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

. 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 classifed 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 efciency of the variant, we solve the SAT problem and obtain a uniform solution in polynomial time with the maximal length of rules 1. Te results of our work are refected in the following two aspects: frst, 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 SAT problem and obtain better results in terms of computational complexity. Hence, in terms of computational efciency, our work is a notable improvement.


Introduction
Membrane computing, a new type of bioinspired computing model, was frst proposed by Pȃun [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][4][5][6][7][8][9], and many variants have been proven Turing universal. In the theoretical research of membrane computing, various computationally hard problems were solved [10][11][12][13][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 feld, 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 feld of machine learning [25][26][27], and some researchers combine these two research areas and obtain excellent results [28]. Currently, many scholars are focusing on developments in this feld. 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]. Tis 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 efciency was not involved. Recently, Jiang et al. solved the SAT problem with CCSSA P systems to study its computational efciency [35]; however, in the study, although a uniform SAT 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], SAT 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 fat maximal parallelism [38], such a rule can only be activated once. In our work, we adopt a new method combined sequential manner with fat 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 classifed 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).
Te 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. Specifcally, we solve SAT 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 refects excellent performance. Our approach improves upon the current research method.
Te structure of this paper is as follows: Section 2 mainly gives the defnition of CCSS P systems. In Section 3, based on the SAT problem, the computational efciency 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.

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].

Te Model of CCSS P Systems
where (i) O is an alphabet of objects (ii) K represents the set of channel states (not necessarily exist in the set of O ) (iii) E represents an infnite number of objects in the environment (iv) μ means the membrane structure denoted by a tree, and all nodes 1, . . . , m in the tree correspond to the labels of membranes. Relative to each membrane i, p(i) denotes the outside region of membrane i; denote the multisets of objects initially located in membrane i (vi) s i (1 ≤ i ≤ m) represent channel states initially located on the membrane with label i, which is the channel between membrane i and p(i) } represent a series of rules, including symport rules and division rules. In what follows, O + denotes the set of strings composed by the symbol in O but without λ; and given a string x, the length of x, which is the quantity of symbols in this string, is denoted by |x|.

Symport Rules
where S, S ′ ∈ K, x ∈ O + , |x| > 0. Te length of a symport rule is |x|. 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 S and multiset x exists inside (resp., outside) membrane i, (S, (x, out), S ′ ) (resp., (S, (x, in), S ′ )) can be applied. By changing the channel state to S ′ , multiset x is transferred to region p(i) (resp., membrane i), where p(i) represents the outer membrane of membrane i; if membrane i is the skin membrane (it does not have the upper neighbor membrane), p(i) represents the environment.

Division Rules
At a given moment, relative to object a in membrane i (except for the skin membrane), a division rule is executed, and two membranes with the same label will appear. Simultaneously, object b including c appear in new membranes to replace the previous object a; other objects in the initial membrane can be copied to the two newly generated membranes. It is important to emphasize that objects a, b, and c can represent the same or diferent 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.

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 diferent channels; moreover, rules that can be executed must be executed on the corresponding channel. For example, multiset of objects u 2 v 2 exists in membrane 1, and channel state S 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 R 1 and R 2 are activated simultaneously, and each rule can be executed multiple times. As a result, all copies of objects u and v will be transferred to the environment. Te computing process is shown in Figure 1.
Next, we consider the method combined sequential manner with fat maximal parallelism. When the system starts running, one rule (R 1 or R 2 ) 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 R 1 is only allowed to be applied. Eventually, one copy of object u is transferred to the environment; (ii) Rule R 2 is only allowed to be applied. Eventually, one copy of object v is transferred to the environment. Te computing process is shown in Figure 2.
Tere is at most one channel on each membrane (between two adjacent regions). On each channel, we use the strategy of sequential manner combined with fat maximal parallelism to apply rules. Te confguration of CCSS P systems is infuenced 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 confguration to characterize it. As the computation continues, a series of confgurations can be generated with the execution of rules at each step, and we call it a transition between two arbitrary confgurations. A series of transitions is called a computation. Notably, the application of rules is based on sequential manner combined with fat maximal parallelism and follows the principle of nondeterminism [1]. Finally, when system operation stops, the halting confguration can be obtained; at that moment, the system halts, and no rules are applied, and the computing result is sent to the output region.

Recognizer CCSS P Systems.
In membrane computing, the recognizer P systems are used to solve decision problems. where (iii) i in (resp., i out ) represents the input (resp., output) region.
In such a system, the other parameters are defned as in Defnition 1. A recognizer CCSS P system can start from the initial confguration with an input multiset; eventually, YES occurs in the region i out .
Defnition 3. X � (I X , θ X ) denotes a decision problem, where I X is the instance, and θ X represents a predicate of the instances. Te problem can be solved in polynomial time, if the following holds: (i) Π is polynomially uniform by Turing machines (ii) Relative to I X , there is a pair (cod, s) of polynomialtime computable functions such that (a) suppose u corresponds to an instance, u ∈ I X , and s(u) is a natural number; additionally, Discrete Dynamics in Nature and Society cod(u) represents an input multiset of CCSS P systems; (b) relative to (X, cod, s), such a system is sound, that is, with regard to u ∈ I X , if CCSS P systems have an accepting computation, then θ X (u) � 1; (c) relative to (X, cod, s), such a system is complete, that is, suppose u ∈ I X , relative to a problem, if θ X (u) � 1, computations of Π(s(u)) with cod(u) is an accepting one; (d) such a system is polynomially bounded, that is, CCSS P systems stop computation and reach the halting confguration after p(|u|) steps (p is a polynomial function).

Defnition 4.
Te maximum rule length of a CCSS P system is equivalent to that of symport rules in the system. PMC CPS−CSSR(k) 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 k. Proof. A SAT formula consists of n Boolean variables and m clauses:
We codify c with the multiset (1 ≤ j ≤ m and 1 ≤ i ≤ n).
To solve the SAT, a recognizer P system Π SAT(m,n) is constructed: where (i) O is the set of objects: (ii) K is the set of channel states: . . , e m . (vii) i in � 2 and i out � 0. (viii) Te rules in R 1 are as follows: Figure 2: Applying sequential manner combined with fat maximal parallelism. 4 Discrete Dynamics in Nature and Society , (YES, out), P 2mn+2m+11n+2 ). (ix) Te rules in R 2 are as follows: Division rules: Symport rules: (x) Te rules in R 3 are as follows: Division rules: (xi) Te rules in R 4 are as follows: Division rules: Symport rules: Te whole computing process is primarily classifed into two main phases: the generation phase and the checking/ output phase. Tere are many rules involved in the generation phase, which is relatively complicated; hence, we will describe the generation phase in detail.
Te generation phase generates all possible assignments of all variables and detects satisfable clauses with these assignments. Te 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. Te 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 confguration, the multiset of objects g 1 b 1 e 1 , . . . , e m exists in membrane 4. Because the system includes object g 1 present in membrane 4, the system executes the division rule R 34,i , which can generate two cells with the same label, and object g 2 will appear in the two membranes. Since division rules have higher priority, they can be used in membrane 4 because of g 2 . Finally, after n steps, 2 n membranes will appear, and object g n+1 will be generated in each membrane. Subsequently, relative to channel state Q 1 on cell 4, rule R 35,i is activated and can be executed m times; the channel state will increase by 1 at each step, and fnally, object Q m+1 will appear. At that moment, rule R 36 can be executed, sending object g n+1 to the membrane with label 1. Based on the channel state on membrane 4, the system will execute R 37 at the next step, and object b 1 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. Te following computing process assigns values (e.g., "true" or "false") to the variables x 1 of all clauses in a SAT formula, and the rules from R 6,1 to R 33,2 are applied. First, R 6,i is activated, and then, division rule R 5,i is executed. Objects t 1 and f 1 appear in the two newly generated membranes. After a division rule is used, f i is sent to the region outside membrane 2 by applying rule R 7,i , and the channel state of this membrane will be F i . Next, rules R 8,i and R 9,i can be activated; objects f i and g n+1 generated from membrane 4 will enter membrane 2. Ten, only rule R 10,i is used, and the channel state of this membrane changes to T i ′ . Next, R 11,i and R 12,i are simultaneously executed, b 1 (resp., f 1 ) enters membrane 2 based on channel state T i ′ (resp., F i ′ ), and T i,1 (resp., F i,1 ) will become the new channel state.
Te 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 diferent membranes simultaneously. In label 2, one rule in set {R 13,i,j , R 14,i,j , and R 15,i,j } is executed frst and then one rule in set {R 19,i,j , R 21,i,j , and R 22,i,j } will be used; this process checks the current variable assignment in each clause labelled "true" in the sequential manner Discrete Dynamics in Nature and Society combined with fat maximal parallelism. A similar computation is executed in another membrane with label 2: one rule in set {R 16,i,j , R 17,i,j , and R 18,i,j } is applied frst and then one rule in set {R 20,i,j , R 23,i,j , and R 24,i,j } will be used; this process checks the current variable assignment in each clause labelled "false." In the computing process above, e j 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 m + 1, which means that each clause associated with the current variable has been checked, the system executes rule R 25,i (resp., R 26,i ); notably, t i (resp, f i ) would exist in membrane 1, and the channel state of the corresponding membrane is changed to S 2 . Te newly generated object t i is moved to membrane 3 by employing rule R 32,i , and then, division rule R 31,i is activated based on the infuence of t i ; simultaneously, object b 2 appears in the newly generated membrane. Next, rule R 33,i is employed, object b 2 is sent to membrane 1, and the next iteration proceeds in the computation.
Te subsequent computation is similar to the previous process, that is, values are assigned to other variables (from x 2 to x n ) in the SAT formula, and the corresponding process used above for variable x 1 is implemented. Finally, all variables corresponding to clauses in the SAT formula are compared with assignments, and the result "true" for the corresponding clauses is obtained. Te system uses the strategy of sequential manner combined with fat maximal parallelism; notably, relative to some rules (e.g., R 6,i and R 8,i ), 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 n-th variable is executed, the application of rules R 25,n and R 26,n ends, and the above computing process ceases.
Although t n and f n appear in membrane 1 at that moment, the rules in membrane 3 (e.g., R 31,i , R 32,i , and R 33,i ) are not applied.
When S n+1 exists on the channel state of membrane 2, the system begins the computation for the checking/output phase. At this point, rule R 27 is activated, and object e 1 is transferred out of membrane 2; additionally, Z 2 would be the new channel state. Next, rule R 28,j is applied as long as objects e 2 , e 3 , . . . , e m 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 Z m+1 , the SAT formula has a satisfable solution; at that moment, rules R 29 and R 30 can be applied to send NO to membrane 2 and YES to the membrane with label 1. If the channel state of membrane 2 does not reach Z m+1 , the conclusion is no satisfable solution; in this case, rules R 29 and R 30 are not executed, and NO remains in membrane 1. At step 2mn + 2m + 11n, if NO is in membrane 1, rule R 3 is activated, and NO appears in the environment as the computing result. Terefore, if an SAT solution is unsatisfactory, the entire computation requires 2mn + 2m + 11n + 1 steps; otherwise, rule R 3 will not be executed. Additionally, when the system remains at step 2mn + 2m + 11n + 1, rule R 4 is activated, and YES appears in the environment as the computing result. Terefore, if an SAT solution is satisfed, the whole computation requires 2mn + 2m + 11n + 2 steps.      Discrete Dynamics in Nature and Society (iv) Initial number of objects: m + 5; (v) Te total number of rules: 14mn + 3m + 23n + 11; (vi) Te maximal length of rules: 1.
In [35], the SAT 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. Te proof indicates that our work has improved upon the current research method.

Case Study
In what follows, a case study is explored to demonstrate the previous proof. A formula of the SAT is denoted by c � ( Figure 4 denotes the initial confguration. When the system begins operation, rules involving membrane 1 and membrane 4 are applied in parallel. In membrane 4, many objects e 1 , e 2 , . . . , e m are generated because of the division rule, which is useful later. Tere are three variables in the formula, which correspond to the iterations in the computing process. During the frst iteration, the division rule is also used in membrane 2, and two assignments ("true" and     Discrete Dynamics in Nature and Society 7 "false") for variable x 1 occur in each membrane; then, the system checks the two assignments of variable x 1 in each clause, and fnally, the confguration is obtained ( Figure 5). Te execution process for variable x 2 is similar to that for variable x 1 , as shown in Figure 6. When the fnal iteration is completed, Figure 7 denotes the confguration of the system. Te details of the above computing process for variables x 1 , x 2 , and x 3 can be found in Figure 3 of Section 3.1.

which has 3 clauses and 3 variables, and the input multiset is denoted by
Finally, we determine whether multiset e 1 e 2 e 3 exists in each membrane with label 2. In this instance, it is obvious that the formula has satisfable solutions. Hence, when the system stops, YES is output to the environment as the computing result (Figure 8).

Conclusions
In this work, we have constructed a novel variant, namely, CCSS P systems; with respect to communication rules, only symport rules are employed. Te computational efciency of this variant has been explored. Te proof indicates that the SAT 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. Tus, 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 efciency.
In this work, during system operation, we use sequential manner combined with fat 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 timefreeness, the execution time of each rule may be diferent. Terefore, 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-efciency computations. However, the parallelization of certain computing processes is not particularly excellent. Readers can attempt to improve parallelism among diferent 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
Te authors declare that they have no conficts of interest.