mentioned that for

¡£¥¨§

So, a natural way of describing degrees of conﬁdence the 2-element set of truth values , an ar-

# %

© ¥§¦ ¥©¨ more adequately is to use intervals ¤ instead bitrary logical operation can be repre- of real numbers. In this representation, real numbers can sented as a composition of unary and binary operations.

be viewed as particular – degenerate – cases of intervals Speciﬁcally, in this case, an arbitrary logical operation

¥ ¥

© . The idea of using intervals have been originally can be represented as a composition of negation , proposed by Zadeh himself and further developed by conjunction , and disjunction . Bandler and Kohout [2], Turks¨ ¸en [27], and others; for a In [6], we proved that the same is true for the case when

recent survey, see, e.g., [24]. is a ﬁnite Boolean algebra. Speciﬁcally, we prove

In interval-valued fuzzy approach, to describe each de- that for such sets , an arbitrary logical operation can be gree of conﬁdence, we must describe two real numbers: represented as a composition of negation, conjunction the lower endpoint and the upper endpoint of the corre- (“intersection”), disjunction (“union”), constants, and a sponding “conﬁdence interval”. special unary operation called absolute truth:

We can go one step further and take into consideration Deﬁnition 1. For an arbitrary Boolean algebra ,

-¥ 1

that the endpoints of the corresponding interval are also we deﬁne an absolute truth operation as follows:

- 1 -¥ 1 ¥ not precisely known. Thus, each of these endpoints is, and for all .

in actuality, an interval itself. So, to describe a degree of

-31 The function is similar to the delta-function (see, conﬁdence, we now need four real numbers: two to de- e.g., [30]), which is deﬁned, crudely speaking, as a

scribe the lower endpoint, and two to describe the upper function which is different from 0 only at one point ¡ one.

. It is even more similar to Kronecker’s “delta”

¥ ¡

In general, we get a multi-D fuzzy logic. A natural ques- which is deﬁned as if and

-¥ 1 &%' ( tion is: can every (continuous) operation on a multi-D when # : we can easily see that . fuzzy logic be represented as a composition of (continuous) unary and and binary operations?

# %

Proposition [6]. For every ﬁnite Boolean algebra , can be represented as a composition of

and for every positive integer , every -ary operation unary and binary operations. This result is based on the

# %

can be represented as a composition of , following known result:

¥ , , , and constants. Theorem (Kolmogorov). Every continuous function of This representation has the form three or more variables can be represented as a compo-

¡ sition of continuous functions of one or two variables.

£¢ £ ::¦: £¢ £ £¦::¦: £ £

# # #

-¥¤ ¥¦¤1 - ¥ 1 -¥§¤ ¥ 1

£::¦: £ #

# lution to the conjecture of Hilbert, formulated as the

-¥ ¥ 1©¨

where is taken over all tuples ¤ , and

£¢ £

thirteenth problem [8]: one of 22 problems that Hilbert

¥ 1 - is deﬁned as

has proposed in 1900 as a challenge to the XX century

¢ £

mathematics.

- ¥ 1 £ -- ¥ 1 )- ¥ 11

£¢ £

This problem can be traced to the Babylonians, who

¥ 1 ¥

and is equal to - when , and to

£¢ £

found (see, e.g., [3]) that the solutions of quadratic

¥ 1 ¥

- when .

"! # "$

equations ¥© (viewed as function of

! $ This representation is similar to the known possibil- three variables ¥ , , and ) can be represented as super- ity to represent an arbitrary function as a linear compositions of functions of one and two variables, namely, bination of delta-functions, so-called “sifting property” arithmetic operations and square roots. Much later, sim-

([30], pp. 11 and ff.): ilar results were obtained for functions of ﬁve variables

! $ , %

¥ , , , , , that represent the solution of quartic equa-

£¦:::£

-¥¤ 1

¥©'&( )! '*+ )$ 6,£, )%

tions . But then, Ga-

lois proved in 1830 that for higher order equations, we

:¦:: £¦::¦:£ £ £¦::¦:£ ::: £

# # #

- ¥ ¥ 1¦ ¥ ¥

¤ ¤

¤ cannot have such a representation. This negative result

%© % has caused Hilbert to conjecture that not all functions of

where several variables can be represented by functions of two

£::¦:£ : ::¦:

or fewer variables. Hilbert’s conjecture was refuted by

# # #

¥ 1 -¥ ¥ 1 - ¥ 1 -

¤ ¤ ¤ Kolmogorov (see, e.g., [14], Chapter 11) and his student

V. Arnold. For the case when each variable only takes integer value, we can have another analogue of the same result, It is worth mentioning that Kolmogorov’s result is not

with Kronecker’s “delta” instead of a delta-function: only of theoretical value: it was used to speed up actual

£¦:::£

computations (see, e.g., [4, 7, 11, 12, 16, 17]).

- 1

::¦: :¦:: £::¦:£ :

% % -¥§¤ ¥ 1

3.2. New Result

% % Let us show that one can generalize Kolmogorov’s the- 2.2. New Result orem and prove that a similar representation holds for

multi-D logics as well.

Let us show that a similar representation can be used to Let - be a positive integer, and let be a closure of

prove this result for an arbitrary ﬁnite set . a simply connected bounded open set in .0/ (e.g., of a

Theorem 1. For every ﬁnite set , and for every posi- convex set). Such a set will be called a multi-D set

$#"%

of truth values. For example, for interval-valued fuzzy ! tive integer , every -ary operation can

sets,

£ ¡43 3 3 ¥¨§8:

be represented as a composition of unary and binary

¥ !121 ¥ ! operations. -

(For reader’s convenience, all the proofs are placed in Theorem 2. For every multi-D set of truth values ,

the special Proofs section.) and for every positive integer , every continuous -

$# % ! ary operation can be represented as a composition of continuous unary and binary operations. 3. Multi-D Logics

3.1. What Was Known Before 4. Proofs

In the Introduction, we have already mentioned that for Proof of Theorem 1. To describe the desired represen-

¡¤£¦¥

- and denote them and . We will then deﬁne three in the -dimensional space, where by

binary operations:

£¦:::¦£ £

¦ ¦

¢- 1

¢ £ £¢ £

¥ ! 1 - ¥ ! 10 £ ¥ ! ¦ - is deﬁned as when ,

we denoted the -th component of the point

£¢ £

£::¦:£

¦© ¦ ©

- ¥ ! 1 £ ¥ !

- 1 . /

and when . . Therefore, each -valued function

# %

/ . / "! -

can be represented as real-valued

¥ 3 3

#&%

¥ ¥ ¥

is deﬁned in such a way that

. ! -

functions , .

¨ ¥ !

for all ¥ ; when and , we can take

! .

arbitrary values for ¥ , e.g., we can assume that Each of these functions maps ele-

in this case, ¥ . ments from (i.e., components) into a real num-

ber. Therefore, each of these functions can be repre-

¥ ¥ ¥

is deﬁned in such a way that

sented as a real-valued function of real variables

# #

£:¦::£ : £¦:::¦£ £:¦::£

¤ ¤

¦ ¦ ¦ ¦

¨ ¥ !

for all ¥ ; when and , we can take

- ¤

¤ Each of these func-

/ /

arbitrary values for ¥ , e.g., we can assume that

tions can be (due to Kolmogorov’s theorem) repre-

! in this case, ¥ . sented as a composition of functions of one and two variables. So, to represent the original function of Now, it can easily checked that an arbitrary operation

£¦:::¦£ variables from as a composition of functions of one

- 1

¤ from to can be represented as or two variables from , we can do the following:

£::¦: £

£¢ £ ::¦: £¢ £ £¦::¦: £ £

¦ ¦ ¦©

# # #

- 1

First, we apply, to each input ¤ ,

£:¦::¦£

¦ ¦

- - 1 - 1

functions ¤ of one -valued variable

£::¦: £

£:¦:: £

¦ ¦ ¦

- ¤ 1©¨

- ¥ ¤ ¥ 1 ¨ where is taken over all tuples . which transform an element into

Q.E.D. corresponding “degenerate” elements /

£:¦::£ £)::¦:£

¦ ¦

Proof of Theorem 2. This proof is similar to the one ¦

- 1 £ - 1

¤ ¤

presented in [28]. ¤

£:¦::£ £ :¦::£

¦ ¦

First, since the set is bounded, we can embed it into a ¦

- 1 £ - 1

£ ¤£ ::¦:¥£ £

¢¡ ¡ ¢¡ ¡

box , i.e., a set of all points

£:¦::£ 3 :¦:: 3

£¦::¦: £ :

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦

- 1 1 1 1 1

¤ ¤ ¡ ¡

- 1 - 1

for which , , .

/ /

An arbitrary continuous function on a compact set /

When we apply these - functions to input ele-

can be extended to the entire box , so we can assume ¦©

- - 1

ments, we get degenerate elements

£¦:::¦£

that maps into .

¦ ¦

- - 1 , for all from 1 to and for all from

Let us prove that every continuous function 1 to .

# %

! . / of three or more variables can be repre-

sented as a composition of continuous unary and binary Next, we follow the operations from Kolmogorov’s

% theorem with these degenerate elements, and get the

. / § ! . / ¨ ! functions and . “degenerate”-valued functions

Kolmogorov theorem proves that such a representation

¥ ¥

£¦::¦: £

¦ ¦ -

is possible for - . Based on the case , we

- 1 £ ¤

can now prove the theorem for all - , by using the fol-

£::¦:£ £¦:::£ £:¦:: £ £

# #

¤ ¤

¦ ¦ ¦

lowing argument (its idea is similar to the one presented ¦

-4 - 1 - 1 1

¤ ¤

in [18]): :::£

£:¦:: £

¦ ¦ © ¦ ©

- 1

We have a function of variables

£:¦::£

¦ ¦

- 1

£:¦:: £ £

¤ ¤

¦ ¦ ¦

- 1 ¨

£::¦:£ £¦:::£ £:¦::£ £

# #

¤ ¤

¦ ¦ ¦ ¦

-0 - 1 - 1 1

/ / ::¦:£

as the desired compositions of .0/ -valued functions of

£:¦:: £ :

# #

¦ ¦ ¦

. /

1¨

- one or two -valued variables.

£:¦::£

¦ ¦

¦ © ¦©

- 1 1

For each input - , the value Finally, we use combination functions , . . . ,

£ £¦::¦:£

¦ ¦

- 1

to combine the functions ¤ into a

£:¦::£

/ /

¦ ¦ ¦

- 1 single function . Namely, these functions work as fol-

lows:

£::¦: £ £ £::¦: £

¦ ¦ ¦

of this function is a point ¦

- - 1 - 1 1 £

£¦:::£

£ £:¦::£ £

¦ ¦

¦ ¦ ¦

- 1

- ¤ 1

£::¦:£ £:¦::¦£ £:¦::£ :::£

# #

¤ ¤

¦ ¦ ¦ ¦

1 - 1 1 -0 ¤- /

£¦::¦:£ £ £¦::: £ £::¦: £ £ £¦::¦:

¦ ¦ ¦ ¦ ¦ ¦

-- 1 - 11

¤ ¤

justiﬁed by the known possibility to represent an arbi-

¤ ¤

£ £¦:::¦£ £

£¦::¦:£ trary -ary logical operation as a composition of unary

¦ ¦ ¦ ¦

- ¤ ¤ 1

/ and binary ones. A similar representation result is true

¡£¥ ¡£¥

::¦:£ ©

£ £ £¦::¦: £ £

£:¦:: £ based fuzzy logic is only an approximation to the actual

¦ ¦ ¦ ¦ ¦ ¦

- - ¤ ¤ 1 - 11 ¤

¤ human reasoning about uncertainty. A more accurate

/ / /

/ /

£¦::¦:£ £ :

¦ ¦

¦ description of human reasoning requires that we take

- 1

¤ ¤

/ ¦ / into consideration the uncertainty with which we know

We apply these combination functions to the values ¡£¥

£:¦::¦£

the values from the interval . This additional un- ¡£¥

produced by the functions ¤ , to get the re-

£ ::¦: £

/ certainty leads to two modiﬁcations of the -based

- - 1 1

sults ¤ , , ,

£ :¦::

* *

fuzzy logic: ﬁnite-valued logic and multi-D logic.

- 1

, As a result, we get:

¦ We show that for both modiﬁcations, an arbitrary -ary

£ £ £¦::¦: £ £

- 1 -0 1 ¤

¤ logical operation can be represented as a composition

of unary and binary ones. Thus, the above justiﬁcation

£ £ £ £¦:::£ £

- 1 -0 ¤ 1

for using only unary and binary logical operation as ba-

* * * * *

::¦:£ sic ones is still valid if we take interval uncertainty into

£¦:::¦£ £¦:::¦£ £

£ consideration.

- ¤ 1 -0 ¤ ¦1

* ¦ and ﬁnally,

Acknowledgments

£ £:¦:: £

- 1 -4 1

¤ ¤

/ / / / /

¦ This work was supported in part by NASA under co-

£::¦:£ :

¦ ¦

- 1 operative agreement NCC5-209, by Future Aerospace

Science and Technology Program (FAST) Center for

£¦::¦: £

¦ ¦

- 1 Thus, the function has been repre- Structural Integrity of Aerospace Systems, effort spon- sented as a composition of functions of one or two . / - sored by the Air Force Ofﬁce of Scientiﬁc Research, Air valued variables. The statement is proven. Force Materiel Command, USAF, under grant number F49620-00-1-0365, and by Grant No. W-00016 from This representation is not exactly what we want, be- the U.S.-Czech Science and Technology Joint Fund.

cause the corresponding unary and binary functions

% %

/ /

§ ! . ¨ ! . and can take values outside the original set of truth values . To complete the proof, we must therefore “compress” the corresponding References functions – e.g., by applying appropriate linear trans- formation to each coordinate in their ranges. After we [1] J. Agusti´i et al. “Structured local fuzzy logics in

compressed all these functions, we get not the original MILORD”, In: L. Zadeh, J. Kacrpzyk (eds.) Fuzzy function , but the “compressed” one, so, to get , we Logic for the Management of Uncertainty, Wiley, must apply “un-compression” (inverse linear transfor- N.Y., 1992, pp. 523–551. mation). [2] W. Bandler and L.J. Kohout, “Uniﬁed theory of

By deﬁnition, this “un-compression” ¡ is a unary oper- multi-valued logical operations in the light of the ation which transforms the original set into a larger checklist paradigm”, Proc. of IEEE Conference

set, so to need to make ¡ into a unary logical operation. on Systems, Man, and Cybernetics, Halifax, Nova

Since the set is simply connected, it is a retract of Scotia, Oct. 1984. /

. (see, e.g., [15], Ch. 8), i.e., there exists a continu-

% ¦

¦ [3] C.B. Boyer and U.C. Merzbach, A History of

!2. / ¢ - 1

ous mapping ¢ for which for all

¦ Mathematics, Wiley, N.Y., 1991.

¨ . Thus, as the desired unary logical operation, we

¦ ¦

&- 1 £ ¢ -¤¡+- 1 1 ¢

can take ¡ : the use of does not change [4] H.L. Frisch, C. Borzi, G. Ord, J.K. Percus, and

the result which was already in , but brings all the G.O. Williams, “Approximate Representation of values outside back into the set . Q.E.D. Functions of Several Variables in Terms of Func- tions of One Variable”, Physical Review Lett., 1989, Vol. 63, No. 9, pp. 927–929. 5. Conclusion [5] L. Godo, R. Lopez de Mantaras, C. Sierra, and A. Verdaguer, “MILORD: The Architecture and Traditionally, in logic, only unary and binary opera- management of Linguistically expressed Uncer- tions are used as basic ones. In traditional (2-valued) tainty”, International Journal of Intelligent Sys- logic, the use of only unary and binary operations is tems, 1989, Vol. 4, pp. 471–501. [6] I.R. Goodman and V. Kreinovich, “On Repre- [20] H.T. Nguyen and V. Kreinovich, “Possible new sentation and Approximation of Operations in directions in mathematical foundations of fuzzy Boolean Algebras”, International Journal of Intel- technology: a contribution to the mathematics of ligent Systems, 2001, Vol. 16 (to appear). fuzzy theory”, In: Nguyen Hoang Phuong and A. Ohsato (eds.), Proceedings of the Vietnam-Japan [7] R. Hecht-Nielsen, “Kolmogorov’s Mapping Neu- Bilateral Symposium on Fuzzy Systems and Appli- ral Network Existence Theorem”, IEEE Int’l Conf. cations VJFUZZY’98, HaLong Bay, Vietnam, 30th on Neural Networks, San Diego, 1987, Vol. 2, pp. September–2nd October, 1998, pp. 9–32. 11–14. [21] H.T. Nguyen, V. Kreinovich, and V. Shekhter, “On [8] D. Hilbert, “Mathematical Problems, lecture de- the Possibility of Using Complex Values in Fuzzy livered before the Int’l Congress of Mathematics Logic For Representing Inconsistencies”, Interna- in Paris in 1900,” translated in Bull. Amer. Math, tional Journal of Intelligent Systems, 1998, Vol. Soc., 1902, Vol. 8, pp. 437–479. 13, No. 8, pp. 683–714. [9] G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall, Upper [22] H.T. Nguyen, V. Kreinovich, and D. Sprecher, Saddle River, NJ, 1995. “Normal forms for fuzzy logic – an application of Kolmogorov’s theorem” International Journal [10] A. N. Kolmogorov, “On the Representation of on Uncertainty, Fuzziness, and Knowledge-Based Continuous Functions of Several Variables by Su- Systems, 1996, Vol. 4, No. 4, pp. 331–349. perposition of Continuous Functions of One Vari- able and Addition”, Dokl. Akad. Nauk SSSR, 1957, [23] H.T. Nguyen, V. Kreinovich, and B. Wu, Vol. 114, pp. 369–373. “Fuzzy/probability fractal/smooth”, Interna- tional Journal of Uncertainty, Fuzziness, and [11] V. Kurkova, “Kolmogorov’s Theorem Is Rele- Knowledge-Based Systems (IJUFKS), 1999, Vol. vant”, Neural Computation, 1991, Vol. 3, pp. 617– 7, No. 4, pp. 363–370. 622. [24] H.T. Nguyen, V. Kreinovich, and Q. Zuo, [12] V. Kurkova, “Kolmogorov’s Theorem and Multi- “Interval-valued degrees of belief: applications of layer Neural Networks”, Neural Networks, 1991, interval computations to expert systems and intel- Vol. 5, pp. 501–506. ligent control”, International Journal of Uncer- [13] C. Langrand, V. Kreinovich, and H.T. Nguyen, tainty, Fuzziness, and Knowledge-Based Systems “Two-dimensional fuzzy logic for expert sys- (IJUFKS), 1997, Vol. 5, No. 3, pp. 317–358. tems”, Sixth International Fuzzy Systems Associ- [25] H.T. Nguyen and E.A. Walker, First Course in ation World Congress, San Paulo, Brazil, July 22– Fuzzy Logic, CRC Press, Boca Raton, FL, 1999. 28, 1995, Vol. 1, pp. 221–224. [26] J. Puyol-Gruart, L. Godo, and C. Sierra. A special- [14] G.G. Lorentz, Approximation of functions, Halt, ization calculus to improve expert systems com- Reinhart, and Winston, N.Y., 1966. munication, Research Report IIIA 92/8, Institut [15] G. McCarty, Topology, Dover, New York, 1988. d’Investigacio´ en Intelligenicia` Artiﬁcial, Spain, May 1992. [16] M. Nakamura, R. Mines, and V. Kreinovich. “Guaranteed intervals for Kolmogorov’s theorem [27] I.B. Turks¨ ¸en, “Interval valued fuzzy sets based on (and their possible relation to neural networks)”, normal forms”, Fuzzy Sets and Systems, 1986, Vol. Interval Computations, 1993, No. 3, pp. 183–199. 20, pp. 191–210. [17] M. Ness, “Approximative versions of Kol- [28] T. Yamakawa and V. Kreinovich, “Why Funda- mogorov’s superposition theorem, proved con- mental Physical Equations Are of Second Order?”, structively”, J. Comput. Appl. Math., 1993. International Journal of Theoretical Physics, [18] V.M. Nesterov, “Interval analogues of Hilbert’s 1999, Vol. 38, No. 6, pp. 1763–1770. 13th problem”, In: Abstracts of the Int’l Confer- [29] L.A. Zadeh, “Fuzzy Sets”, Information and Con- ence Interval’94, St. Petersburg, Russia, March 7– trol, 1965, Vol. 8, pp. 338–353. 10, 1994, pp. 185–186. [30] A.H. Zemanian, Distribution theory and transform [19] H.T. Nguyen and V. Kreinovich, “Kolmogorov’s analysis, Dover, New York, 1987. Theorem and its impact on soft computing”, In: R.R. Yager and J. Kacprzyk, The Ordered Weighted Averaging Operators: Theory and Ap- plications, Kluwer, Boston, MA, 1997, pp. 3–17.