Solving the SAT Problem by Cell-Like P Systems with Channel States and Symport Rules

Wan, Xiaoming; Liu, Chuchuan; Luo, Yueguo

doi:https://doi.org/10.1155/2023/4993855

Discrete Dynamics in Nature and Society

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 4993855 | https://doi.org/10.1155/2023/4993855

Solving the SAT Problem by Cell-Like P Systems with Channel States and Symport Rules

Xiaoming Wan,¹Chuchuan Liu,²and Yueguo Luo³

Academic Editor: Ya Jia

Received30 Mar 2023

Revised22 May 2023

Accepted29 May 2023

Published10 Jun 2023

Abstract

Cell-like P systems with channel states, which are a variant of tissue P systems in membrane computing, can be viewed as highly parallel computing devices based on the nested structure of cells, where communication rules are classified as symport rules and antiport rules. In this work, we remove the antiport rules and construct a novel variant, namely, cell-like P systems with channel states and symport rules, where one rule is only allowed to be nondeterministically applied once per channel. To explore the computational efficiency of the variant, we solve the problem and obtain a uniform solution in polynomial time with the maximal length of rules 1. The results of our work are reflected in the following two aspects: first, communication rules are restricted to only one type, namely, symport rules; second, the maximal length of rules is decreased from 2 to 1. Our work indicates that the constructed variant with fewer rule types can still solve the problem and obtain better results in terms of computational complexity. Hence, in terms of computational efficiency, our work is a notable improvement.

1. Introduction

Membrane computing, a new type of bioinspired computing model, was first proposed by Păun [1]. Like quantum computing and DNA computing, it can be used to construct various models and implement computer algorithms. In [2], Professor Adleman successfully solved the Hamiltonian path problem with 7 vertices. Membrane computing is inspired by biological cells, tissues, and nervous systems and can be implemented as distributed and parallel computing devices. At present, many new variants have been proposed [3–9], and many variants have been proven Turing universal. In the theoretical research of membrane computing, various computationally hard problems were solved [10–14]. Recently, inspired by membrane computing, Roy et al. proposed a new type of neural computing system [15], which will promote the development of membrane computing. In the application field, membrane computing has achieved excellent results in optimization algorithms [16, 17], biology-based approaches [18, 19], mobile robots [20], fault diagnosis [21, 22], and other related applications [23, 24]. Recently, some excellent results have been achieved in the field of machine learning [25–27], and some researchers combine these two research areas and obtain excellent results [28]. Currently, many scholars are focusing on developments in this field. For additional details, some review books including papers (e.g., [29]) and the website https://ppage.psystems.eu/ can be viewed to obtain the latest information.

Currently, there are primarily three types of P systems in membrane computing: cell-like P systems [1], tissue-like P systems [30], and neural-like P systems [31]. This article focuses on cell-like P systems (for additional details, see [1]). In [32], symport/antiport rules were introduced to cell-like P systems, and substances can move between two membranes; moreover, multisets of objects in adjacent regions can exchange positions. Freund et al. proposed the concept of the channel state by combining tissue P systems [33]. When such a P system is running, the states of channels can be changed; in addition, channel states can activate rule execution. In [34], the channel state concept was introduced to cell-like P systems with symport/antiport rules to construct a new variant (abbreviated as CCSSA P systems); furthermore, the computational power of this new variant was explored, but the study of computational efficiency was not involved. Recently, Jiang et al. solved the problem with CCSSA P systems to study its computational efficiency [35]; however, in the study, although a uniform solution was obtained, the maximum rule length in the CCSSA P systems was 2. In terms of computational complexity, in membrane computing, a better method can solve the same NP-hard problem with a maximum rule length shorter than 2. If a membrane system can be constructed to solve the same problem, even with a shorter maximum rule length, then the system provides excellent computational property. For instance, in [36], was solved with a maximum length rule of 8; however, in [37], the maximum length of rules was reduced to 3 with the same model. Hence, the latter paper improved upon the previous research results. Obviously, in membrane computing, the length of rules is an important factor in terms of computational complexity.

In [1], the notion of maximal parallelism was proposed, and it is an attractive strategy: at a given moment, multiple rules on each channel can be selected to use, where one rule may be executed multiple times. Nevertheless, in flat maximal parallelism [38], such a rule can only be activated once. In our work, we adopt a new method combined sequential manner with flat maximal parallelism, that is, on an arbitrary channel, even if more than one rule is employed at a given moment, regardless of the direction of movement of the rules on the channel and the channel states considered in the rules, only one rule can be nondeterministically applied once.

With respect to CCSSA P systems, communication rules are classified into two categories, namely, symport rules and antiport rules. In this work, we consider constructing a variant with fewer rule types; if the new variant can still solve the same problem, it is shown that the constructed variant is powerful enough. Hence, we remove the antiport rules of CCSSA P systems, thereby constructing a novel variant, that is, cell-like P systems with channel states and symport rules (abbreviated as CCSS P systems).

The following are the contributions of our work:(i)We remove the antiport rules of CCSSA P systems, thereby constructing a novel variant, that is, cell-like P systems with channel states and symport rules, where, with respect to communication rules, ony symport rules are used. In such a variant, at a given moment, one rule is only allowed to be nondeterministically applied once per channel.(ii)Membrane division is introduced to the constructed variant, and its computational complexity is studied. Specifically, we solve and obtain a uniform solution. With respect to computational complexity, in the process of obtaining a theoretical proof, the maximal length of rules is 1, which reflects excellent performance. Our approach improves upon the current research method.

The structure of this paper is as follows: Section 2 mainly gives the definition of CCSS P systems. In Section 3, based on the problem, the computational efficiency of the system with respect to solving an NP-complete problem is studied; next, we explore a case study to verify the membrane system introduced in Section 4. Finally, some conclusions are presented, and future work is considered.

2. Cell-Like P Systems with Channel States and Symport Rules

In this section, we remove the antiport rules of CCSSA P systems, thereby constructing a novel variant, namely, CCSS P systems. For the formal language and the automaton theory, one can see [39, 40].

2.1. The Model of CCSS P Systems

Definition 1. A CCSS P system (degree , ) is a tuple:where(i) is an alphabet of objects(ii) represents the set of channel states (not necessarily exist in the set of )(iii) represents an infinite number of objects in the environment(iv) means the membrane structure denoted by a tree, and all nodes in the tree correspond to the labels of membranes. Relative to each membrane , denotes the outside region of membrane ;(v) denote the multisets of objects initially located in membrane (vi) represent channel states initially located on the membrane with label , which is the channel between membrane and (vii) is the output region (viii) represent a series of rules, including symport rules and division rules. In what follows, denotes the set of strings composed by the symbol in but without ; and given a string , the length of , which is the quantity of symbols in this string, is denoted by .

2.1.1. Symport Rules

where , . The length of a symport rule is . Formally, the maximum rule length of a CCSS P system can be denoted by the maximum rule length of symport rules in the system.

At a given moment, when the channel state of a membrane is and multiset exists inside (resp., outside) membrane , (resp., ) can be applied. By changing the channel state to , multiset is transferred to region (resp., membrane ), where represents the outer membrane of membrane ; if membrane is the skin membrane (it does not have the upper neighbor membrane), represents the environment.

2.1.2. Division Rules

where .

At a given moment, relative to object in membrane (except for the skin membrane), a division rule is executed, and two membranes with the same label will appear. Simultaneously, object including appear in new membranes to replace the previous object ; other objects in the initial membrane can be copied to the two newly generated membranes. It is important to emphasize that objects , , and can represent the same or different symbols. Moreover, the priority of division rules is higher than that of other rules, that is, if such a rule can be applied, symport rules cannot be applied at that moment.

2.2. System Operation

Even if more than one rule can be employed on a channel at a given moment, only a rule can be nondeterministically selected. In addition, if a rule can be executed more than one time at a given moment, this rule can only be activated once. When the application of a rule on a channel is completed, another rule can be applied. However, relative to all channels, at any step, rules can be executed in parallel on different channels; moreover, rules that can be executed must be executed on the corresponding channel. For example, multiset of objects exists in membrane 1, and channel state appears on the membrane; in addition, there are two rules associated with the membrane:

First, we consider the strategy of maximal parallelism to run a system. Initially, rules and are activated simultaneously, and each rule can be executed multiple times. As a result, all copies of objects and will be transferred to the environment. The computing process is shown in Figure 1.

Next, we consider the method combined sequential manner with flat maximal parallelism. When the system starts running, one rule ( or ) is only allowed to be nondeterministically applied on the channel; moreover, relative to the rule that is selected to be applied on the channel, it is only allowed to be applied once. Hence, the following two cases will occur: (i) Rule is only allowed to be applied. Eventually, one copy of object is transferred to the environment; (ii) Rule is only allowed to be applied. Eventually, one copy of object is transferred to the environment. The computing process is shown in Figure 2.

There is at most one channel on each membrane (between two adjacent regions). On each channel, we use the strategy of sequential manner combined with flat maximal parallelism to apply rules. The configuration of CCSS P systems is influenced by the following factors: the membrane structure, the multiset of objects in each region, and the channel states between adjacent regions. Before such a system starts running, we use the initial configuration to characterize it. As the computation continues, a series of configurations can be generated with the execution of rules at each step, and we call it a transition between two arbitrary configurations. A series of transitions is called a computation. Notably, the application of rules is based on sequential manner combined with flat maximal parallelism and follows the principle of nondeterminism [1]. Finally, when system operation stops, the halting configuration can be obtained; at that moment, the system halts, and no rules are applied, and the computing result is sent to the output region.

2.3. Recognizer CCSS P Systems

In membrane computing, the recognizer P systems are used to solve decision problems.

Definition 2. A recognizer CCSS P system (degree , ) is a tuple:where(i) represents an input alphabet ;(ii), ;(iii) (resp., ) represents the input (resp., output) region. In such a system, the other parameters are defined as in Definition 1. A recognizer CCSS P system can start from the initial configuration with an input multiset; eventually, occurs in the region .

Definition 3. denotes a decision problem, where is the instance, and represents a predicate of the instances. The problem can be solved in polynomial time, if the following holds:(i) is polynomially uniform by Turing machines(ii)Relative to , there is a pair of polynomial-time computable functions such that(a)suppose corresponds to an instance, , and is a natural number; additionally, represents an input multiset of CCSS P systems;(b)relative to , such a system is sound, that is, with regard to , if CCSS P systems have an accepting computation, then ;(c)relative to , such a system is complete, that is, suppose , relative to a problem, if , computations of with is an accepting one;(d)such a system is polynomially bounded, that is, CCSS P systems stop computation and reach the halting configuration after steps ( is a polynomial function).

Definition 4. The maximum rule length of a CCSS P system is equivalent to that of symport rules in the system. represents that the class of decision problems can be solved by a family of recognizer CCSS P systems in a uniform manner in polynomial time, where the maximal length of rules is .

3. A Uniform Solution to the Problem Based on CCSS P Systems

3.1. Constructing CCSS P Systems to Solve the

Theorem 5. .

Proof. A formula consists of Boolean variables and clauses:where , , and ; is the negation of a propositional variable .
A formula is encoded by :We codify with the multiset ( and ).To solve the , a recognizer P system is constructed:where(i) is the set of objects:(ii) is the set of channel states:(iii) denotes input objects(iv)(v)(vi).(vii) and .(viii)The rules in are as follows: (ix)The rules in are as follows: Division rules: Symport rules: (x)The rules in are as follows: Division rules: Symport rules: (xi)The rules in are as follows: Division rules: Symport rules: . .The whole computing process is primarily classified into two main phases: the generation phase and the checking/output phase. There are many rules involved in the generation phase, which is relatively complicated; hence, we will describe the generation phase in detail.
The generation phase generates all possible assignments of all variables and detects satisfiable clauses with these assignments. The computing process of this phase is shown in Figure 3. Initially, the computation begins by using the membranes with label 1 and label 4. Relative to these two membranes, a parallel computing process is used. The function of membrane 1 is to count the steps during system operation so that the computing result can be output at a given moment (we will explain this computing process later). Next, we only consider the computations involving membranes with label 4. At the initial configuration, the multiset of objects exists in membrane 4. Because the system includes object present in membrane 4, the system executes the division rule , which can generate two cells with the same label, and object will appear in the two membranes. Since division rules have higher priority, they can be used in membrane 4 because of . Finally, after steps, membranes will appear, and object will be generated in each membrane. Subsequently, relative to channel state on cell 4, rule is activated and can be executed times; the channel state will increase by 1 at each step, and finally, object will appear. At that moment, rule can be executed, sending object to the membrane with label 1. Based on the channel state on membrane 4, the system will execute at the next step, and object in membrane 4 will be sent to membrane 1; these objects will be used in the subsequent computing process.
After the above computation involving membrane 4 ends, the rules involving membrane 2 begin to be applied. The following computing process assigns values (e.g., “true” or “false”) to the variables of all clauses in a formula, and the rules from to are applied. First, is activated, and then, division rule is executed. Objects and appear in the two newly generated membranes. After a division rule is used, is sent to the region outside membrane 2 by applying rule , and the channel state of this membrane will be . Next, rules and can be activated; objects and generated from membrane 4 will enter membrane 2. Then, only rule is used, and the channel state of this membrane changes to . Next, and are simultaneously executed, (resp., ) enters membrane 2 based on channel state (resp., ), and (resp., ) will become the new channel state.
The subsequent computing process is primarily used to compare the assignment of the current variables with the corresponding variables in all clauses, and it runs in parallel related to two different membranes simultaneously. In label 2, one rule in set {, , and } is executed first and then one rule in set {, , and } will be used; this process checks the current variable assignment in each clause labelled “true” in the sequential manner combined with flat maximal parallelism. A similar computation is executed in another membrane with label 2: one rule in set {, , and } is applied first and then one rule in set {, , and } will be used; this process checks the current variable assignment in each clause labelled “false.” In the computing process above, is an object that has been generated in membrane 4. When rules are executed, the value of the second subscript of the channel state corresponding to membrane 2 increases by 1; when the value increases to , which means that each clause associated with the current variable has been checked, the system executes rule (resp., ); notably, (resp, ) would exist in membrane 1, and the channel state of the corresponding membrane is changed to . The newly generated object is moved to membrane 3 by employing rule , and then, division rule is activated based on the influence of ; simultaneously, object appears in the newly generated membrane. Next, rule is employed, object is sent to membrane 1, and the next iteration proceeds in the computation.
The subsequent computation is similar to the previous process, that is, values are assigned to other variables (from to ) in the formula, and the corresponding process used above for variable is implemented. Finally, all variables corresponding to clauses in the formula are compared with assignments, and the result “true” for the corresponding clauses is obtained. The system uses the strategy of sequential manner combined with flat maximal parallelism; notably, relative to some rules (e.g., and ), even if more than one copy of a object can be used to by a rule simultaneously, such a rule cannot be executed multiple times and only used once at the step. Finally, when the -th variable is executed, the application of rules and ends, and the above computing process ceases. Although and appear in membrane 1 at that moment, the rules in membrane 3 (e.g., , , and ) are not applied.
When exists on the channel state of membrane 2, the system begins the computation for the checking/output phase. At this point, rule is activated, and object is transferred out of membrane 2; additionally, would be the new channel state. Next, rule is applied as long as objects , exist in a membrane with label 2, and these objects can be removed from membrane 2; simultaneously, the value of the corresponding channel state will be continuously increased. If the channel state of membrane 2 reaches , the formula has a satisfiable solution; at that moment, rules and can be applied to send to membrane 2 and to the membrane with label 1. If the channel state of membrane 2 does not reach , the conclusion is no satisfiable solution; in this case, rules and are not executed, and remains in membrane 1. At step , if is in membrane 1, rule is activated, and appears in the environment as the computing result. Therefore, if an solution is unsatisfactory, the entire computation requires steps; otherwise, rule will not be executed. Additionally, when the system remains at step , rule is activated, and appears in the environment as the computing result. Therefore, if an solution is satisfied, the whole computation requires steps.

3.2. Some Formal Details

(i)Size of the set : ;(ii)Size of the set : ;(iii)Initial number of membranes: 4;(iv)Initial number of objects: ;(v)The total number of rules: ;(vi)The maximal length of rules: 1.

Corollary 6. .

In [35], the problem was solved with the maximum rule length 2, where communication rules include symport rules and antiport rules.

In this work, however, on the one hand, communication rules are restricted to only one type, namely, symport rules; on the other hand, the maximal length of rules is decreased from 2 to 1. The proof indicates that our work has improved upon the current research method.

4. Case Study

In what follows, a case study is explored to demonstrate the previous proof. A formula of the is denoted by , which has 3 clauses and 3 variables, and the input multiset is denoted by .

Figure 4 denotes the initial configuration. When the system begins operation, rules involving membrane 1 and membrane 4 are applied in parallel. In membrane 4, many objects , are generated because of the division rule, which is useful later. There are three variables in the formula, which correspond to the iterations in the computing process. During the first iteration, the division rule is also used in membrane 2, and two assignments (“true” and “false”) for variable occur in each membrane; then, the system checks the two assignments of variable in each clause, and finally, the configuration is obtained (Figure 5).

The execution process for variable is similar to that for variable , as shown in Figure 6. When the final iteration is completed, Figure 7 denotes the configuration of the system. The details of the above computing process for variables , , and can be found in Figure 3 of Section 3.1.

Finally, we determine whether multiset exists in each membrane with label 2. In this instance, it is obvious that the formula has satisfiable solutions. Hence, when the system stops, is output to the environment as the computing result (Figure 8).

5. Conclusions

In this work, we have constructed a novel variant, namely, CCSS P systems; with respect to communication rules, only symport rules are employed. The computational efficiency of this variant has been explored. The proof indicates that the problem is solved by applying symport rules and membrane division. With regard to the system we constructed, the maximal length of rules is 1; moreover, the rule types of communication rules decreased from 2 to 1, that is, only symport rules are applied. Thus, in terms of computational complexity, our method improves upon the current research method.

Membrane separation and cell separation have obtained some satisfactory results in the existing literature (e.g., [37]). In this work, we adopt membrane division in the proposed variant; nevertheless, readers can perform membrane separation to potentially construct a new variant and explore its computational efficiency.

In this work, during system operation, we use sequential manner combined with flat maximal parallelism as the main strategy. Inspired by some actual biological phenomena, other methods have been introduced in membrane computing, such as time-freeness [41], local synchronization [42], rule synchronization [43], asynchronism [44], and minimal parallel [45] approaches. Especially for time-freeness, the execution time of each rule may be different. Therefore, based on the variant of this article, readers can introduce time-freeness to CCSS P systems and construct a more robust computing system, which is worthy of further study.

In our work, multiple membranes can work in parallel to perform high-efficiency computations. However, the parallelization of certain computing processes is not particularly excellent. Readers can attempt to improve parallelism among different membranes to construct a system with enhanced performance.

Data Availability

No datasets were analyzed or generated during the course of the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Science and Technology Research Program of the Chongqing Municipal Education Commission (Grant no. KJZD-K202003201).

References

G. Păun, “Computing with membranes,” Journal of Computer and System Sciences, vol. 61, no. 1, pp. 108–143, 2000.
View at: Publisher Site | Google Scholar
L. Adleman, “Molecular computation of solutions to combinatorial problems,” Science, vol. 266, no. 5187, pp. 1021–1024, 1994.
View at: Publisher Site | Google Scholar
Z. Gazdag and G. Kolonits, “A new method to simulate restricted variants of polarizationless P systems with active membranes,” Journal of Membrane Computing, vol. 1, no. 4, pp. 251–261, 2019.
View at: Publisher Site | Google Scholar
H. Peng, T. Bao, X. Luo et al., “Dendrite P systems,” Neural Networks, vol. 127, pp. 110–120, 2020.
View at: Publisher Site | Google Scholar
H. Peng and J. Wang, “Coupled neural P systems,” IEEE Transactions on Neural Networks and Learning Systems, vol. 30, no. 6, pp. 1672–1682, 2019.
View at: Publisher Site | Google Scholar
B. Song, X. Zeng, M. Jiang, and M. J. Pérez-Jiménez, “Monodirectional tissue P systems with promoters,” IEEE Transactions on Cybernetics, vol. 51, no. 1, pp. 438–450, 2021.
View at: Publisher Site | Google Scholar
B. Song, X. Zeng, and A. Rodríguez-Patón, “Monodirectional tissue P systems with channel states,” Information Sciences, vol. 546, pp. 206–219, 2021.
View at: Publisher Site | Google Scholar
T. Wu, Z. Zhang, and L. Pan, “On languages generated by cell-like spiking neural P systems,” IEEE Transactions on NanoBioscience, vol. 15, no. 5, pp. 455–467, 2016.
View at: Publisher Site | Google Scholar
Y. Zhao, X. Liu, M. Sun, J. Qu, and Y. Zheng, “Time-free cell-like P systems with multiple promoters/inhibitors,” Theoretical Computer Science, vol. 843, pp. 73–83, 2020.
View at: Publisher Site | Google Scholar
J. Cooper and R. Nicolescu, “Alternative representations of P systems solutions to the graph colouring problem,” Journal of Membrane Computing, vol. 1, no. 2, pp. 112–126, 2019.
View at: Publisher Site | Google Scholar
P. Guo, J. Zhu, H. Chen, and R. Yang, “A linear‐time solution for all‐SAT problem based on P system,” Chinese Journal of Electronics, vol. 27, no. 2, pp. 367–373, 2018.
View at: Publisher Site | Google Scholar
Y. Luo, H. Tan, Y. Zhang, and Y. Jiang, “The computational power of timed P systems with active membranes using promoters,” Mathematical Structures in Computer Science, vol. 29, no. 5, pp. 663–680, 2019.
View at: Publisher Site | Google Scholar
Y. Luo, Y. Zhao, and C. Chen, “Homeostasis tissue-like P systems,” IEEE Transactions on NanoBioscience, vol. 20, no. 1, pp. 126–136, 2021.
View at: Publisher Site | Google Scholar
B. Song, K. Li, D. Orellana-Martín, L. Valencia-Cabrera, and M. J. Pérez-Jiménez, “Cell-like P systems with evolutional symport/antiport rules and membrane creation,” Information and Computation, vol. 275, Article ID 104542, 2020.
View at: Publisher Site | Google Scholar
K. Roy, A. Jaiswal, and P. Panda, “Towards spike-based machine intelligence with neuromorphic computing,” Nature, vol. 575, no. 7784, pp. 607–617, 2019.
View at: Publisher Site | Google Scholar
G. Singh and K. Deep, “Effectiveness of new multiple-PSO based membrane optimization algorithms on CEC 2014 benchmarks and Iris classification,” Natural Computing, vol. 16, no. 3, pp. 473–496, 2017.
View at: Publisher Site | Google Scholar
G. Zhang, H. Rong, F. Neri, and M. J. Pérez-Jiménez, “An optimization spiking neural p system for approximately solving combinatorial optimization problems,” International Journal of Neural Systems, vol. 24, no. 5, Article ID 1440006, 2014.
View at: Publisher Site | Google Scholar
P. Frisco, M. Gheorghe, and M. J. Pérez-Jiménez, Applications of Membrane Computing in Systems and Synthetic Biology, Springer-Verlag, Heidelberg, Germany, 2014.
M. García-Quismondo, M. Levin, and D. Lobo, “Modeling regenerative processes with membrane computing,” Information Sciences, vol. 381, pp. 229–249, 2017.
View at: Publisher Site | Google Scholar
C. Buiu and A. G. Florea, “Membrane computing models and robot controller design, current results and challenges,” Journal of Membrane Computing, vol. 1, no. 4, pp. 262–269, 2019.
View at: Publisher Site | Google Scholar
J. Wang, H. Peng, W. Yu et al., “Interval-valued fuzzy spiking neural P systems for fault diagnosis of power transmission networks,” Engineering Applications of Artificial Intelligence, vol. 82, pp. 102–109, 2019.
View at: Publisher Site | Google Scholar
G. Zhang, M. J. Pérez-Jiménez, and M. Gheorghe, Real-life Applications with Membrane Computing, Springer-Verlag, Heidelberg, Germany, 2017.
R. Xian, R. Lugu, H. Peng, Q. Yang, X. Luo, and J. Wang, “Edge detection method based on nonlinear spiking neural systems,” International Journal of Neural Systems, vol. 33, no. 1, Article ID 2250060, 2023.
View at: Publisher Site | Google Scholar
X. Chen, H. Peng, J. Wang, and F. Hao, “Supervisory control of discrete event systems under asynchronous spiking neuron P systems,” Information Sciences, vol. 597, pp. 253–273, 2022.
View at: Publisher Site | Google Scholar
S. Xia, G. Wang, Z. Chen, Y. Duan, and Q. liu, “Complete random forest based class noise filtering learning for improving the generalizability of classifiers,” IEEE Transactions on Knowledge and Data Engineering, vol. 31, no. 11, pp. 2063–2078, 2019.
View at: Publisher Site | Google Scholar
S. Xia, Y. Liu, X. Ding, G. Wang, H. Yu, and Y. Luo, “Granular ball computing classifiers for efficient, scalable and robust learning,” Information Sciences, vol. 483, pp. 136–152, 2019.
View at: Publisher Site | Google Scholar
S. Xia, D. Peng, D. Meng et al., “Ball k-means: fast adaptive clustering with No bounds,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 44, pp. 87–99, 2022.
View at: Publisher Site | Google Scholar
J. Hu, H. Peng, J. Wang, and W. Yu, “kNN-P: a kNN classifier optimized by P systems,” Theoretical Computer Science, vol. 817, pp. 55–65, 2020.
View at: Publisher Site | Google Scholar
B. Song, K. Li, D. Orellana-Martín, M. J. Pérez-Jiménez, and I. PéRez-Hurtado, “A survey of nature-inspired computing: membrane computing,” ACM Computing Surveys, vol. 54, no. 1, pp. 1–31, 2021.
View at: Publisher Site | Google Scholar
C. Martín-Vide, G. Păun, J. Pazos, and A. Rodríguez-Patón, “Tissue P systems,” Theoretical Computer Science, vol. 296, no. 2, pp. 295–326, 2003.
View at: Publisher Site | Google Scholar
M. Ionescu, G. Păun, and T. Yokomori, “Spiking neural P systems,” Fundamenta Informaticae, vol. 71, no. 2, pp. 279–308, 2006.
View at: Google Scholar
A. Păun and G. Păun, “The power of communication: P systems with symport/antiport,” New Generation Computing, vol. 20, no. 3, pp. 295–305, 2002.
View at: Publisher Site | Google Scholar
R. Freund, G. Păun, and M. J. Pérez-Jiménez, “Tissue P systems with channel states,” Theoretical Computer Science, vol. 330, no. 1, pp. 101–116, 2005.
View at: Publisher Site | Google Scholar
B. Song, L. Pan, and M. J. Pérez-Jiménez, “Cell-like P systems with channel states and symport/antiportrules,” IEEE Transactions on NanoBioscience, vol. 15, no. 6, pp. 555–566, 2016.
View at: Publisher Site | Google Scholar
S. Jiang, Y. Wang, and Y. Su, “A uniform solution to SAT problem by symport/antiport P systems with channel states and membrane division,” Soft Computing, vol. 23, no. 12, pp. 3903–3911, 2019.
View at: Publisher Site | Google Scholar
L. Pan and M. J. Pérez-Jiménez, “Computational complexity of tissue-like P systems,” Journal of Complexity, vol. 26, no. 3, pp. 296–315, 2010.
View at: Publisher Site | Google Scholar
M. J. Pérez-Jiménez and P. Sosík, “An optimal Frontier of the efficiency of tissue P systems with cell separation,” Fundamenta Informaticae, vol. 138, no. 1–2, pp. 45–60, 2015.
View at: Publisher Site | Google Scholar
L. Pan, G. Păun, and B. Song, “Flat maximal parallelism in P systems with promoters,” Theoretical Computer Science, vol. 623, pp. 83–91, 2016.
View at: Publisher Site | Google Scholar
M. Minsky, Computation: Finite and Infinite Machines, Prentice-Hall, Englewood Cliffs, New Jersey, 1967.
G. Rozenberg and A. Salomaa, Handbook of Formal Languages, Springer, Berlin, Germany, 1997.
M. Cavaliere and D. Sburlan, “Time-independent P systems,” Lecture Notes in Computer Science, Springer-Verlag, Berlin, Germany, 2005.
View at: Google Scholar
L. Pan, A. Alhazov, H. Su, and B. Song, “Local synchronization on asynchronous tissue P systems with symport/antiport rules,” IEEE Transactions on NanoBioscience, vol. 19, no. 2, pp. 315–320, 2020.
View at: Publisher Site | Google Scholar
B. Song and L. Pan, “Rule synchronization for tissue P systems,” Information and Computation, vol. 281, Article ID 104685, 2021.
View at: Publisher Site | Google Scholar
P. Frisco, G. Govan, and A. Leporati, “Asynchronous P systems with active membranes,” Theoretical Computer Science, vol. 429, pp. 74–86, 2012.
View at: Publisher Site | Google Scholar
G. Ciobanu, L. Pan, G. Păun, and M. J. Pérez-Jiménez, “P systems with minimal parallelism,” Theoretical Computer Science, vol. 378, no. 1, pp. 117–130, 2007.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2023 Xiaoming Wan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies