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Uncertain systems are universal approximators
Journal of Uncertainty Analysis and Applications volume 2, Article number: 13 (2014)
Abstract
Uncertain inference is a process of deriving consequences from uncertain knowledge or evidences via the tool of conditional uncertain set. Based on uncertain inference, uncertain system is a function from its inputs to outputs. This paper proves that uncertain systems are universal approximators, which means that uncertain systems are capable of approximating any continuous function on a compact set to arbitrary accuracy. This result can be viewed as an existence theorem of an optimal uncertain system for a wide variety of problems.
Introduction
Fuzzy systems were developed from fuzzy set theory initialized by Zadeh [1] in 1965. In recent decades, fuzzy systems have been successfully applied to a wide variety of practical problems, such as fuzzy control, fuzzy identification, fuzzy expert system. Generally speaking, there are two types of widely used fuzzy systems: Mamdani fuzzy systems [2] and Takagi-Sugeno fuzzy systems [3]. The main difference between these fuzzy systems lies on inference rules, such as Zadeh’s compositional rule of inference, Lukasiewicz’s inference rule [4], Mamdani’s inference rules [2, 5] for Mamdani fuzzy systems. On the other hand, Wang [6] proved that fuzzy systems are universal approximators which is an important theoretical basis for the application of it.
Although fuzzy systems are successfully used in engineering problems, it isn’t perfect in its theoretical framework. For example, membership functions are adopted to describe the fuzzy sets used in fuzzy inference rules, however, the overuse maximum operations in fuzzy set theory is impeached. Surveys have shown that the inputs and outputs of practical systems should not be fuzzy sets. This promotes Liu [7] to define uncertain set and introduced uncertain systems based Liu’s inference rules. Then, Gao Gao and Ralescu [8] extended the Liu’s inference rule to multi-antecedent. As an application of uncertain system, Gao [9] used the uncertain controller to successfully control the inverted pendulum. Besides, uncertain logic was introduced by Liu [10] via uncertain set. Universal approximation capability of uncertain systems is the basis of almost all the theoretical research and practical applications of uncertain systems. In this paper, we will prove that uncertain systems are universal approximators, and this result can be viewed as an existence theorem of an optimal uncertain system for a wide variety of problems.
The rest of the paper is organized as follows. Section ‘Preliminaries’ introduces some basic concepts and results on uncertain sets as preliminaries. The framework of Liu’s inference rule and uncertain systems are introduced in Section ‘Inference rule and uncertain systemsInference rule and uncertain systems’. It is shown that uncertain systems are universal approximators in Section ‘Uncertain systems are universal approximators’. Section ‘Conclusions’ concludes this paper with a brief summary.
Preliminaries
Liu [11] founded an uncertainty theory that is based on normality, duality, subadditivity and product axioms. Gao [12] studied some properties of uncertain measure. Since then, considerable work has been done based on uncertainty theory. Liu [13] founded uncertain programming to model optimization problems in uncertain environments. Liu [14] proposed the concept of uncertain process and uncertain calculus. Li and Liu [15] proposed uncertain logic. And Liu [16] studied entailment based on uncertain proposition logic. Recently, uncertain systems are suggested by Liu [7] in 2010 via uncertain inference and uncertain set theory. Uncertainty theory has become a new tool to describe belief degree and has applications in practical use. For a detailed exposition, the interested reader may consult the book [17] to know more about uncertainty theory. In this section, we will introduce some definitions related on uncertain systems. Let Γ be a nonempty set, and let L be a σ-algebra over Γ. Each element Λ∈Γ is called an event. In order to measure uncertain events, Liu [11] introduced an uncertain measure as a set function satisfying normality, self-duality, and countable subadditivity axioms. Then the triplet is called an uncertainty space.
Let Γ be a nonempty set, anda σ-algebra over Γ. An uncertain measure
(Liu [11]) is a set function defined on
satisfying the following four axioms:
Axiom 1
(Normality Axiom) ;
Axiom 2.
(Duality Axiom) for any event Λ;
Axiom 3.
(Subadditivity Axiom) For every countable sequence of events {Λ i }, we have
Axiom 4.
(Product Axiom [18]) Let be uncertainty spaces for k=1,2,…. The product uncertain measure is an uncertain measure satisfying
where Λ k are arbitrarily chosen events from L k for k=1,2,…, respectively.
Definition 1
(Liu [7]) An uncertain set is a measurable function ξ from an uncertainty space to a collection of sets, i.e., both {B⊂ξ} and {ξ⊂B} are events for any Borel set B.
Let ξ and η be two nonempty uncertain sets. What is the appropriate event that ξ is included in η? We may suggest that {ξ⊂η}={γ|ξ(γ)⊂η} intuitively. The set {B⊄ξ}={γ|B⊄ξ(γ)} and {ξ⊄B}={γ|ξ(γ)⊄B} are also events.
Definition 2.
(Liu [19]) An uncertain set ξ is said to have a membership function μ if for any Borel set B of real numbers, we have
The above equations will be called measure inversion formulas.
Example 1
By a rectangular uncertain set we mean the uncertain set fully determined by the pair (a, b) of crisp numbers with a<b, whose membership function is
Example 2.
By a triangular uncertain set we mean the uncertain set fully determined by the triplet (a, b, c) of crisp numbers with a<b<c, whose membership function is
Example 3.
By a trapezoidal uncertain set we mean the uncertain set fully determined by the quadruplet (a, b, c, d) of crisp numbers with a<b<c<d, whose membership function is
Liu [17] proved that a real-valued function μ is a membership function if and only if 0≤μ(x)≤1.
Definition 3.
(Liu [19]) Let ξ be an uncertain set with membership function μ. Then the set-valued function
is called the inverse membership function of ξ. Sometimes, the set μ−1(α) is called the α-cut of μ.
The rectangular uncertain set ξ=(a, b) has an inverse membership function
The triangular uncertain set ξ=(a, b, c) has an inverse membership function
The trapezoidal uncertain set ξ=(a, b, c, d) has an inverse membership function
Liu [17] proposed the conditional membership function of an uncertain set ξ after some event A has occurred:
and
Definition 4.
(Liu [20]) The uncertain sets ξ1, ξ2,…, ξ n are said to be independent if for any Borel sets B1, B2,…, B n , we have
and
where are arbitrarily chosen from , i=1,2,…, n, respectively.
Definition 5.
(Liu [20]) Let ξ and η be independent uncertain sets with membership functions μ and ν, respectively. Then their union ξ∩η has a membership function
Definition 6.
(Liu [20]) Let ξ and η be independent uncertain sets with membership functions μ and ν, respectively. Then their union ξ∪η has a membership function
Definition 7.
(Liu [20]) Let ξ be an uncertain sets with membership functions μ. Then their union ξc has a membership function
Definition 8.
(Liu [20]) Let ξ1, ξ2,…, ξ n be independent uncertain sets with inverse membership functions μ1(x), μ2(x),…, μ n (x), respectively. If f is a measurable function, then the uncertain set ξ=f(ξ1, ξ2,…, ξ n ) has an inverse membership function,
Inference rule and uncertain systems
Uncertain inference was proposed by Liu [7] as a process of deriving consequences from uncertain knowledge or evidence via the tool of conditional uncertain set. Then Gao, Gao and Ralescu [8] extended Liu’s inference rule to the one with multiple antecedents and with multiple if-then rules. Besides, Gao [9] applied uncertain inference to invert pendulum. Furthermore, uncertain set was extended to the field of logic (Liu [10]) where the uncertain quantifier, uncertain subject and uncertain predicate are described by uncertain sets. In this section, we will introduce the concept and some properties of uncertain inference.
Liu’s Inference Rule [[7]]. Let and
be two concepts. Assume rules ‘if
is an uncertain set ξ then
is an uncertain set η’. From
is a constant a, we infer that
is an uncertain set
which is the conditional uncertain set of η given a∈η. The inference rule is represented by

It has been proved by Liu [7] that the membership function of η∗ satisfying
Liu’s Inference Rule (multiple antecedent [[8]]). Let , and be three concepts. Assume rules ‘if
is an uncertain set ξ and
is η then
is an uncertain set τ.’ From is a constant a and
is a constant b, respectively, we infer that
is an uncertain set.
which is conditional uncertain set of τ of given a∈ξ and b∈η. The inference rule is represented by

It has been proved by Gao, Gao, and Ralescu [8] that
Liu’s Inference Rule (Multiple If-Then Rules). Let and
be two concepts. Assume two rules ‘if
is an uncertain set ξ1 then
is an uncertain set η1’ and ‘if
is an uncertain set ξ2 then
is an uncertain set η2’. From
is a constant a, we infer that
is an uncertain set
The inference rule is represented by

If ξ1, ξ2, η1, η2 are independent uncertain sets with continuous membership functions μ1, μ2, ν1, ν2, respectively, then the inference rule yields
where are uncertain sets whose membership functions are
For a general system, Liu [17] proposed the following inference rule.
Liu’s Inference Rule (general form) [[17]]. Let be m concepts. Assume the rules ‘if is ξi 1 and ⋯ and is ξ
i
m
, then is for i=1,2,…, k. From is a1 and ⋯ and is a
m
, we infer that
is an uncertain set
where the coefficients are determined by
The inference rule is represented by

Assume ξi 1, ξi 2,…, ξ i m , η i are independent uncertain sets with membership functions μi 1, μi 2,…, μ i m , ν i , i=1,2,…, k, respectively. If are constants a1, a2,…, a m , respectively. It has been proved by Liu [17] that
where are uncertain sets whose membership functions are given by
Uncertain systems are universal approximators
An uncertain system, introduced by Liu [7], is a function from its inputs to outputs via Liu’s inference rule, in which five parts are contained: input, a rule-base, Liu’s inference rule, an expected value operator and outputs. An uncertain system is a function f:(α1, α2,⋯, α m )→(β1, β2,⋯, β n ), namely,
Then we get an uncertain system f. Gao [9] has succeeded in controlling inverted pendulum. In this section, we will prove that uncertain systems are universal approximators. That is, uncertain systems are capable of approximating any continuous function on a compact set to arbitrary accuracy. Since there are great flexibilities in constructing rules, we have different methods to construct uncertain systems. The quantity of Liu’s inference rules in an uncertain system is under consideration.
Theorem 1
For any given continuous function g on a compact set D⊂ℜm, and any given ε>0, there exists an uncertain system f such that
Proof
Without loss of generality, assume g is a real-valued function with two variables α1 and α2 and D=[0,1]2. Since D is a compact set and g is a continuous function, we know that g is uniformly continuous on D. Thus, for any given number ε>0, there exists a number δ>0 such that
where ||(α1, α2)−(α 1′, α 2′)||<δ. Let k be the minimum integer larger than . Let
Note that Di, j are disjoint sets and ||(α1, α2)−(i/k, j/k)||≤δ, wherever (α1, α2)∈Di, j. Let ξ i and η j be independent rectangular uncertain sets with membership functions μ i and ν i
where i, j=1,2,…, k, respectively. Next, we construct a rule-base like,
where i, j=1,2,…, k, respectively. Hence, we get an uncertain system f. From a detailed investigation using definition of uncertain system and Liu’s inference rule, we know that the uncertain system is
Then we have
The theorem is proved.
Theorem 1 shows that the uncertain systems can approximate continuous functions. Next, we will extend the result in the sense of mean square convergence.
Theorem 2.
For any given function g on a compact set D⊂ℜm and any given ε>0, there exists an uncertain system f such that
Proof.
Without loss of generality, let g be a real-valued function with two variables. Note that D is a compact set, and we have . Then there exists a continuous function h on D such that
It follows from Theorem 1 that there exists an uncertain system f satisfying
where . Hence,
Theorem 2 shows that uncertain systems are capable of L2 approximating functions which are not required continuously, including simple functions and piecewise continuous function.
Conclusions
In this paper, we prove that uncertain systems are universal approximators, and a method to construct an uncertain system to approximate any continuous function on a compact set to arbitrary accuracy is given. Hence, uncertain systems are capable of approximating any continuous function on a compact set. Furthermore, uncertain systems are capable of approximating functions, which are not continuous, in the sense of mean square convergence.
References
Zadeh L: Fuzzy sets. Inf. Control. 1965, 8: 338–353.
Mamdani EH: Applications of fuzzy algorithms for control of a simple dynamic plant. Proc. IEEE 1974, 121(12):1585–1588.
Takagi T, Sugeno M: Fuzzy identification of system and its applications to modeling and control. IEEE Trans. Syst. Man Cybernatics 1985, 15(1):116–132.
Lukasiewicz J: Selected Works - Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, Warsaw 1970.
Mamdani EH: Application of fuzzy logic to approximate reasoning using linguistic synthesis. IEEE Trans. Comput. 1977, C-26(12):1182–1191.
Wang L: Fuzzy systems are universal approximators. In: IEEE International Conference on Fuzzy Systems. San Diego, 8–12 March 1992;
Liu B: Uncertain set theory and uncertain inference rule with application to uncertain control. J. Uncertain Syst 2010, 4(2):83–98.
Gao X, Gao Y, Ralescu DA: On Liu’s inference rule for uncertain systems. Int. J. Uncertainty Fuzziness Knowledge-Based Syst 2010, 18(1):1–11. 10.1142/S0218488510006349
Gao Y: Uncertain inference control for balancing an inverted pendulum. Fuzzy Optimization Decis. Making 2012, 11(4):481–492. 10.1007/s10700-012-9124-y
Liu B: Uncertain logic for modeling human language. J. Uncertain Syst 2011, 5(1):3–20.
Liu B: Uncertainty Theory. Springer, Berlin; 2007.
Gao X: Some properties of continuous uncertain measure. Int. J. Uncertainty Fuzziness Knowledge-Based Syst 2009, 17(3):419–426. 10.1142/S0218488509005954
Liu B: Theory and Practice of Uncertain Programming. Springer, Berlin; 2009.
Liu B: Fuzzy process, hybrid process and uncertain process. J. Uncertain Syst 2008, 2(1):3–16.
Li X, Liu B: Hybrid logic and uncertain logic. J. Uncertain Syst 2009, 3(2):83–94.
Liu B: Uncertain entailment and modus ponens in the framework of uncertain logic. J. Uncertain Syst 2009, 3(4):243–251.
Liu B: Uncertainty Theory: a Branch of Mathematics for Modeling Human Uncertainty. Springer, Berlin; 2010.
Liu B: Some research problems in uncertainty theory. J. Uncertain Syst 2009, 3(1):3–10.
Liu B: Membership functions and operational law of uncertain sets. Fuzzy Optimization Decis. Making 2012, 11(4):387–410. 10.1007/s10700-012-9128-7
Liu B: A new definition of independence of uncertain sets. Fuzzy Optimization Decis. Making 2013, 12(4):451–461. 10.1007/s10700-013-9164-y
Acknowledgements
This work was supported by National Natural Science Foundation of China Grant No. 61304182.
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Peng, Z., Chen, X. Uncertain systems are universal approximators. J. Uncertain. Anal. Appl. 2, 13 (2014). https://doi.org/10.1186/2195-5468-2-13
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DOI: https://doi.org/10.1186/2195-5468-2-13
Keywords
- Uncertainty theory
- Uncertain inference
- Uncertain system
- Universal approximation