# Efficient Algorithms for Data Processing under Type-3 (and Higher) Fuzzy Uncertainty

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## Abstract

**:**

## 1. Outline

## 2. Why Data Processing

## 3. Need for Fuzzy Uncertainty and Need for Higher-Order Fuzzy Uncertainty

#### 3.1. Need for Fuzzy Uncertainty

**Definition**

**1**

#### 3.2. Fuzzy Numbers

**Definition**

**3**

**.**A fuzzy set $m:\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}\to [0,1]$ is called a fuzzy number if it satisfies the following two conditions:

- We have $m\left(x\right)\to 0$ when $x\to -\infty $ and when $x\to +\infty $.
- There exists a number ${x}_{0}$ such that $m\left(x\right)$ is (non-strictly) increasing for $x\le {x}_{0}$ and (non-strictly) decreasing for $x\ge {x}_{0}$.

#### 3.3. “And”- and “Or”-Operations (T-Norms and T-Conorms)

**Definition**

**4**

**.**An “and”-operation (t-norm) is a function ${f}_{\&}:[0,1]\times [0,1]\to [0,1]$ that satisfies the following properties for all a, b, ${a}^{\prime}$, ${b}^{\prime}$, and c:

- ${f}_{\&}(a,b)={f}_{\&}(b,a)$ (commutativity);
- ${f}_{\&}(a,{f}_{\&}(b,c))={f}_{\&}({f}_{\&}(a,b),c)$ (associativity);
- if $a\le {a}^{\prime}$ and $b\le {b}^{\prime}$, then ${f}_{\&}(a,b)\le {f}_{\&}({a}^{\prime},{b}^{\prime})$ (monotonicity);
- ${f}_{\&}(0,a)=0$ and ${f}_{\&}(1,a)=a$.

**Definition**

**5**

**.**An“or”-operation (t-conorm) is a function ${f}_{\vee}:[0,1]\times [0,1]\to [0,1]$ that satisfies the following properties for all a, b, ${a}^{\prime}$, ${b}^{\prime}$, and c:

- ${f}_{\vee}(a,b)={f}_{\vee}(b,a)$ (commutativity);
- ${f}_{\vee}(a,{f}_{\vee}(b,c))={f}_{\vee}({f}_{\vee}(a,b),c)$ (associativity);
- if $a\le {a}^{\prime}$ and $b\le {b}^{\prime}$, then ${f}_{\vee}(a,b)\le {f}_{\vee}({a}^{\prime},{b}^{\prime})$ (monotonicity);
- ${f}_{\vee}(0,a)=a$ and ${f}_{\vee}(1,a)=1$.

#### 3.4. Operations on Fuzzy Sets

**Definition**

**6**

**.**Let U be a set and let ${m}_{1}:U\to [0,1]$ and ${m}_{2}:U\to [0,1]$ be fuzzy sets; then:

- by the intersection ${m}_{\cap}={m}_{1}\cap {m}_{2}$ of these fuzzy sets, we mean the set ${m}_{\cap}\left(x\right)=min({m}_{1}\left(x\right),$${m}_{2}\left(x\right))$;
- by the union ${m}_{\cup}={m}_{1}\cup {m}_{2}$ of these fuzzy sets, we mean the set ${m}_{\cup}\left(x\right)=max({m}_{1}\left(x\right),{m}_{2}\left(x\right))$.

#### 3.5. Data Processing under Fuzzy Uncertainty

- We know that the quantity-of-interest y is a function $y=f({x}_{1},{x}_{2},\dots )$ of several auxiliary quantities ${x}_{1},{x}_{2},\dots $
- We also know, for each i, the membership function ${m}_{i}\left({x}_{i}\right)$ that describes, for each real number ${x}_{i}$, the degree to which this number is a possible value of the i-th input.

**Definition**

**7**

**.**Let ${U}_{1},{U}_{2},\dots ,U$ be sets, let ${m}_{i}:{U}_{i}\to [0,1]$ be fuzzy sets, and let $f:{U}_{1}\times {U}_{2}\times \dots \to U$ be a function. By the result $m=f({m}_{1},{m}_{2},\dots )$ of applying the function f to fuzzy sets ${m}_{i}$ we mean a fuzzy set $m:U\to [0,1]$ defined by the Equation (1).

#### 3.6. Need for Type-2 Fuzzy Technique

**Definition**

**8**

**Definition**

**9**

#### 3.7. Operations on Interval-Valued and General Type-2 Fuzzy Sets

**Definition**

**10**

**.**Let U be a set and let ${m}_{1}:U\to F\left([0,1]\right)$ and ${m}_{2}:U\to F\left([0,1]\right)$ be type-2 fuzzy sets; then:

- by the intersection ${m}_{\cap}={m}_{1}\cap {m}_{2}$ of these type-2 fuzzy sets, we mean the type-2 fuzzy set ${m}_{\cap}\left(x\right)=min({m}_{1}\left(x\right),{m}_{2}\left(x\right))$, where, for each x, the result $min({m}_{1}\left(x\right),{m}_{2}\left(x\right))$ of applying the function $f(a,b)=min(a,b)$ to fuzzy sets ${m}_{1}\left(x\right)$ and ${m}_{2}\left(x\right)$ is defined by Definition 7.
- by the union ${m}_{\cup}={m}_{1}\cup {m}_{2}$ of these type-2 fuzzy sets, we mean the set ${m}_{\cup}\left(x\right)=max({m}_{1}\left(x\right),$${m}_{2}\left(x\right))$, where, for each x, the result $max({m}_{1}\left(x\right),{m}_{2}\left(x\right))$ of applying the function $f(a,b)=max(a,b)$ to fuzzy sets ${m}_{1}\left(x\right)$ and ${m}_{2}\left(x\right)$ is defined by Definition 7.

#### 3.8. Data Processing under Type-2 Fuzzy Uncertainty

- We know that the quantity-of-interest y is a function $y=f({x}_{1},{x}_{2},\dots )$ of several auxiliary quantities ${x}_{1},{x}_{2},\dots $.
- We also know, for each i, the membership function ${m}_{i}\left({x}_{i}\right)$ that describes, for each real number ${x}_{i}$, the (fuzzy-valued) degree to which this number is a possible value of the i-th input.

**Definition**

**11**

**.**Let ${U}_{1},{U}_{2},\dots ,U$ be sets, let ${m}_{i}:{U}_{i}\to F\left([0,1]\right)$ be type-2 fuzzy sets, and let $f:{U}_{1}\times {U}_{2}\times \dots \to U$ be a function. By the result $m=f({m}_{1},{m}_{2},\dots )$ of applying the function f to type-2 fuzzy sets ${m}_{i}$ we mean a fuzzy set $m:U\to F\left(\right[0,1\left]\right)$ defined by the Equation (1), in which the right-hand side is understood according to Definition 7.

#### 3.9. Need for Type-3 and Higher-Order Fuzzy Techniques

**Definition**

**12**

**.**Let U be a set, and let ${F}_{2}\left([0,1]\right)$ denote the set of all type-2 fuzzy subsets of the interval $[0,1]$. By a type-3 fuzzy subset of U, or, for short, a type-3 fuzzy set, we mean a function $m:U\to {F}_{2}\left([0,1]\right)$.

#### 3.10. Operations on Type-3 Fuzzy Sets

**Definition**

**13**

**.**Let U be a set and let ${m}_{1}:U\to {F}_{2}\left([0,1]\right)$ and ${m}_{2}:U\to {F}_{2}\left([0,1]\right)$ be type-3 fuzzy sets; then:

- by the intersection ${m}_{\cap}={m}_{1}\cap {m}_{2}$ of these type-3 fuzzy sets, we mean the type-3 fuzzy set ${m}_{\cap}\left(x\right)=min({m}_{1}\left(x\right),{m}_{2}\left(x\right))$, where, for each x, the result $min({m}_{1}\left(x\right),{m}_{2}\left(x\right))$ of applying the function $f(a,b)=min(a,b)$ to type-2 fuzzy sets ${m}_{1}\left(x\right)$ and ${m}_{2}\left(x\right)$ is defined by Definition 11.
- by the union ${m}_{\cup}={m}_{1}\cup {m}_{2}$ of these type-3 fuzzy sets, we mean the type-3 fuzzy set ${m}_{\cup}\left(x\right)=max({m}_{1}\left(x\right),{m}_{2}\left(x\right))$, where, for each x, the result $max({m}_{1}\left(x\right),{m}_{2}\left(x\right))$ of applying the function $f(a,b)=max(a,b)$ to type-2 fuzzy sets ${m}_{1}\left(x\right)$ and ${m}_{2}\left(x\right)$ is defined by Definition 11.

#### 3.11. Is This Worth Considering?

#### 3.12. Data Processing under Type-3 Fuzzy Uncertainty

- We know that the quantity-of-interest y is a function $y=f({x}_{1},{x}_{2},\dots )$ of several auxiliary quantities ${x}_{1},{x}_{2},\cdots $.
- We also know, for each i, the membership function ${m}_{i}\left({x}_{i}\right)$ that describes, for each real number ${x}_{i}$, the (type-2-fuzzy-valued) degree to which this number is a possible value of the i-th input.

**Definition**

**14.**

#### 3.13. What about Higher Order Types?

**Definition**

**15.**

**Definition**

**16.**

- by the intersection ${m}_{\cap}={m}_{1}\cap {m}_{2}$ of these type-L fuzzy sets, we mean the type-L fuzzy set ${m}_{\cap}\left(x\right)=min({m}_{1}\left(x\right),{m}_{2}\left(x\right))$, where, for each x, the result $min({m}_{1}\left(x\right),{m}_{2}\left(x\right))$ of applying the function $f(a,b)=min(a,b)$ to type-$(L-1)$ fuzzy sets ${m}_{1}\left(x\right)$ and ${m}_{2}\left(x\right)$ is defined by Definition 14 (for $L=4$) or Definition 17 (for other L).
- by the union ${m}_{\cup}={m}_{1}\cup {m}_{2}$ of these type-L fuzzy sets, we mean the type-L fuzzy set ${m}_{\cup}\left(x\right)=max({m}_{1}\left(x\right),{m}_{2}\left(x\right))$, where, for each x, the result $max({m}_{1}\left(x\right),{m}_{2}\left(x\right))$ of applying the function $f(a,b)=max(a,b)$ to type-$(L-1)$ fuzzy sets ${m}_{1}\left(x\right)$ and ${m}_{2}\left(x\right)$ is defined by Definition 14 (for $L=4$) or Definition 17 (for other L).

**Definition**

**17.**

#### 3.14. Need for Data Processing under Such Uncertainty

## 4. Effective Algorithms for Data Processing under Type-1 Fuzzy Uncertainty: Reminder

#### 4.1. How to Actually Perform Data Processing: Analysis of the Problem

**Definition**

**18**

**.**Let U be a set, let $m:U\to [0,1]$ be a fuzzy set, and let $\alpha \in [0,1]$ be a real number. Then, by the$\alpha $-cut of m, we mean the following set:

- when $\alpha >0$, we take $\{x:m(x)\ge \alpha \}$;
- when $\alpha =0$, we take $\overline{\{x:m(x)>0\}}$.

#### 4.2. Comment

#### 4.3. Resulting Algorithm

- First, if the information about the inputs ${x}_{i}$ is stored in the form of the usual membership functions ${m}_{i}\left({x}_{i}\right)$, we compute, for each i and for each value $\alpha \in \{0,0.1,\dots ,1.0\}$, the corresponding $\alpha $-cut$${\mathbf{x}}_{i}\left(\alpha \right)=\{{x}_{i}:{m}_{i}\left({x}_{i}\right)\ge \alpha \}.$$(Recall that for $\alpha =0$, we will have to use a slightly more complex formula.)
- Then, for each value $\alpha $ from the above list, we use an interval computation algorithm to compute the range $\mathbf{y}\left(\alpha \right)=f({\mathbf{x}}_{1}\left(\alpha \right),{\mathbf{x}}_{2}\left(\alpha \right),\dots ).$ These ranges form the $\alpha $-cut representation of the desired membership function $m\left(y\right)$.
- Finally, if we want to represent this membership function in the usual form, we compute $m\left(y\right)=max\{\alpha :y\in \mathbf{y}(\alpha \left)\right\}$.

#### 4.4. How Many Computation Steps Do We Need

## 5. Data Processing under Interval-Valued Fuzzy Uncertainty: Reminder

#### 5.1. Formulation of the Problem

#### 5.2. Interval Case: Analysis of the Problem

#### 5.3. Interval Case: Resulting Algorithm

- Based on each of these membership functions, for each i and for each value $\alpha $ from the given list, we compute the orrepsonding $\alpha $-cuts as:$${\underline{\mathbf{x}}}_{i}\left(\alpha \right)=\{{x}_{i}:{\underline{m}}_{i}\left({x}_{i}\right)\ge \alpha \}\mathrm{and}{\overline{\mathbf{x}}}_{i}\left(\alpha \right)=\{{x}_{i}:{\overline{m}}_{i}\left({x}_{i}\right)\ge \alpha \}.$$
- We compute the $\alpha $-cuts $\underline{\mathbf{y}}\left(\alpha \right)$ and $\overline{\mathbf{y}}\left(\alpha \right)$ for the endpoints $\underline{m}\left(y\right)$ and $\overline{m}\left(y\right)$ of the interval-valued membership function $[\underline{m}\left(y\right),\overline{m}\left(y\right)]$ as follows:$$\underline{\mathbf{y}}\left(\alpha \right)=f({\underline{\mathbf{x}}}_{1}\left(\alpha \right),{\underline{\mathbf{x}}}_{2}\left(\alpha \right),\dots )\mathrm{and}\overline{\mathbf{y}}\left(\alpha \right)=f({\overline{\mathbf{x}}}_{1}\left(\alpha \right),{\overline{\mathbf{x}}}_{2}\left(\alpha \right),\dots ).$$
- Finally, the compute the endpoints $\underline{m}\left(y\right)$ and $\overline{m}\left(y\right)$ of the desired interval-valued membership function $[\underline{m}\left(y\right),\overline{m}\left(y\right)]$ as$$\underline{m}\left(y\right)=max\{\alpha :y\in \underline{\mathbf{y}}\left(\alpha \right)\}\mathrm{and}\overline{m}\left(y\right)=max\{\alpha :y\in \overline{\mathbf{y}}\left(\alpha \right)\}.$$

#### 5.4. How Many Computation Steps Do We Need

## 6. Data Processing under General Type-2 Fuzzy Uncertainty: Reminder

#### 6.1. Formulation of the Problem

#### 6.2. General Type-2 Case: Analysis of the Problem

#### 6.3. General Type-2 Case: Resulting Algorithm

- First, for each i and for each value $\beta $ from the given list, we compute the $\beta $-cuts$$[{\underline{m}}_{i}\left({x}_{i}\right)\left(\beta \right),{\overline{m}}_{i}\left({x}_{i}\right)\left(\beta \right)]\stackrel{\mathrm{def}}{=}\{t:{m}_{i}({x}_{i},t)\ge \beta \}.$$
- Then, for each i and for each pair of values $(\alpha ,\beta )$ from the given list, we compute the $\alpha $-cuts$${\underline{\mathbf{x}}}_{i}(\alpha ,\beta )\stackrel{\mathrm{def}}{=}\{{x}_{i}:{\underline{m}}_{i}\left({x}_{i}\right)\left(\beta \right)\ge \alpha \}\mathrm{and}{\overline{\mathbf{x}}}_{i}(\alpha ,\beta )\stackrel{\mathrm{def}}{=}\{{x}_{i}:{\overline{m}}_{i}\left({x}_{i}\right)\left(\beta \right)\ge \alpha \}.$$
- For each $\alpha $ and $\beta $, we then use an interval computation algorithm to compute:$$\underline{\mathbf{y}}(\alpha ,\beta )=f({\underline{\mathbf{x}}}_{1}(\alpha ,\beta ),{\underline{\mathbf{x}}}_{2}(\alpha ,\beta ),\dots )\mathrm{and}$$$$\overline{\mathbf{y}}(\alpha ,\beta )=f({\overline{\mathbf{x}}}_{1}(\alpha ,\beta ),{\overline{\mathbf{x}}}_{2}(\alpha ,\beta ),\dots ).$$
- Based on these intervals, for each $\beta $, we compute$$\underline{m}\left(y\right)\left(\beta \right)=sup\{\alpha :y\in \underline{\mathbf{y}}(\alpha ,\beta )\}\mathrm{and}\overline{m}\left(y\right)\left(\beta \right)=sup\{\alpha :y\in \overline{\mathbf{y}}(\alpha ,\beta )\}.$$
- Finally, we compute the desired membership function$$m(y,t)=max\{\beta :t\in [\underline{m}\left(y\right)\left(\beta \right),\overline{m}\left(y\right)\left(\beta \right)]\}.$$

#### 6.4. How Many Computation Steps Do We Need

## 7. Data Processing under Type-3 (and Higher Order) Fuzzy Uncertainty: A New Algorithm

#### 7.1. Formulation of the Problem

#### 7.2. Type-3 Case: Analysis of the Problem

#### 7.3. Type-3 Case: Resulting Algorithm

- First, for every i and for all $\gamma $ from the selected list of values, we compute:$$[{\underline{m}}_{i}({x}_{i},t)\left(\gamma \right),{\overline{m}}_{i}({x}_{i},t)\left(\gamma \right)]\stackrel{\mathrm{def}}{=}\{s:{m}_{i}({x}_{i},t,s)\ge \gamma \}.$$
- Then, for each i, $\beta $, and $\gamma $, we compute:$$[{\underline{m}}_{i}^{-}\left({x}_{i}\right)(\beta ,\gamma ),{\underline{m}}_{i}^{+}\left({x}_{i}\right)(\beta ,\gamma )]\stackrel{\mathrm{def}}{=}\{t:{\underline{m}}_{i}({x}_{i},t)\left(\gamma \right)\ge \beta \}\mathrm{and}$$$$[{\overline{m}}_{i}^{\phantom{\rule{0.166667em}{0ex}}-}\left({x}_{i}\right)(\beta ,\gamma ),{\overline{m}}_{i}^{\phantom{\rule{0.166667em}{0ex}}+}\left({x}_{i}\right)(\beta ,\gamma )]\stackrel{\mathrm{def}}{=}\{t:{\overline{m}}_{i}({x}_{i},t)\left(\gamma \right)\ge \beta \}.$$
- Then, for each i, $\alpha $, $\beta $, and $\gamma $, we compute$${\underline{\mathbf{x}}}_{i}^{-}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{{x}_{i}:{\underline{m}}_{i}^{-}\left({x}_{i}\right)(\beta ,\gamma )\ge \alpha \},$$$${\underline{\mathbf{x}}}_{i}^{+}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{{x}_{i}:{\underline{m}}_{i}^{+}\left({x}_{i}\right)(\beta ,\gamma )\ge \alpha \},$$$${\overline{\mathbf{x}}}_{i}^{\phantom{\rule{0.166667em}{0ex}}-}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{{x}_{i}:{\overline{m}}_{i}^{\phantom{\rule{0.166667em}{0ex}}-}\left({x}_{i}\right)(\beta ,\gamma )\ge \alpha \},$$$${\overline{\mathbf{x}}}_{i}^{\phantom{\rule{0.166667em}{0ex}}+}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{{x}_{i}:{\overline{m}}_{i}^{\phantom{\rule{0.166667em}{0ex}}+}\left({x}_{i}\right)(\beta ,\gamma )\ge \alpha \}.$$
- For each $\alpha $, $\beta $, and $\gamma $, we then use an interval computation algorithm to compute:$${\underline{\mathbf{y}}}^{-}(\alpha ,\beta ,\gamma )=f({\underline{\mathbf{x}}}_{1}^{-}(\alpha ,\beta ,\gamma ),{\underline{\mathbf{x}}}_{2}^{-}(\alpha ,\beta ,\gamma ),\dots ),$$$${\underline{\mathbf{y}}}^{+}(\alpha ,\beta ,\gamma )=f({\underline{\mathbf{x}}}_{1}^{+}(\alpha ,\beta ,\gamma ),{\underline{\mathbf{x}}}_{2}^{+}(\alpha ,\beta ,\gamma ),\dots ),$$$${\overline{\mathbf{y}}}^{\phantom{\rule{0.166667em}{0ex}}-}(\alpha ,\beta ,\gamma )=f({\overline{\mathbf{x}}}_{1}^{\phantom{\rule{0.166667em}{0ex}}-}(\alpha ,\beta ,\gamma ),{\overline{\mathbf{x}}}_{2}^{\phantom{\rule{0.166667em}{0ex}}-}(\alpha ,\beta ,\gamma ),\dots ),$$$${\overline{\mathbf{y}}}^{\phantom{\rule{0.166667em}{0ex}}+}(\alpha ,\beta ,\gamma )=f({\overline{\mathbf{x}}}_{1}^{\phantom{\rule{0.166667em}{0ex}}+}(\alpha ,\beta ,\gamma ),{\overline{\mathbf{x}}}_{n}^{\phantom{\rule{0.166667em}{0ex}}+}(\alpha ,\beta ,\gamma ),\dots ).$$
- Next, for each y, $\beta $, and $\gamma $, we compute$${\underline{m}}^{-}\left(y\right)(\beta ,\gamma )=max\{\alpha :y\in {\underline{\mathbf{y}}}^{-}(\alpha ,\beta ,\gamma )\},$$$${\underline{m}}^{+}\left(y\right)(\beta ,\gamma )=max\{\alpha :y\in {\underline{\mathbf{y}}}^{+}(\alpha ,\beta ,\gamma )\},$$$${\overline{m}}^{\phantom{\rule{0.166667em}{0ex}}-}\left(y\right)(\beta ,\gamma )=max\{\alpha :y\in {\overline{\mathbf{y}}}^{\phantom{\rule{0.166667em}{0ex}}-}(\alpha ,\beta ,\gamma )\},$$$${\overline{m}}^{\phantom{\rule{0.166667em}{0ex}}+}\left(y\right)(\beta ,\gamma )=max\{\alpha :y\in {\overline{\mathbf{y}}}^{\phantom{\rule{0.166667em}{0ex}}+}(\alpha ,\beta ,\gamma )\}.$$
- For each y, t, and $\gamma $, we compute$$\underline{m}(y,t)\left(\gamma \right)=max\{\beta :t\in [{\underline{m}}^{-}\left(y\right)(\beta ,\gamma ),{\underline{m}}^{+}\left(y\right)(\beta ,\gamma )]\}and$$$$\overline{m}(y,t)\left(\gamma \right)=max\{\beta :t\in [{\overline{m}}^{\phantom{\rule{0.166667em}{0ex}}-}\left(y\right)(\beta ,\gamma ),{\overline{m}}^{\phantom{\rule{0.166667em}{0ex}}+}\left(y\right)(\beta ,\gamma )]\}.$$
- Finally, for all y, t and s, we compute$$m(y,t,s)=max\{\gamma :s\in [\underline{m}(y,t)\left(\gamma \right),\overline{m}(y,t)\left(\gamma \right)]\}.$$

#### 7.4. What about Higher Order Fuzzy Sets?

#### 7.5. How Many Computational Steps Do We Need

- For type-1, for each y, the desired information $m\left(y\right)$ consists of a single number. In this case, if we use 11 values of $\alpha $, we need to use an interval computation algorithm 11 times.
- For type-2, for each y, we need to find the values $m(y,t)$ corresponding to different values $t\in [0,1]$. If we use 11 values for t, we thus need at least 11 times more computations than in the type-1 case—and indeed, we need order of $11\times 11$ calls to an interval computation algorithm—namely, $2\times {11}^{2}$ calls.
- For type-3, for each y, we need to find the values $m(y,t,s)$ corresponding to different values $t,s\in [0,1]$. If we use 11 values of each of the variables t and s, we thus need at least ${11}^{2}$ times more computations than in the type-1 case—and indeed, we need order of ${11}^{2}\times 11={11}^{3}$ calls to an interval computation algorithm—namely, ${2}^{2}\times {11}^{3}$ calls.
- In general, for type-L, for each y, we need to find the values $m(y,{t}_{1},\dots ,{t}_{L-1})$ corresponding to different values ${t}_{1},\times ,{t}_{L-1}\in [0,1]$. If we use 11 values for each of the variables ${t}_{i}$, we thus need at least ${11}^{L-1}$ times more computations than in the type-1 case—and indeed, as one can show by induction over L, we need order of ${11}^{L-1}\times 11={11}^{L}$ calls to an interval computation algorithm—namely, ${2}^{L-1}\times {11}^{L}$ calls.

## 8. Conclusions and Future Work

#### 8.1. Conclusions

#### 8.2. Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Kreinovich, V.; Kosheleva, O.; Melin, P.; Castillo, O.
Efficient Algorithms for Data Processing under Type-3 (and Higher) Fuzzy Uncertainty. *Mathematics* **2022**, *10*, 2361.
https://doi.org/10.3390/math10132361

**AMA Style**

Kreinovich V, Kosheleva O, Melin P, Castillo O.
Efficient Algorithms for Data Processing under Type-3 (and Higher) Fuzzy Uncertainty. *Mathematics*. 2022; 10(13):2361.
https://doi.org/10.3390/math10132361

**Chicago/Turabian Style**

Kreinovich, Vladik, Olga Kosheleva, Patricia Melin, and Oscar Castillo.
2022. "Efficient Algorithms for Data Processing under Type-3 (and Higher) Fuzzy Uncertainty" *Mathematics* 10, no. 13: 2361.
https://doi.org/10.3390/math10132361