Crossover in mutation oriented norm evolution

Abstract

Norms are a coordination mechanism. They control agents’ behavior in a multiagent system (MAS) and need to evolve to cope with changing environments. Mutation oriented norm evolution is a strategies for allowing norms to evolve. However, this strategy simply adds some possible trigger condition constraints on the norms, which means that some agents are unable to perform actions. To address this problem, this paper presents a new strategy for norm evolution based on an improved crossover operator. First, this paper presents a power-set approach to improve the integrity of norm evolution. This approach can help ensure that all possible combinations of norms are considered during the analysis, providing a deeper understanding of how norms interact and evolve within a norm set. Then, to improve the efficiency of norm evolution, a trade-off between efficiency and completeness is proposed. This approach reduces the search space and improves efficiency, as not every power set combination needs to be searched; it also ensures completeness. Finally, the crossover operator in this strategy is improved based on the trade-off approach. Specifically, the triggers and expectations of one mutated norm enrich the triggers and expectations of other norms. All of these factors enrich the normative conditions through the trade-off approach. A MAS can take immediate action to adapt to new requirements or problems encountered, and quickly make normative changes and learn to respond appropriately to a new situation. The MAS is able to more clearly understand and learn about causality in the environment during norm evolution, and understand the connection between behavior and outcomes. The proposed strategy is applied to a case study of an unmanned vehicle system. The experimental results show that the trade-off approach has greater completeness and effectiveness in norm evolution. This strategy achieves a more complete and effective autonomous norm evolution. It helps the system achieve its goals better and reach better performance in terms of adaptability, hel** it to function better in complex multiagent environments.

Introduction

Norms are the keys to the effective operation of agents in the system. Some functions of the systems are performed though the mutual connection and interactions of several parts, which are governed by norms. An agent [1] is an individual or thing that perceives the environment of the system and can act. Norm generation is usually a process that starts from scratch. Norms develop through interactions between agents. A norm can be expressed in the forms ‘If A, then B’ [2,3,4], ‘Event-Condition-Action (ECA)’ [5, 6] or ‘Trigger-Action’ [6, 8,9,10]. A MAS is composed of multiple agents interacting in a changing environment. These agents form a large-scale, complex system [11, 12]. As circumstances change, the original norms may become obsolete or no longer apply. The activities of agents under the original norms are not sufficient to address changes in the environment. Therefore, norms need to evolve to adapt to the changing environment.

At present, a considerable amount of literature has been published on norm evolution. Song et al. [13,14,15] proposed an adaptive finite time prescribed performance control strategy. An adaptive self-triggering control law is designed to determine the next trigger time according to the current information. In addition, a balance between tracking performance and communication costs is achieved between the controller and the actuator. The above research is related to norm evolution in MASs. In a MAS, the behavior of the agent is adjusted by a similar method to adapt to changes in the environment and improve the overall performance of the system. By designing adaptive norms and control strategies, the stable operation of a MAS can be maintained. The communication and cooperation between agents can be optimized to balance system performance and resource consumption. Mutation oriented norm evolution (Mone) is a strategy for autonomous norm evolution [16]. By monitoring the norm execution sequence, [16] mined the influences of norm execution, constructed an influence transmission model, and used these influences to explore the improvement of norms. A method for improving norms through crossover operations was proposed. Specifically, the norm execution sequence increases with increasing MAS running time. The influences between norm executions are observed in sequence. Mone simply adds some possible conditions to the trigger and increases the trigger condition constraints of the norms. As a result, the triggers of these norms are not easy to satisfy, and agents cannot perform certain actions. In addition, this aspect of Mone has not been fully characterized thus far. According to the needs of actual projects, this paper implements Mone inn an unmanned system with multiple agents. Based on the theory of Mone, this paper presents a trade-off between efficiency and completeness as well as an improved crossover operator. The proposed methods are applied to an unmanned vehicle system (UVS) to realize norm evolution effectively and quickly; this approach is applied to a weeding scenario, as shown in Fig. 1. This type of unmanned vehicle is usually equipped with sensors to reduce manual labor and increase the efficiency of agricultural production. In the case of weeding tasks in automated agriculture, task data can be collected for real-time monitoring and analysis to further optimize operational efficiency and decision-making processes. For an unmanned vehicle to reach a weeding site efficiently according to the on-site environment, relevant experts determine the norms of the supporting actions. Finally, simulation experiments prove that the improved methods are more effective in evolving norms to adapt to the environment. The contributions of this paper can be summarized as follows:

• First, to address the inapplicability of Mone in changing environments, this paper presents a power set approach to split the trigger conditions of the evolved norm. This method improves the completeness of norm evolution.

• Subsequently, considering the balance between effectiveness and completeness, this paper proposes a trade-off approach. This method improves the efficiency of the evolved norms within the system in adapting to the environment.

• The ‘crossover’ operator is subsequently improved based on the trade-off method. Adding some expectation conditions to trigger conditions ensures that the evolved norms will only trigger when the expectation conditions and trigger conditions are fulfilled at the same time. Adding some trigger conditions to the expectation conditions ensures that the evolved norms are only judged when the expectation conditions and trigger conditions are fulfilled at the same time. This method fully utilizes the advantages of different norms and improves their accuracy and reliability.

• Finally, simulation experiments demonstrate that the method proposed in this paper outperforms Mone in terms of the efficiency, completeness, and effectiveness of norm evolution.

Fig. 1

Weeding unmanned vehicle

The remainder part of the paper is organized as follows: Sect. “Related work” introduces related works on norm evolution. Section “Background” introduces the process of Mone and the basic knowledge related to the work of this paper. In Sect. “Improved methods of evolutionary norms”, notations is introduced, as are the proposed power-set method, trade-off method, and improved crossover operator, and then corresponding examples are given. The pseudocodes of related works is provided. In Sect. “Case study”, UVS is applied to a weeding scenario. According to the field environment, the proposed method is used to determine the evolved norms, and the experimental results are presented and analyzed. Section “Discussion” discusses the advantages and disadvantages of the proposed strategy. Finally, Sect. “Conclusions and future work” summarizes the significance of the study and discusses future work.

Related work

MAS, decision and norm

In a MAS, agents can communicate, collaborate, or compete with each other to achieve a common goal. There has been significant research on the intelligence of MAS in various respects, such as collaboration [17], decision-making [18, 13, 15] proposed an adaptive finite-time prescribed performance control strategy. This strategy uses an improved fractional-order filter to address the problem of large computational load while eliminating filter error. On this basis, considering the influence of bandwidth limitations, an adaptive self-triggering control law was designed to determine the next trigger time according to the current information. Finally, the system output was limited to a small area in a limited time. An adaptive finitely specified performance control strategy would determine the next trigger time according to the current information and system state so that the system output could converge to the predetermined area in a limited time. In a MAS, the behavior of the agent can be adjusted by a similar method to adapt it to changes in the environment and improve the overall performance of the system. Song et al. [34]. In the norm splitting process, the set of trigger conditions of the original norm can be considered a set, and then all possible subsets can be generated using the power set method. Each subset represents a trigger condition for a norm. The subset of norms generated through the power set approach can be used to explore different combinations of norms to fulfill a specific need or condition. In complex systems, the health state and security level of the system can be predicted. The belief norm base is a method for predicting hidden behavior, but there are still two problems that need to be solved. First, when the observed information does not exist, the output of the belief norm-base may not be able to draw accurate conclusions, affecting the accuracy of the prediction model. Second, interference factors such as noise and sensor quality will reduce the reliability of the information collection, which will affect the reliability of hiding behavior. To solve these problems, Zhou et al. [34] proposed a power-set hidden belief norm base model called attribute reliability. In this model, the influence of interference factors on hiding behavior is considered to represent attribute reliability, and the recognition framework of the power set is adopted. By considering attribute reliability, the model can make more accurate predictions in the presence of global ignorance and unreliable hiding behavior. This model can predict hidden faults in complex systems more accurately, than other models can, allowing appropriate measures to be taken to avoid the occurrence of faults. However, the most prominent bottleneck in applying power set methods is the exponential growth of their power set subsets as the number of set elements increases. In norm unfolding, rough set theory can be used to analyze the dependencies and redundancies between norms to generate a minimal and efficient set of norms. Chen et al. [35] proposed dividing the dataset into a smaller number of subsets. Through rough set theory, norms that overlap or duplicate other norms in terms of conditions and behavior can be identified and removed. This approach prevents redundant norms form being incorporated into the norm set, reduces the number of norms, and improves the efficiency and readability of the MAS. By utilizing this method to expand norms and eliminate redundancy, a more concise and efficient set of norms can be generated.

Through the autonomous learning capability of MASs, they can continuously refine and improve the norms by learning and analyzing the environment and tasks in real time [25, 36, 37]. When agents in a MAS encounter various abnormal situations (e.g., mutations, conflicts), coordination between agents is quickly achieved through intelligent exception handling mechanisms and conflict resolution mechanisms. Moreover, norms will continue to evolve, and optimize to adapt to the actual situation and the rapidly changing environment. As the number and complexity of norms increase, it is challenging to manage them and to continuously drive their innovation and evolution.

Background

In this section, first, the basic concepts of the Mone algorithm and the mechanism of norm evolution are introduced. Then, the basics related to the work of this paper are introduced. Finally, based on the evolution mechanism of Mone, the inapplicability of the algorithm is analyzed.

Brief introduction to Mone

• (1) Norm

  A norm regulates the actions of agents by triggering conditions and actions. It is defined as a trigger-action pair. The operation of a system requires a set of norms, which are denoted as $\hbox {N} = \{r_1, r_2, \ldots \}$. Norm $r_i \in N$ has two parts: a trigger and action. Therefore, $r_i$ is expressed as $r_i = \{Tri_i, Act_i\}$. When the same norm is executed, the result of the action may be different. Therefore, the definition of a norm is extended with expectations to verify the results of actions namely $r_i = \{Tri_i, Act_i, Exp_i\}$.

  Where:

  • $Tri_i = \{tri_{i_1}, tri_{i_2}, \ldots \}$ is a set of trigger conditions. It is used to judge whether relevant actions can be carried out. Note that $tri_{i_j}$ is a logical judgment expression, such as $tri_{i_1} = \{a > 15\}, tri_{i_2} = \{b < 20\}$.

  • $Act_i = \{act_{i_1}, act_{i_2}, \ldots \}$ is a set of actions. Note that $act_{i_j}$ is a function or behavior, such as $act_{i_1} = \{a = a + 5\}, s_{i_2} = \{b = b - 5\}$.

  • $Exp_i = \{exp_{k_1}, exp_{k_2}, \ldots \}$ is a set of logical judgment expressions after $Act_i$ is executed, such as $exp_{i_1} = \{a > 67\}, exp_{i_2} = \{b < 15\}$.

• (2) State

  The state of a system is represented by a set S, S = $\{s_1, s_2, \dots \}$. $s_i$ is the instantaneous state. There is a strict time sequence between instantaneous states. An instantaneous state $s_i$ has multiple expressions namely $s_i = \{s_{i_1}, s_{i_2}, \ldots$}. For example, $s_{i_1} = \{a\ is\ 16\}$, $s_{i_2} = \{b\ is\ 17\}$.

• (3) Triggered norms

  In an instantaneous state $s_i$, all logical expressions in $Tri_i$ need to be judged. The state within the logical judgment expression is represented by the instantaneous state $s_i$. If $tri_{i_j}$ is satisfied, the judgment result is T; otherwise, the judgment result is F. If the conjunctive normal form for these results is T, norm $r_i$ is triggered; otherwise, norm $r_i$ is not triggered.

  For example, in the instantaneous states $s_{i_1} = \{a\ is\ 16\}, s_{i_2} = \{b\ is\ 17\}$, the trigger conditions of the norms are $tri_{i_1} = \{a > 15\}$ and $tri_{i_2} = \{b < 20\}$ respectively. According to the judgment result of the expression, norm $r_i$ is triggered.

• (4) Obeyed norms and mutated norms

  After norm $r_i$ is triggered, $Act_i$ is executed. In the instantaneous state $s_j$, all logical expressions in $Exp_i$ need to be judged. The state within the logical judgment expression is represented by the instantaneous state $s_j$. If $exp_{i_j}$ is satisfied, the judgment result is T; otherwise, the judgment result is F. If the conjunctive normal form for these results is T, then norm $r_i$ is called an ‘obeyed norm’; If these judgment results include at least one F, then norm $r_i$ is called a ‘mutated norm’.

  For instance, in the instantaneous states $s_{j_1} = \{a\ is\ 21\}, s_{i_2} = \{b\ is\ 12\}$, the expectations of the norms are $exp_{i_1} = \{a > 67\}$ and $exp_{i_2} = \{b < 15\}$, respectively. According to the judgment result of the expression, norm $r_i$ is a mutated norm.

• (5) LEG

  A layered execution graph (LEG) is a path formed by recording the execution sequence of norms. It records triggered norms, the execution results of the norms, and the corresponding instantaneous states. The LEG can detect and trace the influence of the execution sequences of the norms. These influences provide clues for improving these norms.

• (6) Norm evolution

  In a system, norm evolution is based on the analysis of the mutual action of norms. This can explain mutated norms and aid the crossover operation between these norms. This process can address exceptions of a state. As a result, the number of trigger conditions for the norm is increases, and the constraints on the norm’s trigger increase.

  Specifically, it is assumed that a system has several triggered norms, and the obtained norm execution sequence is $r_a$, $r_b$, $r_c$. First, all mutated norms are detected through the LEG; in this case, they are norms $r_a$ and $r_c$. It can be determined that the exception of $r_c$ is caused by the exception of $r_a$. Second, the mutated expectation of norm $r_c$ are extracted, namely, $exp_{c_3}$, $exp_{c_5}$, and $exp_{c_6}$. Then, they are taken as antonyms, namely, not $exp_{c_3}$, not $exp_{c_5}$, and not $exp_{c_6}$, which is the new knowledge of this system. Next, these pieces of new knowledge are added to the trigger of another mutated norm through the crossover operator. Finally, an improved trigger is obtained: $Tri_a = \{tri_{a_1}, tri_{a_2}, \ldots$, not $exp_{c_3}$, not $exp_{c_5}$, not $exp_{c_6}$}. After the norms evolve, there is no change in the time complexity or space complexity. The time complexity mainly includes the time complexity of triggering and executing a norm. The time complexity is T(n) = O(2n), and the space complexity is S(n) = O(n).

  Genetic algorithms (GAs) are heuristic optimization algorithms used to find the best solution in the search space, which is different from the traditional local search algorithm [38]. GAs search for potential solution spaces by simulating the process of biological evolution through the genetic operations of individuals in a population, such as selection, crossover, and mutation. GAs maintain a population and retain information about multiple individuals. This algorithm can avoid falling into local optimality. In the process of evolution, GAs can explore the solution space in the global scope through continuous selection and crossover, which helps to find better solutions. Although GAs is not guaranteed to find the global optimal solution, it can usually achieve better results in problems with complex solution spaces and strong multi-peaks, so as to avoid falling into the local optimal. Therefore, GAs have certain advantages in finding the optimal solution or approximate optimal solution to complex problems. In order to solve the problem of over-learning or under-learning in the optimization process of distance measurement in local weighted learning, Bai et al. [38] propose an propose an improved genetic algorithm. In this algorithm, a new fitness function is defined. This function can assign the maximum fitness to the optimal distance measure. They use the optimal distance measure to identify the motion features of ships. GAs share similarities with Mone. Both the GA and Mone transform the problem into another form, and both use crossover for evolution. The crossover of the GA involves replacing and recombining the partial structure of two-parent individuals to generate a new individual. The crossover of Mone is an operation that improves and recombines the partial structure of the two-parent norms to evolve the parent. Their differences lie in the following three points: the GA evolves individuals, while Mone evolves norms; the GA generates new individuals, and Mone evolves the original norms. The purpose of the GA is to replace a recombination, and the purpose of Mone is to improve the recombination.

Preliminaries

• (1) Experimental platform

  BBS is a problem-solving model that applies a blackboard structure. It consists of a knowledge source (KS), blackboard (BB), and control shell (CS). Previous studies have provided detailed descriptions of BBS, including [39,40,41]. Since BBS is executed in sequence, the state change in the BB is clear, so BBS is used for the following research on the evolution of norms. Due to its rapid data sharing ability, this approach is widely used in remotely distributed and heterogeneous information system integration to solve problems such as the high-speed sharing of massive amounts of business information. The Generic Blackboard Open Source (GBBopen¹) software was developed and implemented in the Common Lisp language environment. GBBopen is open-source software; users can improve it by modifying the source code. Therefore, the subsequent experiments in this paper were implemented on this platform.

• (2) Norm in GBBopen

  In GBBopen, the KS attributes include ‘trigger-events’, ‘precondition-function’, ‘execution-function’, and ‘pos tcondition-function’. The ‘trigger-events’ bind events to a norm. The ‘trigger-events’ activate the norm by subscribing to the event. Each of the remaining attributes is bound to an executable function. Figure 2 shows the relationship between norm and the KS. Figure 2 illustrates that each component of a norm corresponds to a KS attribute of GBBopen. For example, the trigger of a norm may correspond to ‘trigger-events’ and ‘precondition-function’ of KS. Figure 2 indicates that the KS executes the bound function from top to bottom to determine the trigger, action, and expectation of a norm.

• (3) Power set

  Taking the power set is one of the basic operations of a set, and it yields a set composed of all the subsets of the set. Let set A = {a, b}. The power set of A is denoted by P (A). Then, P(A) = {$\phi$, {a}, {b}, {a, b}}. The number of elements in a set is called the cardinality of the set. The cardinality of A is denoted as Card(A). Therefore, Card(A) = 2. The cardinality of the power set of A is expressed as Card(P(A)), and Card(P(A)) $= 2^2 = 4$.

Fig. 2

Relationship between KS and norm

Based on the previous example illustration of the Mone evolution mechanism, $r_a$ evolves autonomously, and $Tri_a$ of norm $r_a$ evolves from {$tri_{a_1}, tri_{a_2}, \ldots$} to {$tri_{a_1}, tri_{a_2}, \ldots$, not $exp_{c_3}$, not $exp_{c_5}$, not $exp_{c_6}$} by the mechanism of Mone. Mone simply adds some possible conditions to the trigger condition, increasing the trigger condition constraints of the norm and making the norm trigger more difficult to satisfy. If the subsequent state within the system does not reach $Tri_a$ of the evolved norm, then it causes the agent fails to execute the action. Therefore, in the next section, this paper first introduces some symbols and gives examples in conjunction with the terminology used in Mone. Then, a strategy of norm evolution is proposed to address the inapplicability of Mone.

Improved methods of evolutionary norms

In this section, some notations is first introduced and illustrated with examples, such as ‘Norm’, ‘Mutation’, and ‘Crossover’ in Mone. Subsequently, because of the inapplicability of Mone, this paper improves upon several aspects of Mone(e.g., efficiency and completeness), and explanations and examples are provided. Finally, pseudocode for the improved method is provided.

Notations description

• (1) Notation $(\prec )$.

  It is assumed that the execution time of a triggered norm is ignored; arguably, the order of triggered norms is equivalent to the order of executed norms. This paper defines the notation ‘$\prec$’ as a sequence in which multiple norms are triggered and uses this notation in what follows. At a moment $t_x$, norm $r_i$ is triggered in an instantaneous state $s_i \in S$. At another moment $t_y$, norm $r_j$ is triggered in an instantaneous state $s_j \in S$. The following simplifying notation is used:

  $$\begin{aligned} r_i \prec r_j \end{aligned}$$

  (1)

  In the formula (1), norm $r_j$ is triggered after norm $r_i$. The time interval between $t_x$ and $t_y$ is instantaneous, which can be expressed as $\Delta t_y-t_x \rightarrow 0$. Assuming that the instantaneous state $s_i \in S$ of the system triggers multiple norms $r_i$ and $r_k$ $\in$ N at time $t_x$, norms $r_i$ and $r_k$ are triggered at the same time. Thus, the order of the triggered norms can be expressed as $\{r_i, r_k\} \prec r_j$ or $r_i \prec r_j$, $r_k \prec r_j$.

• (2) Notation $(\thicksim )$.

  This paper defines the notation ‘$\thicksim$’ as the antonym of a condition. Assuming that the logical symbols ‘a = b’ and ‘$a \le b$’ are used to denote some conditions, the results of the antonyms of the above conditions are ‘$a \ne b$’ and ‘$a > b$’, respectively. For the expectation $Exp_i$ of norm $r_i$, where $Exp_i = \{exp_{i_1}, exp_{i_2}, \ldots \}$, taking the antonym of $Exp_i$ is equivalent to taking the antonym of each expectation condition $exp_{i_j}\in Exp_i$; the antonym of $Exp_i$ can therefore be expressed as:

  $$\begin{aligned} \thicksim Exp_i&=\ \thicksim \{exp_{i_1}, exp_{i_2}, \ldots \} \\&= \{\thicksim exp_{i_1}, \thicksim exp_{i_2}, \ldots \} \end{aligned}$$

  Through the above operations, the flexibility and adaptability of the system increase, and the execution of the expected conditions is adjusted dynamically so that the system can flexibly adapt to different scenarios and requirements.

• (3) Notation $(\cup )$.

  The notation ‘$\cup$’ indicates that an element in a set is added to another set to form a new set, i.e., an element and a set are integrated into a new set.

  It is assumed that there are norms $r_i, r_j$, and $r_i \prec r_j$. Add the expectation condition $exp_{j_k} \in Exp_j$ to the trigger $Tri_i$; then, the integrated trigger $Tri'_i$ is:

  $$\begin{aligned} Tri'_i = Tri_i \cup exp_{j_k} \end{aligned}$$

  (2)

  Note from Eq. (2) that $Tri_i$ and $exp_{j_k}$ are integrated into a new trigger $Tri_i$ ($Tri'_i$) through the notation ‘$\cup$’, indicating that extracting an expectation condition $exp_{j_k} \in Exp_j$ enriches the trigger condition $Tri_i$ of $r_i$. By adding trigger conditions, one can limit the scope of the norm and control the execution of the norm. In addition, flexibility and dynamics can be introduced so that the norm is triggered only in certain situations. This approach prevents the norm from being incorrectly executed in irrelevant circumstances and improves the accuracy and reliability of the norm.

• (4) Notation $(\ll )$.

  The notation ‘$\ll$’ denotes that the expressions on both sides of it are compared, and the expression on the left side is much smaller than that on the right side. For instance, 1 $\nless$ 10,000 implies that 1 is much less than 10,000.

• (5) Notation $(\nless )$.

  The notation ‘$\nless$’ indicates that the expression on the left side is slightly less than that on the right side. For example, 1 $\nless$ 1.67 indicates that 1 is slightly less than 1.67.

Correlation methods

Determining an appropriate trade-off between efficiency and completeness is a common problem in computational approaches to norm evolution. Efficiency refers to the computational resources, including time and memory, required to execute a strategy. This subsection first present a power set approach to splitting the trigger conditions of evolutionary norms. Subsequently, a trade-off method is proposed, considering the balance between validity and completeness. Finally, based on the trade-off method, the crossover operator of Mone is improved.

• (1) Power set method

  The power set is a set family composed of all the subsets (including the complete set and the empty set) of the original set. The trigger conditions of norms are further refined by the power set method. Suppose norms $r_i$ and $r_j$ are mutated norms, where $r_i \prec r_j$, $Tri_j = \{tri_{j_1}, tri_{j_2}, tri_{j_3}, tri_{j_4}, \dots \}$, and $Exp_j = \{exp_{j_1}, exp_{j_2}, exp_{j_3}, exp_{j_4}, exp_{j_5}, \ldots \}$. The expectation conditions of the mutated norms are extracted, which are denoted by $Exp\_mu_j$. This paper assumes that $Exp\_mu_j = \{exp_{j_1}, exp_{j_2}, exp_{j_3}, exp_{j_4}, exp_{j_5}, \ldots \}$, where $Exp\_mu_j \subset Exp_j$.

  With the notation $(\thicksim )$ defined above, to take the antonym of the set $Exp\_mu_j$, the following formula can be used:

  $$\begin{aligned} \thicksim Exp\_mu_j= & {} \{\thicksim exp_{j_1}, \thicksim exp_{j_2}, \thicksim exp_{j_3}, \\{} & {} \thicksim exp_{j_4}, \thicksim exp_{j_5}, \thicksim \dots \} \end{aligned}$$

  The result of the improved trigger $Tri_i$ obtained by Mone is as follows:

  $$\begin{aligned} Tri'_i= & {} \{Tri_i, \thicksim exp_{j_1}, \thicksim exp_{j_2}, \thicksim exp_{j_3}, \\{} & {} \thicksim exp_{j_4}, \thicksim exp_{j_5}, \thicksim \dots \} \end{aligned}$$

  Form the definition of the power set, taking the power set of $\thicksim Exp\_mu_j$, the result is:

  $$\begin{aligned}&P(\thicksim Exp\_mu_j) \\&\quad = \{\phi , \{\thicksim exp_{j_1}\}, \{\thicksim exp_{j_2}\}, \{\thicksim exp_{j_3}\},\\&\qquad \{\thicksim exp_{j_4}\}, \{\thicksim exp_{j_5}\},\\&\qquad \{\thicksim exp_{j_1}, \thicksim exp_{j_2}\}, \{\thicksim exp_{j_1}, \thicksim exp_{j_3}\},\\&\qquad \{\thicksim exp_{j_1}, \thicksim exp_{j_4}\}, \{\thicksim exp_{j_1}, \thicksim exp_{j_5}\},\\&\qquad \{\thicksim exp_{j_2}, \thicksim exp_{j_3}\}, \{\thicksim exp_{j_2}, \thicksim exp_{j_4}\},\\&\qquad \{\thicksim exp_{j_2}, \thicksim exp_{j_5}\},\\&\qquad \{\thicksim exp_{j_3}, \thicksim exp_{j_4}\}, \{\thicksim exp_{j_3}, \thicksim exp_{j_5}\},\\&\qquad \{\thicksim exp_{j_4}, \thicksim exp_{j_5}\},\\&\qquad \{\thicksim exp_{j_1}, \thicksim exp_{j_2}, \thicksim exp_{j_3}\},\\&\qquad \{\thicksim exp_{j_1}, \thicksim exp_{j_2}, \thicksim exp_{j_4}\},\\&\qquad \dots \\&\qquad \{\thicksim exp_{j_1}, \thicksim exp_{j_2}, \thicksim exp_{j_3}, \thicksim exp_{j_4},\},\\&\qquad \{\thicksim exp_{j_1}, \thicksim exp_{j_2}, \thicksim exp_{j_3}, \thicksim exp_{j_5},\},\\&\qquad \{\thicksim exp_{j_1}, \thicksim exp_{j_2}, \thicksim exp_{j_4}, \thicksim exp_{j_5},\},\\&\qquad \{\thicksim exp_{j_1}, \thicksim exp_{j_3}, \thicksim exp_{j_4}, \thicksim exp_{j_5},\},\\&\qquad \dots \} \end{aligned}$$

  The set family $P(\thicksim Exp\_mu_j)$ obtained by the power set method, and the element of $pem_j \in P(\thicksim Exp\_mu_j)$ in the set family is added to trigger $Tri_i$ with the notation ‘$\cup$’. Ultimately, the trigger conditions of the generated norms for norm $r_i$ are updated as follows:

  $$\begin{aligned} Tri'_{i_1}&= \{tri_{i_1}, tri_{i_2}, \dots \}, \\ Tri'_{i_2}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_1}\}, \\ Tri'_{i_3}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_2}\}, \\ Tri'_{i_4}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_3}\}, \\ Tri'_{i_5}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_4}\}, \\ Tri'_{i_6}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_5}\}, \\ Tri'_{i_7}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_1}, \thicksim exp_{j_2}\}, \\ Tri'_{i_8}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_1}, \thicksim exp_{j_3}\}, \\ Tri'_{i_9}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_1}, \thicksim exp_{j_4}\}, \\ Tri'_{i_{10}}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_1}, \thicksim exp_{j_5}\}, \\&\dots , \\ Tri'_{i_{17}}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_1}, \thicksim exp_{j_2},\thicksim exp_{j_3}\}, \\ Tri'_{i_{18}}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_1}, \thicksim exp_{j_2}, \thicksim exp_{j_4}\},\\ Tri'_{i_{19}}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_1}, \thicksim exp_{j_2},\thicksim exp_{j_5}\}, \\ Tri'_{i_{20}}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_1}, \thicksim exp_{j_3},\thicksim exp_{j_4}\}, \\ Tri'_{i_{21}}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_1}, \thicksim exp_{j_3},\thicksim exp_{j_5}\}, \\ Tri'_{i_{22}}&= \{tri_{i_1}, tri_{i_2}, \dots ,\thicksim exp_{j_1}, \thicksim exp_{j_4},\thicksim exp_{j_5}\}, \\&\dots , \\ Tri'_{i_{\dots }}&= \{tri_{i_1}, tri_{i_2}, \dots \thicksim exp_{j_1}, \thicksim exp_{j_2}, \thicksim exp_{j_3}, \dots \}\\ r'_{i_1}&= \{Tri'_{i_1}, Act_i, Exp_i\}\\ r'_{i_2}&= \{Tri'_{i_2}, Act_i, Exp_i\}\\ r'_{i_3}&= \{Tri'_{i_3}, Act_i, Exp_i\}\\ r'_{i_4}&= \{Tri'_{i_4}, Act_i, Exp_i\}\\ r'_{i_5}&= \{Tri'_{i_5}, Act_i, Exp_i\}\\ r'_{i_6}&= \{Tri'_{i_6}, Act_i, Exp_i\}\\ r'_{i_7}&= \{Tri'_{i_7}, Act_i, Exp_i\}\\ r'_{i_8}&= \{Tri'_{i_8}, Act_i, Exp_i\}\\ r'_{i_9}&= \{Tri'_{i_9}, Act_i, Exp_i\}\\ r'_{i_{10}}&= \{Tri'_{i_{10}}, Act_i, Exp_i\}\\&\dots \\ r'_{i_{\dots }}&= \{Tri'_{i_{\dots }}, Act_i, Exp_i\} \end{aligned}$$

  First, the expected condition of the exception is negated. Then, it is expanded in a power set manner to clarify and analyze each condition. Finally, each element of the power set is added to a norm through the union. According to the definition of power set cardinality, the power set cardinality of set $\thicksim Exp\_mu_j$ is denoted by Card($2^{\thicksim Exp\_mu_j}$). Since the elements of $Tri'_{i_1}$ are the same as the elements of $Tri_i$, so $Tri'_{i_1}$ is removed. Thus far, the number of trigger conditions for the generated norms about norm $r_i$ is Card($2^{\thicksim Exp\_mu_j}$)-1, and the related actions and expectations are taken as $Act_i$ and $Exp_i$. Finally, the number of generated norms about norm $r_i$ is Card($2^{\thicksim Exp\_mu_j}$)-1, which improves the completeness of norm evolution.

  The power set approach works by considering all possible subsets of a given set of norms. This ranges from the case where there are no norms at all to the case where all norms are included, contributing to the completeness of norm evolution. The power set of a set is the set of all subsets, including the empty set and the set itself. When applied to norm evolution, this approach can help ensure that all possible combinations of norms are considered during the analysis. This approach can be particularly useful when designing systems that are adaptable to change and able to accommodate new norms, providing a deeper understanding of how norms interact and evolve within a norm set.

• (2) Trade-off method

  Although the power set method solves the problem of completeness, the number of power set elements exponentially increases. This results in the low efficiency of triggering conditions for refining norms, occupies more memory space, and affects the time needed to judge the triggering conditions of all the norms.

  The trade-off method is designed to strike a balance between efficiency and completeness by searching for norms with ambiguous triggers rather than norms with harsh triggers. This reduces the search space and improves efficiency without having to search for every power set combination. This method also ensures completeness.

  From the definition of set cardinality, the power set cardinality is calculated. The space complexity of the system is S(n) = O($2^n$), and the time complexity is T(n) = O(2n). To improve the efficiency of the refinement trigger conditions and reduce the space complexity of the system, this paper presents a method that trades of efficiency and completeness.

  The trigger condition includes conditions for multiple constraints. For example:

  $$\begin{aligned} Tri_i \cup \thicksim exp_{j_1} \left\{ \begin{array}{l} Tri_i \cup \thicksim exp_{j_1}, exp_{j_2}\\ Tri_i \cup \thicksim exp_{j_1}, \thicksim exp_{j_2}\\ Tri_i \cup \thicksim exp_{j_1}, exp_{j_3}\\ Tri_i \cup \thicksim exp_{j_1}, \thicksim exp_{j_3}\\ Tri_i \cup \thicksim exp_{j_1}, exp_{j_4}\\ Tri_i \cup \thicksim exp_{j_1}, \thicksim exp_{j_4}\\ Tri_i \cup \thicksim exp_{j_1}, exp_{j_5}\\ Tri_i \cup \thicksim exp_{j_1}, \thicksim exp_{j_5}\\ Tri_i \cup \thicksim exp_{j_1}, exp_{j_2}, exp_{j_3}\\ Tri_i \cup \thicksim exp_{j_1}, \thicksim exp_{j_2}, exp_{j_3}\\ Tri_i \cup \thicksim exp_{j_1}, exp_{j_2}, \thicksim exp_{j_3}\\ Tri_i \cup \thicksim exp_{j_1}, \thicksim exp_{j_2}, \thicksim exp_{j_3}\\ Tri_i \cup \thicksim exp_{j_1}, exp_{j_2}, exp_{j_4}\\ Tri_i \cup \thicksim exp_{j_1}, \thicksim exp_{j_2}, exp_{j_4}\\ Tri_i \cup \thicksim exp_{j_1}, exp_{j_2}, \thicksim exp_{j_4}\\ Tri_i \cup \thicksim exp_{j_1}, \thicksim exp_{j_2}, \thicksim exp_{j_4}\\ Tri_i \cup \thicksim exp_{j_1}, exp_{j_2}, exp_{j_5}\\ Tri_i \cup \thicksim exp_{j_1}, \thicksim exp_{j_2}, exp_{j_5}\\ Tri_i \cup \thicksim exp_{j_1}, exp_{j_2}, \thicksim exp_{j_5}\\ Tri_i \cup \thicksim exp_{j_1}, \thicksim exp_{j_2}, \thicksim exp_{j_5}\\ Tri_i \cup \thicksim exp_{j_1}, exp_{j_3}, exp_{j_4}\\ Tri_i \cup \thicksim exp_{j_1}, \thicksim exp_{j_3}, exp_{j_5}\\ Tri_i \cup \thicksim exp_{j_1}, exp_{j_3}, \thicksim exp_{j_5}\\ Tri_i \cup \thicksim exp_{j_1}, \thicksim exp_{j_3}, \thicksim exp_{j_5}\\ \dots \\ Tri_i \cup \thicksim exp_{j_1}, \thicksim exp_{j_2}, \thicksim exp_{j_3}, \\ \quad \thicksim exp_{j_4}, \thicksim exp_{j_5}\\ \end{array} \right. \end{aligned}$$

  (3)

  Therefore, to improve the effectiveness of the norms that will be triggered, trigger conditions with too many constraints are removed. Ultimately, the trigger conditions of the generated norms about norm $r_i$ are updated as follows:

  $$\begin{aligned} Tri'_{i_1}&= \{tri_{i_1}, tri_{i_2},\dots \}\nonumber \\ Tri'_{i_2}&= \{tri_{i_1}, tri_{i_2},\dots , \thicksim exp_{j_1}\}\nonumber \\ Tri'_{i_3}&= \{tri_{i_1}, tri_{i_2},\dots , \thicksim exp_{j_2}\}\nonumber \\ Tri'_{i_4}&= \{tri_{i_1}, tri_{i_2},\dots , \thicksim exp_{j_3}\}\nonumber \\ Tri'_{i_5}&= \{tri_{i_1}, tri_{i_2},\dots , \thicksim exp_{j_4}\}\nonumber \\ Tri'_{i_6}&= \{tri_{i_1}, tri_{i_2},\dots , \thicksim exp_{j_5}\}\nonumber \\&\quad \dots \nonumber \\ Tri'_{i_{\dots }}&= \{tri_{i_1}, tri_{i_2},\dots , \thicksim exp_{j_5}, \dots \}\end{aligned}$$

  (4)
  $$\begin{aligned} r'_{i_1}&= \{Tri'_{i_1}, Act_i, Exp_i\}\nonumber \\ r'_{i_2}&= \{Tri'_{i_2}, Act_i, Exp_i\}\nonumber \\ r'_{i_3}&= \{Tri'_{i_3}, Act_i, Exp_i\}\nonumber \\ r'_{i_4}&= \{Tri'_{i_4}, Act_i, Exp_i\}\nonumber \\ r'_{i_5}&= \{Tri'_{i_5}, Act_i, Exp_i\}\nonumber \\&\quad \dots \nonumber \\ r'_{i_{\dots }}&= \{Tri'_{i_{\dots }}, Act_i, Exp_i\} \end{aligned}$$

  (5)

  The trade-off method reduces precise conditions to vague ones, making the trigger conditions more general and eliminating the need to match each condition exactly in detail. Based on these results, with the increase in the number of elements in $\thicksim Exp\_mu_j$, $r'_i$ increases linearly and occupies a smaller amount of memory space. According to the above analysis, the time complexity of the system is T(n) = O(2n), and the space complexity is S(n) = O(2n). This approach can also improve the efficiency of the refined trigger conditions of the norms.

• (3) Improved Crossover Operator $(\otimes )$

  The above trade-off method solves the problem of completeness and refines the trigger condition of norm efficiency at the same time, but only the evolution of mutated norms is affected. To make the evolution of norms more complete, based on the trade-off method, this paper improves upon the crossover operator in Mone. Specifically, the trigger condition and expectation of one norm enrich the expectation and trigger of another norm. This operator will realize the autonomous evolution norm more completely and effectively. The notation ‘$\otimes$’ represents an operator that fuses multiple norms into one norm. Suppose there are two norms $r_i, r_j \in N$, and $r_i \prec r_j$. Whether $r_j$ is a mutated or obeyed norm, the generated norm $r_k = \{Tri_k, Act_k, Exp_k\}$ can be expressed as:

  $$\begin{aligned} r_k&= r_i \otimes r_j \nonumber \\&= \left\{ \begin{array}{ll} \{Tri_i \cup exp_{j_k}, Act_i, Exp_i \\ \quad \text {if } r_i\text {mutated norm},r_j\text {obeyed norm}.\\ \{Tri_i \cup \thicksim exp_{j_k}, Act_i, Exp_i \cup \thicksim tri_{j_l}\}\\ \quad \text {if } r_i\text {mutated norm},r_j\text {mutated norm}. \end{array} \right. \end{aligned}$$

  (6)

  First, the trigger conditions of the generated norms about norm $r_i$ are updated as follows:

  $$\begin{aligned} Tri'_{i_1}&= \{tri_{i_1}, tri_{i_2},\dots \}\nonumber \\ Tri'_{i_2}&= \{tri_{i_1}, tri_{i_2},\dots , \thicksim exp_{j_1}\}\nonumber \\ Tri'_{i_3}&= \{tri_{i_1}, tri_{i_2},\dots , \thicksim exp_{j_2}\}\nonumber \\ Tri'_{i_4}&= \{tri_{i_1}, tri_{i_2},\dots , \thicksim exp_{j_3}\}\nonumber \\ Tri'_{i_5}&= \{tri_{i_1}, tri_{i_2},\dots , \thicksim exp_{j_4}\}\nonumber \\ Tri'_{i_6}&= \{tri_{i_1}, tri_{i_2},\dots , \thicksim exp_{j_5}\}\nonumber \\ \dots \nonumber \\ Tri'_{i_{\dots }}&= \{tri_{i_1}, tri_{i_2},\dots , \thicksim exp_{j_5}, \dots \} \end{aligned}$$

  (7)

  Then, the expectation conditions of the generated norms about norm $r_i$ are updated as follows:

  $$\begin{aligned} Exp'_{i_1}&= \{exp_{i_1}, exp_{i_2},\dots \}\nonumber \\ Exp'_{i_2}&= \{exp_{i_1}, exp_{i_2},\dots , \thicksim tri_{j_1}\}\nonumber \\ Exp'_{i_3}&= \{exp_{i_1}, exp_{i_2},\dots , \thicksim tri_{j_2}\}\nonumber \\ Exp'_{i_4}&= \{exp_{i_1}, exp_{i_2},\dots , \thicksim tri_{j_3}\}\nonumber \\ Exp'_{i_5}&= \{exp_{i_1}, exp_{i_2},\dots , \thicksim tri_{j_4}\}\nonumber \\&\quad \dots \nonumber \\ Exp'_{i_{\dots }}&= \{exp_{i_1}, exp_{i_2},\dots , \dots \} \end{aligned}$$

  (8)

  Finally, the generated norms about norm $r_i$ are updated as follows:

  $$\begin{aligned} r'_{i_{1_1}}&= \{Tri'_{i_1}, Act_i, Exp'_{i_1}\}\\ r'_{i_{2_1}}&= \{Tri'_{i_2}, Act_i, Exp'_{i_1}\}\\ r'_{i_{3_1}}&= \{Tri'_{i_3}, Act_i, Exp'_{i_1}\}\\ r'_{i_{4_1}}&= \{Tri'_{i_4}, Act_i, Exp'_{i_1}\}\\ r'_{i_{5_1}}&= \{Tri'_{i_5}, Act_i, Exp'_{i_1}\}\\ r'_{i_{6_1}}&= \{Tri'_{i_6}, Act_i, Exp'_{i_1}\}\\&\quad \dots \\ r'_{i_{1_3}}&= \{Tri'_{i_1}, Act_i, Exp'_{i_3}\}\\ r'_{i_{2_3}}&= \{Tri'_{i_2}, Act_i, Exp'_{i_3}\}\\ r'_{i_{3_3}}&= \{Tri'_{i_3}, Act_i, Exp'_{i_3}\}\\ r'_{i_{4_3}}&= \{Tri'_{i_4}, Act_i, Exp'_{i_3}\}\\ r'_{i_{5_3}}&= \{Tri'_{i_5}, Act_i, Exp'_{i_3}\}\\ r'_{i_{6_3}}&= \{Tri'_{i_6}, Act_i, Exp'_{i_3}\}\\&\quad \dots \\ r'_{i_{1_5}}&= \{Tri'_{i_6}, Act_i, Exp'_{i_5}\}\\ r'_{i_{2_5}}&= \{Tri'_{i_6}, Act_i, Exp'_{i_5}\}\\ r'_{i_{3_5}}&= \{Tri'_{i_6}, Act_i, Exp'_{i_5}\}\\ r'_{i_{4_5}}&= \{Tri'_{i_6}, Act_i, Exp'_{i_5}\}\\ r'_{i_{5_5}}&= \{Tri'_{i_6}, Act_i, Exp'_{i_5}\}\\ r'_{i_{6_5}}&= \{Tri'_{i_6}, Act_i, Exp'_{i_5}\}\\&\quad \dots \end{aligned}$$

  In Eq. (6), norms $r_i$ and $r_j$ are passed through the operator $\otimes$ to form a new norm $r_k$, where the expectation condition is $exp_{j_k} \in Exp_j$ and the trigger condition is $tri_{j_l} \in Tri_j$. First, the trigger condition is added to the expected condition, which means that a certain trigger condition must be met before a condition is expected to occur. The expected condition is then added to the trigger condition, which means that certain conditions or actions are triggered after certain expected conditions are met. The improved crossover operator ($\otimes$) implies not only adding some possible conditions to the trigger conditions but also adding possible conditions to the expectations.

  An improved crossover operator is proposed that specifically adds desired conditions to triggering conditions and triggering conditions to expected conditions. This approach is intended to enhance the relevance and validity of the specification by ensuring that the specification is triggered under the appropriate conditions and leads to the desired result. Specifically, by adding the desired conditions to the triggering conditions, the crossover operator renders the specification more context-aware and situationally appropriate. Adding triggers to the expected conditions means that the system anticipates certain situations and prepares specifications accordingly. The improved crossover operator creates specifications that are more closely aligned with the specific goals and constraints of the MAS. This approach balances the need for stability in the behavior of the specification (by maintaining the trigger conditions) with the need for innovation (by adding new expected conditions); the system does not become too chaotic or rigid, and the relevance of the specification is increased. The improved crossover operator allows the MAS to continually develop norms that are robust to changes in the environment, and by combining the trigger conditions and expected conditions in new ways, norms can be generated that handle more complex scenarios. This feature is important for solving problems that require a nuanced understanding of causality in the environment. This forward-looking approach allows the MAS to better adapt to potential future environmental states.

Algorithm and details

According to the above descriptions and representations,the pseudocodes of the proposed improved method are provided. This paper also summarizes the main steps of these pseudocodes.

• (1) Algorithm 1: System operation oon the GBBopen platform

  Step 1. Initialize or enter global variables such as vc, threshold, and PE. Among these, vc is the number of agents required by the system.

  Step 2. Enter the norm base N related to the agent, including trigger, action, and expectation. Enter the knowledge source KS in the GBBopen platform, including ‘trigger-events’, ‘precondition-function’, ‘rating’, ‘execution-function’, ‘postcondition-function’, and bounded functions.

  Step 3. Determine whether to exit GBBopen by comparing the values of t and threshold, the length of the queue pending-ksas, and the value of quiescence.

  Step 4. Output the result of the recorded norm execution, PE.

• (2) Algorithm 2: Evolution of norms

  Step 1. Enter the PE.

  Step 2. Initialize the global variables, including the abnormal execution norm (mn), normal execution norm list (nl), and abnormal execution norm list (al).

  Step 3. For each $v_i$, check the mutated norm in the PE. If a mutated norm is found, the norm is assigned to mn.

  Step 4. After obtaining mn, judge the execution of the norm. If it is mutated, this norm is added to al. If it is obeyed, this norm is added to nl.

  Step 5. Perform the improved crossover operation (call Algorithm 3).

  Step 6. Call Algorithm 3 to obtain the return value, write it to the.txt file, and load it.

  Step 7. Set the variable to $\phi$ and empty the.txt file.

• (3) Algorithm 3: Improved crossover operator

  Step 1. Initialize the global variable K.

  Step 2. Enter the variables mn, al, and nl.

  Step 3. If $a \in al$, then execute operator $\otimes$. If $n \in nl$, then run the otimes operator.

  Step 4. Output the generated norm.

Algorithm 1

System operation process in GBBopen platform

Algorithm 2

Evolution of norms

Evaluation

Achieving an appropriate trade-off between efficiency and completeness is a common problem in computational approaches to norm evolution. Efficiency refers to the computational resources, including time and memory, required to execute a strategy. In the context of norm evolution, an efficient system will quickly identify and implement valid norms without exhaustively searching for all possible combinations. On the other hand, completeness implies that the policy has considered all possible norm bases to ensure that no potentially valuable norm combinations are overlooked. A complete approach will be guaranteed to find the best possible norm base. However, this approach may be computationally infeasible for larger norm bases due to combinatorial explosion.

Efficiency refers to the time or resource consumption required for a method to process a large norm base. Integrity represents the number of norms that a method can cover when it polls all norms in the norm base. In this paper, the number of evolved norms (NEN) is used as a potential metric to generate the minimum number of norms in the process of evolution. A smaller number of norms helps reduce the computational complexity and improve the computational efficiency. By generating fewer norms, the system can process and apply those norms more quickly, which increases the processing power of the system in an open environment. The cycle time needed for the proposed method to poll all norms is another important index for evaluating the performance of the norm evolution method. The efficiency of the method can be assessed by considering the amount of time needed to poll all the norms in the norm base after the specification evolves. The shorter cycle time representation method can process the norms faster, thus improving the real-time response capability of the system. During evolution, methods should be able to both generate fewer norms and process them quickly to improve the performance and responsiveness of the system. Therefore, comprehensively considering the efficiency and integrity indicators, this paper presents a potential metric for quantifying system evolution: the number of evolved norms (NEN) and the cycle time needed to poll all norms, which can be used to effectively measure the efficiency and integrity of the method. NEN is measured by evolving to obtain the fewest norms while simultaneously achieving greater completeness to improve the processing ability of the MAS in an open environment. When evaluating the specification evolution methods, each method is evaluated by considering the time it takes to poll all the norms in the norm base after completing the norm evolution as an evaluation metric.

Algorithm 3

Improved Crossover operation

Case study

In this section, a case study of an unmanned vehicle system performing a weeding task is simulated. The improved method of norm-autonomous evolution is implemented to improve the evolution of the norms to adapt to the environment. First, the design of the UVS is introduced, including the weeding tasks and attributes of the unmanned vehicles. Then the implementation platform for the UVS is introduced. Subsequently, the steps of the UVS are summarized, including the entered weeding location, norms, and number of unmanned vehicles. Finally, the experimental results of the autonomous evolution of the UVS norms are given and analyzed.

Design of the UVS

This case study simulates the execution of weeding tasks in a rectangular area. In the UVS, the weeding sites of the task are represented by set M, where M = $\{m_1, m_2, \ldots \}$. $m_i \in M$ is the ith weeding site and is represented in two-dimensional rectangular coordinates. The coordinates are in units of m. For example, $m_1 = (3,5)$. The set $\hbox {V} = \{v_1, v_2, \ldots \}$ represents unmanned vehicles. The parameters of the unmanned vehicle $v_i \in V$ are explained as follows. A description of the parameters is shown in Table 1.

$$\begin{aligned} v_i \ =\ < id, p, p\_lst, p\_v, a\_v, r, s, tp, \upsilon , \omega , d, fs > \end{aligned}$$

Table 1 The parameter description of unmanned vehicle $v_i$

In this simulation environment, this paper constructs the motion norms for unmanned vehicles corresponding to the motion control mode. This paper considers both vertical and horizontal scenarios, as well as the corresponding expected behavior. The norms considered here were originally constructed and expressed in natural language through text. This paper organizes these norms into a table. This table provides subscribed events, trigger conditions, expected conditions for judging results, and specific aspects to be considered. Table 2 lists the norm bases of the unmanned vehicles. There are approximately 20 such norms. This paper designs a series of norms for completing collective tasks. The designed norms guide individual unmanned vehicles. All unmanned vehicles share these norms. The vehicles freely choose and follow the designed norms to move toward the mowing area. Ultimately, they accomplish the weeding tasks.

Table 2 The design of the Norms in UVS ($\alpha$, $\beta$ are thresholds; Where $v_i.{d(t)}$ represents the value of $v_i.d$ at time t)

Two kinds of unmanned vehicles are set to reach the weeding site. Norms $r_1$ and $r_2$ judge $v_i$ as follows. The difference between goal following and trajectory following is that goal following considers the current position coordinates of the target and trajectory following proceeds sequentially according to the position coordinates of the target. Norm $r_3$ states that $v_i.fs$ changes by following the vehicle in front of $v_i$. The effective method of following will be passed on to subsequent unmanned vehicles. The lateral and vertical control of $v_i$ is governed by the norms $r_4$ and $r_5$. Norm $r_4$ affects the longitudinal and lateral control of all unmanned vehicles in the UVS by the spacing between adjacent unmanned vehicles. In the considered rectangular area, it is recommended that the linear velocity range of $v_i$ be 0.05 $\sim$ 0.5 m/s. The angular velocity of $v_i$ is between 0 $\sim$ 180 rad/s. In this experiment, the value of parameter $\gamma$ is set to 0.2 m according to practical experience. If $v_i$ is found to be farther from the vehicle in front, its speed is increased to reduce the distance to the vehicle in front, and vice versa. If there is a need to create a gap between unmanned vehicles, the norm $r_5$ is used to instruct $v_i$ to brake quickly in case of an expected accident. Norm $r_6$ is a rule that detects the braking of $v_i$. Norm $r_7$ is a rule for applying the brakes of $v_i$. Norm $r_8$ is a rule that detects $v_i.s$. Norm $r_9$ is a rule that causes $v_i.s$ change from stuck to normal. Norm $r_{10}$ is a rule that detects whether $v_i$ is caught in a trap. Norm $r_{11}$ states that the rear of the trapped $v_i$ bypasses the trap. Norm $r_{12}$ makes $v_i$ safer to in avoiding discovered traps. Norm $r_{13}$ obtains the position of $v_i$. Norm $r_{14}$ causes $v_i$ to regain its position. Norm $r_{15}$ is a rule for detecting whether there is an obstacle in front of $v_i$. Norm $r_{16}$ keeps $v_i$ away from obstacles. Norm $r_{17}$ is a rule for detecting whether $v_i$ has a collision. Norms $r_{18}$, $r_{19}$, $r_{20}$, and $r_{22}$ are used to detect $v_i.\upsilon$ and $v_i.\omega$ of $v_i$ in different situations. Norms $r_{21}$ and $r_{23}$ quickly adjust $v_i.\upsilon$ or $v_i.\omega$ to prevent $v_i$ from moving away from $v_i.tp$.

Steps of the UVS

The implementation platform of the case study is a PC, and the application simulation is implemented and executed on GBBopen. The steps taken by the UVS in performing weeding tasks are summarized as follows:

• Step 1: Initialization of the UVS parameters

  • a. Suppose three weeding locations are input to UVS. The corresponding position coordinates are (12, 14), (8, 7), and (11, 13). The units of their coordinates are m.

  • b. Assume that the number of unmanned vehicles in the UVS is 4. Each unmanned vehicle is mapped as a single point. Their coordinates are $v_1.p = (0,0)$, $v_2.p = (1,3)$, $v_3.p = (3,0)$, and $v_4.p = (0,1)$.

  • c. Set the values of $\alpha$ and $\beta$ in the norm to 0.4 m and 0.3 m, respectively.

• Step 2: Initialize the parameters of the tasks.

  • a. The distance between $v_i.p$ and ${\bar{M}}$ is calculated. According to the distance, form a leader-follower columnar formation. The queue is in the order $v_2$, $v_3$, $v_4$, $v_1$. These vehicles are denoted as L, $F_1$, $F_2$, and $F_3$, respectively.

  • b. The distance between L and M is calculated. The weeding tasks are performed in order, from near to far. They are in the order of (8,7), (11,13), (12,14).

• Step 3: The shared norms are freely triggered by an unmanned vehicle $v_i$ according to its attribute values. The corresponding actions are performed, and the expected conditions are judged to achieve the common goal of weeding.

• Step 4: An unmanned vehicle $v_i$ automatically updates the BB every time it executes a norm (e.g., $v_i.p$, $v_i.d$, $v_i.\upsilon$, $v_i.\omega$).

• Step 5: A weeding point $m_i \in M$ is completed, and $m_i$ is deleted from the set M.

• Step 6: When M = $\phi$, the task ends, which indicates that the UVS has completed the weeding task. Otherwise, return to Step 3.

Analysis and evaluation of the norm evolution results of the UVS

Figure 3 shows a hypothesis for this experiment. Assume $v_2$ becomes stuck in a pit or trap during task execution. Figure 3 records the trajectories of four unmanned vehicles performing weeding tasks. Figure 3 shows that vehicles are arranged in the order $v_2$, $v_3$, $v_4$, $v_1$. These unmanned vehicles first perform weeding tasks through goal following. Because $v_2.d$ is greater than 0.2 m, $v_2$ triggers the norm $r_2$. $v_3$, $v_4$, and $v_1$ trigger the norm $r_3$. Each unmanned vehicle moves toward its respective target point $v_i.tp$ or stops. This is because when $v_i.d$ changes for each unmanned vehicle, the norms $r_4$ and $r_5$ are triggered, and the corresponding actions are executed. The unmanned vehicle gradually becomes denser at sparse trajectory points as each $v_i.d$ decreases. As $v_2.d$ approaches 0.2 m, these unmanned vehicles perform weeding tasks in trajectory-following mode. Figure 4 shows part of the LEG plot for this case study. In this figure, the UVS represents all unmanned vehicles, and the state S is represented by a rectangle. In system state S, the execution of norm N is denoted as a circle. Line segments with arrows indicate the results of norm execution. A solid line indicates that the norm was obeyed. A dotted line indicates a mutated norm. As shown in Fig. 4, both obeyed and mutated norms exist. The norms are obeyed until the state $S_x$. At system state $S_x$, $r_4$ is a mutated norm. In the system state $S_y$, $r_8$, $r_{10}$, $r_{13}$, $r_{18}$, and $r_{19}$ are mutated norms.

Fig. 3

Trajectory of four unmanned vehicles during weeding mission

Fig. 4

LEG of four unmanned vehicles on a weeding mission

Unmanned vehicle $v_2$ becomes stuck in a pit or trap while traveling to (11, 13) after performing the weeding task at (8, 7). When each unmanned vehicle performs the weeding task, the status of the unmanned vehicle, the triggered norms and the results of the implementation of the norms are recorded. $v_2$ is executes norm $r_4$ with system state $S_x$ as $v_2.d = 0.3 \,\hbox {m}$. Because the execution result of $Act_4$ is inconsistent with $Exp_4$, $r_4$ is a mutated norm. The UVS detects that norm $r_4$ is a mutated norm through the LEG. Because the result of running $r_4$ is $v_2.d = 0.3 \,\hbox {m}$, and the status of $v_2$ triggers norm $r_8$, the operator of the evolution norm immediately starts. In state $S_y$, $v_2.s$ = normal; the expectation condition of norm $r_8$ is $v_i.s$ = stuck. Therefore, $r_8$ is a mutated norm. In state $S_y$, unmanned vehicle $v_2$ is not detected as being in a pit or trap, and the expected condition of norm $r_{10}$ is that $v_i$ in a trap. Therefore, norm $r_{10}$ is obeyed. Moreover, suppose that another unmanned vehicle $v_4$ deviates from the target direction and that $v_4$ executes norm $r_4$ with a mutation, resulting in other mutated norms. The expectation condition of norm $r_{13}$ is that $v_i.p$ is obtained. The expectation condition of norm $r_{18}$ is actual $v_i.\upsilon \ll$ planned $v_i.\upsilon$. The expectation condition of norm $r_{19}$ is actual $v_i.\omega \ll$ planned $v_i.\omega$. For unmanned vehicle $v_4$ in state $S_y$, $v_4.p$ is not obtained, the actual $v_4.\upsilon$ is 0.07 m/s, and planned $v_4.\upsilon$ is 0.09 m/s. Therefore, ‘actual $v_4.\upsilon \nless$ planned $v_4.\upsilon$’ holds. Additionally, the actual $v_4.\omega$ is 17 rad/s, and the planned $v_4.\omega$ is 24 rad/s, so ‘actual $v_4.\omega \nless$ planned $v_4.\omega$’ holds. Therefore, norms $r_{13}$, $r_{18}$, and $r_{19}$ are mutated norms.

This paper derives the results of different norm evolution methods separately and compares them for analysis. Table 3 shows the results of implementing the Mone method of norm evolution for norm $r_4$. The details are provided in Sect. “Brief introduction to Mone”. As shown in Table 4, norm $r_4$ adds trigger conditions, increases the trigger condition constraint of the norm and improves the rigor of the norm trigger. If the subsequent states within this system do not reach trigger condition $Tri'_4$, then none unmanned vehicles within the UVS can be controlled for motion. Next, the power set approach is applied to evolve the norm $r_4$. This approach can refine the trigger conditions for the norm. The details of this method are provided in Sect. “Correlation methods”. The trigger conditions of the $2^5$-1 norms need to be judged in the UVS, and the execution efficiency is relatively low. Due to the execution of the norm evolution algorithm, the generated norms are stored in a ‘txt’ file and loaded. The generated norms are stored in the UVS, similar to the original norms; thus, some of the memory of the PC is consumed. The details of the trade-off between efficiency and completeness proposed in this paper are provided in Sect. “Correlation methods”. Table 5 shows that through the trade-off method, norm $r_4$ autonomously evolves five norms, and the execution efficiency is higher than that of the power set method. This trade-off method not only ensures completeness and effectiveness but also reduces the memory of the PC.

Table 6 shows the results of the evolution of the norm $r_4$ with an improved crossover operator based on the trade-off approach. The results of the proposed strategy are described in Table 6; this method ensures integrity and efficiency and provides new resources for evolving other norms. The evolution of the above norms updates the norm base of the UVS and generates new norms that run throughout the system. The updated norms ensure that the same norm exceptions are avoided when subsequent unmanned vehicles perform their tasks. When the task has been performed, the autonomous evolution of norms temporarily stops. As long as the UVS is performing tasks, the norms of the system will continue to evolve from beginning to end.

Table 3 The result of evolution norm $r_4$ is based on Mone

Table 4 The result of evolution norm $r_4$ is based on power set method

Table 5 The result of evolution normm $r_4$ is based on trade-off method

Table 6 The result of evolution norm $r_4$ is based on improved crossover operator

Fig. 5

Comparison of the proposed Mone, power-set method, trade-off method, and improved crossover operator

Figure 5 shows the curve of the effect of different mutation expectation conditions on the number of newly generated norms. The green rectangles represent the Mone method. The purple triangles represent the power-set method. The red diamonds represent the trade-off method. The black circles represent the newly proposed strategy, an improved crossover operator. Figure 5 clearly shows that Mone does not generate new norms as the mutation expectation condition increases. All the other evolutionary methods generate new norms and show an increasing trend. The number of norms generated by the power set method shows exponential growth. However, the trade-off method and the improved crossover operator tend to grow linearly. When the number of expectation conditions for all mutated norms is 1, the number of norms generated by the power set method and trade-off method is 1, while the number of norms generated by the improved crossover operator is 18. When the number of expectation conditions for all mutated norms is 4, The number of norms generated by the power set method is 3.75 times that of the trade-off method. The number of norms generated by the improved crossover operator is five times that of the trade-off method. When the number of expectation conditions for all mutated norms is 7, the number of norms generated by the power set method is 18.14 times that of the trade-off method. The number of norms generated by the improved crossover operator is 3.29 times that of the trade-off method. From the above analysis, it is seen that the power set method refines the trigger conditions of norms and improves the completeness of norm evolution. However, this approach ignores efficiency, while the trade-off method improves efficiency. The improved crossover operator has greater completeness and effectiveness in the direction of autonomous norm evolution, as it not only evolves mutated norms but also evolves obeyed norms.

Fig. 6

Comparing polling norm-base periods the Mone, power-set method, trade-off method, and improved crossover operator

Figure 6 shows the curve of the effect of different expectation conditions on the polling norm-base periods. The green rectangles represent the Mone method. The purple triangles represent the power-set method. The red diamonds represent the trade-off method. The black circles represent the newly proposed strategy, an improved crossover operator.

Figure 6 clearly shows that the Mone method does not change the period of the UVS polling norm. This is because the Mone method does not generate any additional norms as it evolves them. The power-set method changes the period of the UVS polling norm extremely quickly. This is because the power-set method generates an exponential level of norms as it evolves them. The trade-off method slightly increases the period of UVS norm polling. This is because the trade-off method generates linearly increasing norms as it evolves the norms. The improved crossover operation smoothly increases the period of UVS norm polling. This is because the improved crossover operation generates norms at a rate between exponential and linear in the process of evolution.

To study the relationship between the occurrence rate of mutations and the number of evolutionary norms, this paper examines different occurrence rates of mutations through crossover operations. In this experiment, the mutation speed was set at 2 times per minute, 4 times per minute, 6 times per minute, 8 times per minute, and 10 times per minute. The time needed to obtain statistical results in this experiment was 5 min. The number of mutations expected of in this experiment was 5. The experimental results are shown in Table 7.

Table 7 The relationship between the occurrence rate of mutation and the number of evolutionary norms

If the rate of mutation is too high, the evolutionary process may be too fast or even collapse. In contrast, if the mutation rate is too low, the system may not be provided with enough new conditions to adapt to the environmental changes, thus causing the evolutionary process to stagnate. To cope with evolutionary demands at different stages, the probability of mutation occurrence can be adjusted. The appropriate occurrence of anomalies can ensure that the system norms have good exploration and development capabilities and avoid system evolutionary collapse and incomplete evolution. For example, when the number of norms in the system is too high, the expectation conditions of the norms can be adjusted to decrease the probability of mutation occurrence. When the system is in its early stages, the expectation conditions of the norms can be adjusted to increase the probability of anomalies.

Discussion

In this paper, a method is proposed to modify and adjust the norms of the system by detecting anomalous behavior in agents. This approach enables the MAS to maintain normal operation even in the presence of anomalous or faulty conditions. The current system is capable of recording the outcomes of the actions taken by each agent, and further analytical studies are needed to better understand the transitions of multiagent states. Moreover, this system has the ability to construct complex strategies by combining multiple norms, and it can evolve norms during runtime. This combination and evolution mechanism provides the norms in the system with strong adaptability. The system can be scaled when new requirements arise. For example, additional norms can be added to meet the needs of future environments.

The new strategy proposed in this paper has several limitations, although it is generalizable. This section discusses the limitations of the proposed strategy in three stages. In the preevolutionary stage, the MAS needs to have sufficient prior knowledge, i.e., norms. The norms are executed based on predefined trigger conditions. A norm is considered mutated if the result of its execution is not as expected. This situation is considered a special case and requires special treatment. In practice, the proposed strategy usually requires some prior knowledge to guide norm evolution. In the absence of sufficient prior knowledge, this strategy may not be able to evolve effective norms. In the evolutionary stage, the MAS needs to have enough exceptions, i.e., mutated norms. For the MAS, evolutionary norms obtained through anomalies are a way to cope with internal and external changes. Exceptions can include various contingencies, random states, changes in parameters and nonparameters, and changes in constants. The occurrence of these anomalies can affect the behavior and state of the system, and the MAS needs to adapt and evolve to cope with these changes. Uncertainty about the structure and number of anomalies is indeed a challenging problem. Because anomalies can be diverse and random in real-time applications, determining how to model and respond to them is a complex task. A common approach is to use learning algorithms and adaptive mechanisms that enable MASs to adjust their own norms and behaviors in response to real-time conditions and environmental changes. This can include the use of reinforcement learning, genetic algorithms, or other evolutionary algorithms to optimize the adaptability of the system and allow the system to continuously evolve and adapt to new anomalies. Of course, exception handling also depends on the design and application of the specific MAS. For domain-specific problems, specific methods and strategies, such as exception detection and fault tolerance mechanisms, may be required to handle exceptions. In actual operation, the proposed strategy may face difficulties in collecting mutated norms, especially in specific environments or special scenarios. Thus, the scarcity of mutated norms may limit the effectiveness and performance of the proposed strategy. In the process of norm evolution, the number of norm evolutions should be moderate. Overevolution can result in norms that are too complex to understand and maintain. During the experimentation or application of an unmanned vehicle system, a certain number of norms should be maintained. If there are too many norms, the efficiency of searching for norms among triggers can be very low. In the postevolutionary phase, the MAS needs to process the evolved norms, i.e., optimize them. If there are conflicts or overlaps between these norms, this may lead to inconsistency or redundancy in the norms and increase the complexity of the norms in the MAS.

To address these limitations, this paper incorporates other methods to improve the accuracy and reliability of mutation detection. To address the scarcity of mutated norms, one can try to collect real mutated norms in a specific environment or scenario. This process can be performed through different devices, thus ensuring that the exceptions cover a wide range of situations. To avoid the over-evolution of norms, a stop** condition can be set. For example, the number of evolutions in the UVS and a threshold for the number of norms can be set, and the process can be stopped when the performance of the system starts to decrease. By tracking norm changes during evolution, the evolutionary trend of the norms can be analyzed, potential redundancies, conflicts, and overcomplexity can be identified, and timely adjustments can be made. A selection mechanism is used to avoid the accumulation of useless norms by retaining only those norms that contribute the most to overall performance while discarding those that are infrequently used or contribute less to performance. Guidance can be provided manually, if necessary, to ensure the comprehensibility and usefulness of the norm set and to assess the generalizability of the norms. Some heuristics can be used to refine norms, such as merging similar norms or pruning unnecessary norm parts. Priorities can also be set for evolved norms to resolve conflicts or overlaps. Norms with a higher priority can be chosen for judgment in the event of a conflict. The use of model integration and learning can also be considered to solve the problem of conflicting or overlap** norms. By integrating the prediction results from multiple models, the strengths of each model are combined, and conflicts and inconsistencies between them are reduced. The norms are learned and tuned through a number of learning methods to further optimize the performance and accuracy of the model. In addition, since the safety and reliability of unmanned vehicles are of paramount importance, any approach using norm-based evolution needs to be rigorously tested and validated to ensure that it is safe and effective under real-world conditions.

Conclusions and future work

This paper presents a novel strategy for evolving norms, aiming to enhance the efficiency, completeness, and effectiveness of norm evolution in MASs. The proposed strategy enables the tracking of norm implementation in MASs. It introduces a trade-off approach that balances effectiveness and completeness and improves the crossover operator of Mone by considering evolutionary sources. The experimental results demonstrate that the efficiency, completeness, and effectiveness of norm evolution were better than those obtained by Mone in both static and dynamic environments. An important application of the proposed strategy is enabling multiple robots to autonomously perform exploration tasks such as cooperative control, formation control, and formation holding. Such robots could operate without the need for communication or learning to determine their actions. Furthermore, potentially dangerous situations could be detected during the execution of unknown experiments, prompting the automatic evolution of norms to adapt to the environment. The specification evolution strategy proposed in this paper contributes to the autonomy of specification evolution in MASs, enabling the system to adapt and refine its specifications without external intervention. This adaptability is crucial for operating in dynamic environments. This is because conditions in dynamic environments may change rapidly, and predefined norms may quickly become obsolete or inefficient. With the evolutionary strategy proposed in this paper, the system can monitor its performance and the effectiveness of its specifications. Based on this feedback, the system can autonomously adjust its specifications to better fit the current conditions. Over time, the system can evaluate the norms and adjusts its behavior accordingly. This process will iteratively produces norms that are well adapted to the environment. By employing this strategy, the specification evolution strategy enables the MAS to autonomously adapt its specification to changing environments, thus enhancing its robustness and flexibility. Such autonomy is particularly important for systems that operate for long periods of time, in unpredictable environments, or in situations where human intervention is impractical or undesirable.

In a MAS, further develo** and improving the learning algorithm and adaptive ability of agents are important research directions. Future studies can focus on how to design more precise and accurate trigger conditions and desired conditions. This includes develo** conditional forms that better reflect the characteristics of the actual environment and task and determining how to select and define appropriate desired conditions to facilitate the learning and behavioral evolution of the agent. Attention can also be given to MASs in more complex and diverse environments and task conditions. For example, dynamically changing environments, resource competition, and other factors can be considered, and it can be determined how to make agents achieve more efficient, robust, and effective behavior under these conditions. Further research could explore how to design robust norms and mechanisms that enable systems to remain stable in the face of various disturbances and failures and to respond to challenges through adaptability and cooperation. Multiagent systems have wide application potential in ecological protection and environmental management.

In the future, the proposed strategy can be applied to several other scenarios. For example, it can be used in self-driving systems to eliminate the need for human intervention in driving and decision-making when approaching shared spaces such as intersections. In addition, future research could focus on combining AI technology with the needs of ecological conservation to develop intelligent solutions for ecosystem monitoring, species protection, environmental governance, etc. Through in-depth research and practice in the above areas, the performance, intelligence, and adaptability of the multiagent systems can be further improved so that they can play a greater role in practical applications and contribute to solving complex problems and promoting sustainable development.