The α-universal triple I (α-UTI) method is a recognized scheme in the field of fuzzy reasoning, which was proposed by our research group previously. The robustness of fuzzy reasoning determines the quality of reasoning algorithms to a large extent, which is quantified by calculating the disparity between the output of fuzzy reasoning with interference and the output without interference. Therefore, in this study, the interval robustness (embodied as the interval stability) of the α-UTI method is explored in the interval-valued fuzzy environment. To begin with, the stability of the α-UTI method is explored for the case of an individual rule, and the upper and lower bounds of its results are estimated, using four kinds of unified interval implications (including the R-interval implication, the S-interval implication, the QL-interval implication and the interval t-norm implication). Through analysis, it is found that the α-UTI method exhibits good interval stability for an individual rule. Moreover, the stability of the α-UTI method is revealed in the case of multiple rules, and the upper and lower bounds of its outcomes are estimated. The results show that the α-UTI method is stable for multiple rules when four kinds of unified interval implications are used, respectively. Lastly, the α-UTI reasoning chain method is presented, which contains a chain structure with multiple layers. The corresponding solutions and their interval perturbations are investigated. It is found that the α-UTI reasoning chain method is stable in the case of chain reasoning. Two application examples in affective computing are given to verify the stability of the α-UTI method. In summary, through theoretical proof and example verification, it is found that the α-UTI method has good interval robustness with four kinds of unified interval implications aiming at the situations of an individual rule, multi-rule and reasoning chain.

As the core of fuzzy logic [

The other one is fuzzy modus tollens (FMT):

Since Zadeh came up with the compositional rule of inference (CRI) [

Due to the strong logical completeness of the TI method, many scholars have carried out more in-depth research in this direction. Many methods are proposed to enhance the TI method, such as the α-triple I constraint degree method [

Among them, robustness is a crucial property, which refers to the resistance of the method to perturbations. It can be quantified by calculating the gap between the output of the fuzzy reasoning method with interference and the output without interference. The smaller the gap, the smaller the impact of interference. Distance [

Fuzzy implication plays a key role in fuzzy reasoning strategy. Different implications will also have different effects on the robustness of the method. Reference [

The main formula of the α-UTI method for FMP is as below [

The α-UTI method has been widely recognized in the field of fuzzy reasoning as a generalization of the CRI method and the TI method. The α-UTI method is also a generalization of the universal triple I (UTI) method proposed in [

The current research gaps in this field include the following two aspects:

On the one hand, how to give the solution of chain reasoning, a special form of reasoning, is a tough problem in fuzzy reasoning. In this regard, we can consider starting from the α-UTI method to construct a scheme oriented towards chain reasoning.

On the other hand, it is not clear how the interval stability of the α-UTI method is. In particular, can the upper and lower bounds of its reasoning be estimated? This aspect has not yet been discussed. This has aroused our great interest, and also constitutes the research goal of this paper.

In this study, we explore the interval robustness of the α-UTI method, which is embodied by interval stability.

The proposed study includes the following aspects:

To begin with, we discuss the robustness of the α-UTI method in the case of an individual rule, and give an estimate of the upper and lower bounds. Therein, four kinds of unified interval implications mentioned above are employed in turn. It is important to note that each kind of implication actually contains a lot of specific implications, so this study can incorporate many implications into its framework. The process and flow chart of the corresponding algorithm are given. The results show that the α-UTI method is stable in the individual rule case under the interval perturbation.

Moreover, we provide the estimation for upper and lower bounds for the multiple rules case of the α-UTI method. Here, four kinds of unified interval implications are respectively examined in the α-UTI method. The results show that, in the context of the interval perturbation, the α-UTI method is stable for the situation of multiple rules.

In addition, for the problem of chain reasoning, we put forward a strategy based upon the α-UTI method, which is called the α-UTI reasoning chain method. We provide an intelligent algorithm to estimate its upper and lower bounds, and draw the corresponding flow chart. In this case, the four types of interval implications are adopted in turn. It is found the α-UTI reasoning chain method is stable for the problem of chain reasoning.

Lastly, the α-UTI method is analyzed through two application examples in affective computing. Here we propose the scheme of emotional deduction based on the α-UTI method. These examples verify the stability of the α-UTI method.

To sum up, the results show that in three cases of individual rule, multiple rules and reasoning chain, the α-UTI method has good interval stability when it respectively adopts four kinds of unified interval implications mentioned above.

The structure of this paper is as follows.

This section mainly introduces the basic knowledge of fuzzy reasoning, and provides some interval implications and interval connectives.

There are two functions

For any

In order to facilitate the subsequent operation, we give two tokens, namely

For the definition of interval fuzzy implication, there are other definitions, such as the need for incrementality with regard to the second variable. But here conditions

Two important operations are further defined here, in which these two concepts are mutually symmetric.

The interval t-norm and the interval t-conorm are symmetric concepts. As the opposite structure of conventional operations, interval fuzzy negation is defined as below.

On the basis of interval fuzzy negation, if it still has:

then this is a strong interval fuzzy negation.

Subsequently, three important classes of interval fuzzy implication are defined as below.

The interval t-norm can also be regarded as an interval fuzzy implication, called the interval t-norm implication.

In what follows, the idea of interval disturbance is provided.

In this work,

For an individual rule, assume that

Moreover, one has

In summary, we have proved that

Theorem 3.1 gives a estimation of the corresponding α-UTI solutions for the case of the interval perturbation with an individual rule. In other words, it respectively provides an upper bound and a lower bound of the α-UTI solutions for R-interval implications.

Theorem 3.2, Theorem 3.3 and Theorem 3.4 can be achieved in a similar mode, in which the interval t-norm implication, the S-interval implication, the QL-interval implication are employed, respectively.

In light of Theorem 3.1 to Theorem 3.4, in the case of an individual rule, if related operators including the interval t-norm, the interval t-conorm and so on, then the α-UTI method is stable with regard to S-, QL- and interval t-norm implication. The α-UTI method is stable with respect to R-interval implication if the R-interval implication and the interval t-norm are continuous. All in all, the α-UTI method is robust for the individual rule.

Suppose that for the α-UTI method, there exists a perturbation array of input

Besides, the consequent properties are satisfied:

In detail, we have from

Others can be analogously expanded.

Considering that

When the continuity condition is satisfied, then

For Theorem 3.1, we give an intelligent algorithm (see the following Algorithm 1) to deal with such operations. For the other theorems, we can get corresponding algorithms.

For multiple rules, the notation for representing fuzzy intervals is similar for the individual rule. It is denoted that

It follows from Lemma 2.1 and Lemma 2.2 that one has

In a similar structure, we can also find

To sum up, we can find that

Theorem 4.2, Theorem 4.3 and Theorem 4.4 can be proved in an analogous mode, in which four kinds of unified interval implications are employed in turn.

By virtue of Theorem 4.1 to Theorem 4.4, in the case of multiple rules, if operations in

Suppose that for the α-UTI method, there exists a perturbation array of input

Besides, the consequent properties are satisfied

In detail, we have from

Others can be analogously analyzed.

Noting that

When the continuity condition is satisfied, then

Similar to Theorem 3.1 and Algorithm 1, we can obtain corresponding algorithms for Theorems 4.1, 4.2, 4.3 and 4.4.

Here we propose the α-UTI reasoning chain method. Let

To begin with, we employ the α-UTI method to get

Because of multiple reasoning, chain reasoning may lead to errors that may expand, so it is naturally more difficult to maintain stability. So the stability study of the α-UTI reasoning chain method is more important.

Now we discuss the problem of upper and lower bounds of inference results of the α-UTI reasoning chain method. First of all, we analyze the case of the S-interval implication.

Therein two notions are as follows

Let us denote

Thus, it can be obtained

Furthermore, it can be derived

With similar treatment, we can end up with the following formula:

To sum up,

Moreover, we further discover the case of the R-interval implication, where the proof process is similar to that obtained.

Therein two notions are as follows

Along similar lines, we can prove the following two theorems with regard to the interval t-norm implication and the QL-interval implication.

Therein two notions are as follows

Therein two notions are as follows

For Theorem 5.1, we provide an intelligent algorithm (see the following Algorithm 2). For the other theorems, we can similarly obtain corresponding algorithms.

By virtue of Theorem 5.1 to Theorem 5.4, if the underlying operations in

Assume that for the α-UTI reasoning chain method, there is a perturbed queue with input

Besides, the consequent properties are satisfied:

In detail, we have from

Others can be similarly expanded.

From Theorem 5.1 to Theorem 5.4,

When the continuity condition is satisfied, one has

In summary, the α-UTI method is stable in the case of an individual rule, multiple rules and reasoning chain under the interval perturbation. In this section, we employ two examples to verify the interval stability of the α-UTI method.

The input is

For testing, we use a structure with interval perturbation in the input. We use Theorem 3.1 with the R-interval implication

For one situation, we take

For another situation, we let

The interval of the second input is obviously smaller than that of the first input. It can be seen that the α-UTI method will eventually converge under the interval disturbance as the interval disturbance decreases under the reasonable input and rule.

For the background of Example 6.1, the input corresponds to the values of sorrow, anger, and hate, and the output corresponds to the value of joy (noting that these four are all basic emotions in affective computing). This reflects the relationship between several basic emotions. Comparing the two situations in Example 6.1, it can be seen that when the three input emotions are subjected to smaller interval perturbations, the change of output emotions tends to be stable.

Affective computing is receiving extensive attention. Emotion deduction (exploring how to reason about the membership of other emotions from some basic emotions) is significant in many ways (e.g., emotional state transitions when building a large emotional corpus, etc.) and is an essential task in affective computing. Here we give an example of the application of the method in emotion deduction to demonstrate the robustness of the α-UTI method.

The input is

Here, the input

For one situation, we take

For another situation, we let

The context of Example 6.2 is emotional deduction. Emotion corpus is one of the key issues in affective computing. We have done a lot of work in constructing an emotion corpus, but the emotion corpus is often based on basic emotions, such as the eight basic emotions mentioned above. However, in the real world, there are many kinds of emotions, far more than these basic emotions. Obtaining values for other emotions, then, has become a recognized puzzle in the field of affective computing. That is why emotional deduction comes in.

Here we put forward the scheme of emotional deduction based on the α-UTI method, and naturally hope that such emotional deduction scheme is stable. Through the comparison of the two situations in Example 6.2, we can see that when the input interval disturbance is smaller, the obtained result also belongs to a smaller range, so the emotional deduction result is stable for multiple rules. This validates that our proposed emotional deduction scheme based on the α-UTI method is effective and practicable.

In this study, we investigate the interval robustness (embodied by the interval stability) of the α-UTI method of fuzzy reasoning. The main contributions of this paper are reflected in the following aspects:

First of all, the stability of the α-UTI method is examined for an individual rule, and the upper and lower bounds are estimated for the α-UTI solutions. Here the analysis is conducted on the basis of four kinds of unified interval implications. The analysis shows that the α-UTI method exhibits good interval stability for an individual rule.

In addition, the stability of the α-UTI method is found in the context of multi-rule conditions, while the upper and lower bounds of its outcomes are estimated. Therein, four kinds of unified interval implications are adopted. It is observed that the α-UTI method is stable in the case of multiple rules.

Furthermore, the α-UTI reasoning chain method is put forward, containing a chain structure with multiple layers. The corresponding solutions are given and the interval perturbations are analyzed. The upper and lower bounds of the outcomes are estimated, involving four kinds of unified interval implications. The results show that the α-UTI reasoning chain method is stable from the viewpoint of interval perturbation.

Finally, the α-UTI method is studied through two application examples, incorporating an application of the α-UTI method in emotion deduction of affective computing. These examples show that the α-UTI solution converges stably to a value if the continuity condition is effective, which verifies the stability of the α-UTI method.

The novelty of this paper is manifested in the following aspects. To begin with, the estimation of the upper and lower bounds of interval perturbations is a novel problem to be explored for the α-UTI method. Moreover, we propose the α-UTI reasoning chain method as a new multi-layer inference mechanism. Lastly, we investigate the interval robustness of fuzzy reasoning under the interval-valued fuzzy environment.

The merits of this study are as follows. Firstly, we propose the α-UTI reasoning chain method, which consists of a chain structure with multiple layers. This method presents a new scheme to solve the problem of chain reasoning. Secondly, four kinds of important unified interval implications are considered in this work, which have certain universality. Finally, the upper and lower bounds are estimated for the α-UTI method in the interval-valued fuzzy environment. The results indicate that the α-UTI method has good interval robustness in situations involving an individual rule, multi-rule and reasoning chain.

The demerits of this study are outlined as follows. On the one hand, we do not consider the use of specific interval implications in the α-UTI method, especially those that do not belong to these four kinds of interval implications. The interval robustness of this kind of the α-UTI method is not considered. On the other hand, we have explored the interval robustness of the α-UTI method in the interval-valued fuzzy environment. However, other environments, such as the intuitionistic fuzzy environment, have not been discussed. These can be the guidelines for the upcoming work.

In future studies, we will consider incorporating the latest clustering algorithms [

A preliminary version of this work was presented at the 3rd International Conference on Artificial Intelligence Logic and Applications (AILA 2023), and its title is “On interval perturbation of the α-universal triple I algorithm for unified interval implications”.

This work was supported by the National Natural Science Foundation of China under Grants 62176083, 62176084, 61877016, and 61976078, the Key Research and Development Program of Anhui Province under Grant 202004d07020004, the Natural Science Foundation of Anhui Province under Grant 2108085MF203.

Conceptualization, Y.T. and Y.H.; methodology, Y.T. and J.G.; software, Y.T. and J.G.; validation, Y.T., Y.H., and J.G.; formal analysis, J.G.; investigation, Y.T. and J.G.; resources, J.G.; data curation, J.G.; writing—original draft preparation, Y.T. and Y.H.; writing—review and editing, Y.T. and J.G.; visualization, J.G.; supervision, Y.T.; project administration, J.G.; funding acquisition, Y.T. All authors have read and agreed to the published version of the manuscript.

Not applicable.

The authors declare that they have no conflicts of interest to report regarding the present study.