P Systems with Proteins on Active Membranes

Abstract

P systems with active membranes, as a sort of basic P system, include in communication rules and out communication rules, where communication rules are controlled by polarizations. However, the communication of objects among living cells may be controlled by several factors, such as proteins, polarizations, etc. Based on this biological fact, in this article, a new class of P systems, named P systems with proteins on active membranes (known as PAM P systems) is considered, where the movement of objects is controlled by both proteins and polarizations. The computational theory of PAM P systems is discussed. More specifically, we show that PAM P systems achieve Turing universality when the systems use two membranes, one protein and one polarization. Moreover, the PAM P systems, with the help of membrane division rules, make the SAT problem solvable. These results indicate that PAM P systems are also a sort of powerful system.

1. Introduction

Membrane computing, which aims to discuss computational models abstracted from the functioning and structure of biological cells, is one sort of natural computing, where the models are known as P systems [1]. All computing models have in common a set of compartments separated by membranes and organized by a certain structure (tree, graph). In general, three main sorts of P systems are discussed: cell-like [1,2,3,4,5], tissue-like [6,7,8,9,10,11] and neural-like [12,13,14,15,16,17,18,19,20] P systems. Since the first article about membrane computing was published, a large bibliography has been accumulated [5,21,22,23,24,25]. More information on membrane computing is available on the website http://imcs.org.cn/ or http://ppage.psystems.eu (accessed on 8 October 2022), where readers can view and download papers in this field.

P systems with active membranes (known as AM P systems) were first considered in [26], and the structure of these membranes is represented by a tree. It is an obvious characteristic of AM P systems that the movement and evolution of objects are controlled by polarization associated with each membrane, where polarization changes between positive, negative, or neutral. Some further works on AM P systems have been published, such as AM P systems and separation rules [27]: separation rules are imported into AM P systems, and this proves that such P systems are able to make the SAT problem solvable in polynomial time; AM P systems without using polarizations [28]: in this system, polarization is avoided but modification of the labels of the membranes is allowed. It has been proved that the computational power of AM P systems and their variants is powerful; most of them were proved to achieve Turing universality, and they can give the result of NP-complete problems or even PSPACE-complete problems [5,29,30,31,32,33].

P systems with proteins on membranes (known as PM P systems) were first considered in [34]. Such systems include two types of objects: usual objects and proteins. Usual objects are placed in membrane or in the environment; proteins are placed on membranes (note that a multiset of proteins is placed on a membrane). According to whether proteins can be changed or not in systems [34], there are two groups of rules: “res” (representing “restricted”) and “cp” (representing “change protein”). In [35], a special class of PM P system, named P systems with flip-flop proteins on membranes, was considered, where at most two states are allowed for each protein. The SAT problem is solvable with the help of PM P systems and membrane division rules [36]. Readers who want to learn more information about PM P systems can review the references [34,37].

As mentioned above, AM P systems contain communication rules, which are controlled by polarizations. PM P systems contain several types of rules, which are controlled by proteins. Obviously, rules in AM P systems and in PM P systems are controlled by only one factor. However, chemical reactions happening among living cells may be controlled by several factors, such as proteins, polarizations, etc. Based on this biological fact, in this article, a new sort of P system, named P systems with proteins on active membranes (known as PAM P systems), is considered, where the movement of objects is controlled by both protein and polarization.

The computational theory of PAM P systems is studied. More specifically, we indicate that PAM P systems achieve Turing universality when the systems use two membranes, one protein and one polarization. Moreover, the PAM P systems, with the help of membrane division rules, make the SAT problem solvable. The results among the P systems with active membranes, P systems with proteins on membranes and this paper are in Table 1.

Table 1. Results of the P systems with active membranes, P systems with proteins on membranes and this paper, where $NO{P}_m(pr{o}_r;ac{t}_c;list-of-rules)$ represents the family of all sets of numbers produced by P systems with m membranes, r proteins in each membrane, and which use c polarizations and types of rules $list-of-rules$; * represents the case where parameter m or r are boundless; the types of rules $2cpp$ or $3ffp$ combined with the polarizations of membranes become the types of rules $\left(3\right)$ and $\left(4\right)$; the types of rules $\left(b\right)$ and $\left(c\right)$ combined with proteins on membranes become the types of rules $\left(3\right)$ and $\left(4\right)$; the types of rules $\left(3\right)$ and $\left(4\right)$ are defined in this paper; and $NRE$ is the family of all recursively enumerable sets of natural numbers.

P Systems with Active Membranes	P systems with Proteins on Membranes	This Paper
$NO{P}_2(ac{t}_2;\left(a\right),\left(c\right))=NRE$ [38]	$NO{P}_1(pr{o}_2;2cpp)=NRE$ [34]	$NO{P}_2(pr{o}_1;ac{t}_1;\left(3\right),\left(4\right))=NRE$
$NO{P}_2(ac{t}_1;\left(a\right),\left(b^{\prime }\right),\left(c\right))=NRE$ [38]	$NO{P}_1(pr{o}_{\ast };3ffp)=NRE$ [34]

2. Preliminaries and Model Description

2.1. Preliminaries

In this subsection, the concepts of formal languages and automata theory are introduced, which will be employed in this article. Readers who want to obtain more details can refer to [39].

A non-empty finite set constitutes an alphabet denoted by $\Gamma$, where each element in the set is known as a symbol. All strings created by joining any symbols from $\Gamma$ form a set which is symbolized by ${\Gamma }^{\ast }$. We use $\lambda$ to represent the case that no symbol appears in a string, which is called $empty$$string$. The non-empty set of ${\Gamma }^{\ast }$ is symbolized as ${\Gamma }^{+}={\Gamma }^{\ast }\backslash \left\{\lambda \right\}$. $\left|u\right|$ is known as the length of u, which symbolizes the total number of symbols presented in string u.

For an alphabet $\Gamma$, we use the ordered pair $(\Gamma ,f)$ to represent a multiset over $\Gamma$ symbolized by $\mathcal{M}$, where f is the mapping of $\Gamma$ to $\mathbb{N}$ (natural numbers set). In addition, $\mathcal{M}(\Gamma )$ and ${\mathcal{M}}^{+}(\Gamma )$ are known as the set of all multisets and of all non-empty multisets, respectively. Given a $\Gamma =\{a_1,\cdots ,a_k\}$, $\{a_1^{f\left(a_1\right)},\cdots ,a_k^{f\left(a_k\right)}\}$ expresses multiset $\mathcal{M}$.

A tuple $M=(m,H,l_0,l_h,I)$ symbolizes a register machine including m registers, a label set H, an initial instruction $l_0\in H$, the halting instruction $l_h\in H$, and a labeled program-instructions set I whose instructions are of two types: (1) $l_i:(\mathtt{ADD}\left(r\right),l_j,l_k)$ (register r increases by one, after that the instructions $l_j$ or $l_k$ are executed in a non-deterministical way); and (2) $l_i:(\mathtt{SUB}\left(r\right),l_j,l_k)$ (subtract 1 from register r if it is non-zero, after that, execute the instruction $l_j$; otherwise, execute the instruction $l_k$).

$N\left(M\right)$ denotes a set including all numbers that are the result of a register machine M executing instructions. The M works as follows: being empty in all registers at the initial moment, $l_0$ is applied to start the machine and subsequent instructions continue to be applied according to the labels; the register 1 contains a number n corresponding to the result generated by M only when the instruction labeled $l_h$ appears in the system. It is known that all sets of numbers generated by register machines are Turing computable; therefore, both of them recognize the same family of sets of numbers $NRE$ (the family of all recursively enumerable sets of natural numbers) [40].

2.2. PAM P Systems

In the following, the definition of PAM P systems will be given. Readers who want to learn more about AM P systems and PM P systems can review [26] and [34], respectively.

Definition 1.

A P system with proteins on active membranes (PAM P system) of degree $m\ge 1$ is a tuple

$$\Pi =(O,P,\mu ,H,w_1/z_1,\dots ,w_m/z_m,\mathcal{E},R,i_{out}),$$

where:

• O symbolizes an alphabet including all symbols of objects;
• P symbolizes an alphabet including all the symbols of proteins where $O\cap P=\varnothing$;
• μ symbolizes the initial structure of a membrane including a set of membranes with labels in $1,\dots ,m$;
• H symbolizes a size-limited set of labels of membranes;
• $\mathcal{E}\subseteq O$ symbolizes a set including all objects that are initially placed in the environment;
• $w_1\dots w_m,$ symbolize all multisets of objects from O;
• $z_1\dots z_m,$ symbolize all multisets of proteins from P;
• R symbolize a set including a finite number of rules which have the following types:

  (1)

  $a{\left[p\right|]}_i^{e_1}\to b{\left[p^{\prime }\right|]}_i^{e_2}$, $i\in H$, $p,p^{\prime }\in P$, $a,b\in O$, $e_1,e_2\in \{+,-,0\}$

  (this rule can be used only when an object a appears in the parent of membrane i, the protein p is placed on membrane i and its polarization is $e_1$. When such a rule is applied, an object a may be evolved to b, the protein p may be evolved to $p^{\prime }$, and the polarization of the membrane may also be modified);

  (2)

  ${\left[p\right|a]}_i^{e_1}\to {\left[p^{\prime }\right|b]}_i^{e_2}$, $i\in H$, $p,p^{\prime }\in P$, $a,b\in O$, $e_1,e_2\in \{+,-,0\}$

  (this rule can be used only when an object a appears in membrane i, the protein p is placed on membrane i and its polarization is $e_1$. When such a rule is applied, an object a may be evolved to b, the protein p may be evolved to $p^{\prime }$, and the polarization of the membrane may also be modified);

  (3)

  $a{\left[p\right|]}_i^{e_1}\to {\left[p^{\prime }\right|b]}_i^{e_2}$, $i\in H$, $p,p^{\prime }\in P$, $a,b\in O$, $e_1,e_2\in \{+,-,0\}$

  (this rule can be used only when an object a appears in the parent of membrane i, the protein p is placed on membrane i and its polarization is $e_1$. When such a rule is applied, an object a is moved into membrane i, and possibly evolved to b during this process, the protein p may be evolved to $p^{\prime }$, and the polarization of the membrane may also be modified);

  (4)

  ${\left[p\right|a]}_i^{e_1}\to b{\left[p^{\prime }\right|]}_i^{e_2}$, $i\in H$, $p,p^{\prime }\in P$, $a,b\in O$, $e_1,e_2\in \{+,-,0\}$

  (this rule can be used only when an object a appears in the membrane i, the protein p is placed on membrane i and its polarization is $e_1$. When such a rule is applied, an object a is sent out of membrane i, and possibly evolved to b during this process; the protein p may be evolved to $p^{\prime }$; and the polarization of the membrane may also be modified);

  (5)

  $a{\left[p\right|b]}_i^{e_1}\to c{\left[p^{\prime }\right|d]}_i^{e_2}$, $i\in H$, $p,p^{\prime }\in P$, $a,b,c,d\in O$, $e_1,e_2\in \{+,-,0\}$

  (this rule can be used only when an object a is contained in the parent of membrane i, an object b is contained in membrane i, a protein p is placed on membrane i and its polarization is $e_1$. When such a rule is applied, an object a will be sent into membrane i, and possibly evolved to d during this process; simultaneously, an object b will be sent out of membrane i, and possibly evolved to c during this process; the protein p may be evolved to $p^{\prime }$; and the polarization $e_1$ may also be modified to $e_2$);

  (6)

  ${\left[p\right|]}_i^e\to {\left[p^{\prime }\right|]}_i^{e_1}{\left[p^{″}\right|]}_i^{e_2}$, $i\in H$, $p,p^{\prime },p^{″}\in P$, $e,e_1,e_2\in \{+,-,0\}$, $1\le i\le m,i\ne i_{out}$, and i is unable to be the root of the tree

  (when such a rule is applied, membrane i is divided into two membranes without changing label, with protein p evolved to $p^{\prime }$ and $p^{″}$, with polarization e modified to $e_1$ and $e_2$, respectively; all the objects in the parent membrane are duplicated in the two new membranes).

• $i_{out}\in \{0,1,\dots ,m\}$ is the output area (0 is the label of environment).

The maximally parallel way is a common strategy of using rules in a PAM P system, where the applicable rules of a PAM P system are added into a maximal multiset to be employed (no more rules are able to be added) at each step. A PAM P system working in this way has the following limitations: any object can be used in only one rule of any type, and any membrane can be used in only one rule of types (1)–(5); when a rule of type (6) (membrane division rules) is applied, the system cannot employ other rules for that membrane in that step.

For a PAM P system as defined above, a configuration ${\mathcal{C}}_t$ contains the following factors: the membrane structure and polarization of membranes at moment t; all objects presented in each area of this membrane structure; all proteins presented on membranes; and all objects presented in the environment at moment t. We can obtain the initial configuration from ($\mu ,w_1\dots ,w_m,z_1\cdots z_m,\mathcal{E}$), and the polarization of each membrane in the initial state is neutral by default.

A system which starts from initial configuration and works in a maximally parallel way with the restrictions mentioned above can achieve a series of consecutive configurations. If all the rules of a system are unavailable in a configuration, this configuration is known as a halting configuration. A transition denoted by ${\mathcal{C}}_t\Rightarrow {\mathcal{C}}_{t+1}$ represents that one configuration ${\mathcal{C}}_t$ transfers to the next configuration ${\mathcal{C}}_{t+1}$. A (finite or infinite) sequence of transitions between configurations indicates a computation for a PAM P system. Only halting computations whose last configuration is a halting configuration give a result which appears in the output area $i_{out}$ as a multiset of objects.

$N(\Pi )$ symbolizes a set including all numbers produced by a PAM P system $\Pi$. $NO{P}_m(pr{o}_r;ac{t}_c;list-of-rules)$ symbolizes the family of sets of numbers $N(\Pi )$ produced by the system $\Pi$ with the following explanations: (1) the number of membranes are at most m; (2) the total proteins placed on one membrane are no more than r; (3) c represents the number of polarizations which will be used in the system; and (4) $list-of-rules$ includes the specifical types of rules used in the system. When parameters r or m are boundless, we use ∗ instead.

To study the computational efficiency of the PAM P systems, the definition of such systems will be given below [29].

Definition 2.

A recognizer PAM P system of degree $m\ge 1$ is defined as a tuple

$$\Pi =(O,P,\Sigma ,\mu ,H,w_1/z_1,\dots ,w_m/z_m,\mathcal{E},R,i_{in},i_{out}),$$

where:

• O symbolizes an alphabet including all the symbols of usual objects and two special objectsyesandno;
• P symbolizes an alphabet including all the symbols of proteins;
• $\Sigma \subseteq O$ symbolizes an input alphabet;
• μ symbolizes the initial structure of the membranes including a set of membranes with labels in $1,\dots ,m$;
• H symbolizes a size-limited set of labels of membranes;
• $\mathcal{E}\subseteq O$ symbolizes a set including all the objects initially placed in the environment;
• $w_1\dots w_m,$ symbolize multisets of objects from O;
• $z_1\dots z_m,$ symbolize multisets of proteins from P;
• R symbolizes a set including the limited number of rules described above in each membrane i;
• $i_{in}\in \{0,1,\dots ,m\}$ is the input area and $i_{out}=0;$
• all computations halt;
• the halting condition of Π is that the system must send either objectyesor objectno(but not both) into the environment only at the last step of the computation.

In the following, we will give the definition for which the problem is solvable in polynomial time (in a uniform way) by a family of PAM P systems [41].

Definition 3.

A family of PAM P systems $\mathbf{\Pi }=\{\Pi (n\left)\right|n\in \mathbb{N}\}$ can give the polynomial time result of a decision problem $X=(I_X,{\Theta }_X)$ if the following conditions are satisfied:

• the family Π is polynomially uniform by Turing machines;
• there exists a pair $(cod,s)$ of polynomial-time computable functions over $I_X$ such that:

  -

  for each instance $u\in I_X$, $s\left(u\right)$ is a natural number and $cod\left(u\right)$ is an input multiset of system $\Pi \left(s\right(u\left)\right)$;

  -

  for each $n\in \mathbb{N}$, $s^{-1}\left(n\right)$ is a finite set;

  -

  the family Π is polynomially bounded with regard to $(X,cod,s)$;

  -

  the family Π is sound with regard to $(X,cod,s)$;

  -

  the family Π is complete with regard to $(X,cod,s)$.

${\mathbf{PMC}}_{\mathbf{MRC}(list-of-rules)}$ symbolizes a set including all decision problems for which the recognizer PAM P systems work in a maximally parallel way to give a result with polynomial time cost, where list-of-rules represents the types of rules used in the system.

3. Computational Power of PAM P Systems

In what follows, the computational power of PAM P systems will be considered. As we all know, the register machine has Turing universality, so we can prove that the PAM P systems are equivalent to the register machine to prove its Turing universality.

Theorem 1.

$NO{P}_2(pr{o}_1;ac{t}_1;\left(3\right),\left(4\right))=NRE.$

Proof. 

We design the PAM P system $\Pi$ to imitate the register machine M.

$$\Pi =(O,P,{[{[]}_2^0]}_1^0,H,\lambda /l_0,\lambda /p,\mathcal{E},R,1),$$

where

• $O=\left\{a_r\right|1\le r\le m\}\cup \{c,c^{\prime },c^{″}\}$,
• $P=\{l,l^{\prime },l^{″}|l\in H\}\cup \left\{p\right\}$,
• $H=\{1,2\}$,
• $\mathcal{E}=\left\{a_r\right|1\le r\le m\}\cup \left\{c\right\}$.

We assume that the numbers of copies of object $a_r$ are infinite.

To simulate the ADD instruction $l_i:(\mathtt{ADD}\left(r\right),l_j,l_k)$ of M, the R is designed as follows:

$r_1:a_r{\left[l_i\right|]}_1^0\to {\left[l_j\right|a_r]}_1^0$,

$r_2:a_r{\left[l_i\right|]}_1^0\to {\left[l_k\right|a_r]}_1^0$.

This system takes one step to simulate an ADD instruction $l_i$. Rules $r_1$ and $r_2$ are applied non-deterministically. By using $r_1$ or $r_2$, membrane 1 receives one object $a_r$, and protein $l_i$ on membrane 1 is modified to $l_j$ or $l_k$. Consequently, the system increases one copy of object $a_r$ and turns to simulate the instruction $l_j$ or $l_k$.

To simulate the SUB instruction $l_i:(\mathtt{SUB}\left(r\right),l_j,l_k)$ of M, the R is designed as follows:

$r_3:c{\left[l_i\right|]}_1^0\to {\left[l_i^{\prime }\right|c]}_1^0$;

$r_4:{\left[l_i^{\prime }\right|a_r]}_1^0\to a_r{\left[l_i^{″}\right|]}_1^0$;

$r_5:{\left[l_i^{″}\right|c^{″}]}_1^0\to c{\left[l_j\right|]}_1^0$;

$r_6:{\left[l_i^{\prime }\right|c^{″}]}_1^0\to c{\left[l_k\right|]}_1^0$;

$r_7:c{\left[p\right|]}_2^0\to {\left[p\right|c^{\prime }]}_2^0$;

$r_8:{\left[p\right|c^{\prime }]}_2^0\to c^{″}{\left[p\right|]}_2^0$.

The system takes four steps to simulate a SUB instruction $l_i$. At step one, with the influence of protein $l_i$, rule $r_3$ is employed, object c in the environment is sent to membrane 1, and protein $l_i$ is modified to $l_i^{\prime }$ on membrane 1. In the next steps, the running of the system is divided into two cases, according to whether membrane 1 contains objects $a_r$ or not.

• At least one copy of object $a_r$ exists in membrane 1. There are two rules $r_4$ and $r_7$ applied in a parallel way in step two. The system sends a copy of object $a_r$ out of membrane 1 and protein $l_i^{\prime }$ is modified to $l_i^{″}$. In addition, membrane 2 receives an object c which is modified to $c^{\prime }$ in this process. In step three, the system employs rule $r_8$ to send object $c^{\prime }$ back to membrane 1 and modifies object $c^{\prime }$ to $c^{″}$. In step four, rule $r_5$ is applied, and object $c^{″}$ is sent to the environment, which is modified to c in this process; simultaneously, protein $l_i^{″}$ is revised to $l_j$ on membrane 1. Consequently, the system decreases one copy of object $a_r$ and turns to simulating instruction $l_j$.
• No object $a_r$ exists in membrane 1. Only one rule $r_7$ is employed in step two. The system sends object c into membrane 2 and modifies object c to $c^{\prime }$ in this process. In step three, object $c^{\prime }$ is sent out of membrane 2, which is modified to $c^{″}$ in this process. In step four, the system employs rule $r_6$ to send object $c^{″}$ into the environment and modifies $c^{″}$ to c; simultaneously, protein $l_i^{\prime }$ is modified to $l_k$ on membrane 1. Consequently, the system turns to simulating instruction $l_k$.

A system only halts when protein $l_h$ appears on membrane 1. Membrane 1 contains the number of copies of object $a_r$ corresponding to the result generated by $\Pi$. Therefore, $\Pi$ is equivalent to M.    □

4. Solving the SAT Problem Using PAM P Systems

The SAT problem is a classic NP-complete problem [42] with the following description: judging whether a Boolean formula in conjunctive normal form ($CNF$) is satisfiable, that is, whether there is a truth-value assignments that means the value of $CNF$ is true.

In the following, we will prove that working in a maximally parallel way, a family of PAM P systems is able to give the result of the SAT problem in polynomial time.

Theorem 2.

$\mathtt{SAT}\in {\mathbf{PMC}}_{\mathbf{MRC}\left(\right(2),(3),(4),(6\left)\right)}.$

Proof. 

Let $\phi =C_1\wedge \cdots \wedge C_m$ be a Boolean formula, where $C_i=y_{i,1}\vee \cdots \vee y_{i,p_i}$, with $y_{i,j}\in \{x_k,\neg x_k\mid 1\le k\le n\}$, $1\le i\le n,1\le j\le p_i$.

We use a family of PAM P systems $\Pi =\{\Pi (t\left)\right|t\in \mathbb{N}\}$ to work out the result of the SAT problem in polynomial time, where all propositional formulas $\phi$ have m clauses and n variables.

We define $s\left(\phi \right)=⟨n,m⟩$ and

$$cod\left(\phi \right)={\alpha }_{1,1}\dots {\alpha }_{n,1}{\alpha }_{1,2}\dots {\alpha }_{n,2}\dots {\alpha }_{1,m}\dots {\alpha }_{n,m},$$

where, for each $1\le i\le n,1\le j\le m$, we have:

$${\alpha }_{i,j}=\left\{\begin{array}{cc}{d}_{i,j}\hfill & \mathrm{if}{\mathrm{x}}_{\mathrm{i}}\mathrm{appears}\mathrm{in}{\mathrm{C}}_{\mathrm{j}};\hfill \\ d_{i,j}^{{}^{\prime }}\hfill & \mathrm{if}\neg {\mathrm{x}}_{\mathrm{i}}\mathrm{appears}\mathrm{in}{\mathrm{C}}_{\mathrm{j}};\hfill \\ d_{i,j}^{{}^{″}}\hfill & \mathrm{if}{\mathrm{x}}_{\mathrm{i}}\mathrm{and}\neg {\mathrm{x}}_{\mathrm{i}}\mathrm{do}\mathrm{not}\mathrm{appears}\mathrm{in}{\mathrm{C}}_{\mathrm{j}}.\hfill \end{array}\right.$$

For each $m,n\in \mathbb{N}$, we design the recognizer PAM P system

$$\Pi =(O,P,\Sigma ,{\left[\right]}_1^0,H,w_1/z_1,\mathcal{E},R,1,0),$$

where

• $O=\Sigma \cup \left\{m_i\right|1\le i\le nm+m+3\}\cup \left\{c_j\right|1\le j\le m\}\cup \{e,m,\mathtt{yes},\mathtt{no}\}$;
• $P=\left\{{\beta }_i\right|1\le i\le nm+m+3\}\cup \{p_{i,j},p_{i,j}^{+},p_{i,j}^{-}|1\le i\le n+1,1\le j\le m+1\}$;
• $\Sigma =\{d_{i,j},d_{i,j}^{\prime },d_{i,j}^{″}|1\le i\le n,1\le j\le m\}$;
• $H=\left\{1\right\}$;
• $w_1=\{m_1,cod\left(\phi \right)\}$;
• $z_1=\{{\beta }_1,p_{1,1}\}$;
• $\mathcal{E}=\left\{m\right\}$;
• the rules of R are designed as follows:
  • $r_{1,i}\equiv {\left[p_{i,1}\right|]}_1^0\to {\left[p_{i,1}^{+}\right|]}_1^0{\left[p_{i,1}^{-}\right|]}_1^0,1\le i\le n$.
  • $\begin{array}{cc}\hfill r_{2,i,j}\equiv \{& {\left[p_{i,j}^{+}\right|d_{i,j}]}_1^0\to {\left[p_{i,j+1}^{+}\right|c_j]}_2{1}^0,\hfill \\ & {\left[p_{i,j}^{+}\right|d_{i,j}^{{}^{\prime }}]}_1^0\to {\left[p_{i,j+1}^{+}\right|d_{i,j}^{{}^{\prime }}]}_1^0,\hfill \\ & {\left[p_{i,j}^{+}\right|d_{i,j}^{{}^{″}}]}_1^0\to {\left[p_{i,j+1}^{+}\right|d_{i,j}^{{}^{″}}]}_1^0|1\le i\le n,1\le j\le m-1\}.\hfill \end{array}$
  • $\begin{array}{cc}\hfill r_{3,i,j}\equiv \{& {\left[p_{i,j}^{-}\right|d_{i,j}]}_1^0\to {\left[p_{i,j+1}^{-}\right|d_{i,j}]}_1^0,\hfill \\ & {\left[p_{i,j}^{-}\right|d_{i,j}^{{}^{\prime }}]}_1^0\to {\left[p_{i,j+1}^{-}\right|c_j]}_1^0,\hfill \\ & {\left[p_{i,j}^{-}\right|d_{i,j}^{{}^{″}}]}_1^0\to {\left[p_{i,j+1}^{-}\right|d_{i,j}^{{}^{″}}]}_1^0|1\le i\le n,1\le j\le m-1\}.\hfill \end{array}$
  • $\begin{array}{cc}\hfill r_{4,i}\equiv \{& {\left[p_{i,m}^{+}\right|d_{i,m}]}_1^0\to {\left[p_{i+1,1}\right|c_m]}_1^0,\hfill \\ & {\left[p_{i,m}^{+}\right|d_{i,m}^{{}^{\prime }}]}_1^0\to {\left[p_{i+1,1}\right|d_{i,m}^{{}^{\prime }}]}_1^0,\hfill \\ & {\left[p_{i,m}^{+}\right|d_{i,m}^{{}^{″}}]}_1^0\to {\left[p_{i+1,1}\right|d_{i,m}^{{}^{″}}]}_1^0|1\le i\le n\}.\hfill \end{array}$
  • $\begin{array}{cc}\hfill r_{5,i}\equiv \{& {\left[p_{i,m}^{-}\right|d_{i,m}]}_1^0\to {\left[p_{i+1,1}\right|d_{i,m}]}_1^0,\hfill \\ & {\left[p_{i,m}^{-}\right|d_{i,m}^{{}^{\prime }}]}_1^0\to {\left[p_{i+1,1}\right|c_m]}_1^0,\hfill \\ & {\left[p_{i,m}^{-}\right|d_{i,m}^{{}^{″}}]}_1^0\to {\left[p_{i+1,1}\right|d_{i,m}^{{}^{″}}]}_1^0|1\le i\le n\}.\hfill \end{array}$
  • $r_{6,j}\equiv {\left[p_{n+1,j}\right|c_j]}_1^0\to {\left[p_{n+1,j+1}\right|c_j]}_1^0,1\le j\le m$.
  • $r_7\equiv m{\left[p_{n+1,m+1}\right|]}_1^0\to {\left[p_{n+1,m+1}\right|e]}_1^0$.
  • $r_8\equiv {\left[p_{n+1,m+1}\right|e]}_1^0\to e{\left[p_{n+1,m+1}\right|]}_1^0$.
  • $r_{9,i}\equiv {\left[{\beta }_i\right|m_i]}_1^0\to {\left[{\beta }_{i+1}\right|m_{i+1}]}_1^0,1\le i\le nm+m+2$.
  • $r_{10}\equiv e{\left[{\beta }_{nm+m+3}\right|]}_1^0\to {\left[{\beta }_{nm+m+3}\right|yes]}_1^0$.
  • $r_{11}\equiv m{\left[{\beta }_{nm+m+3}\right|]}_1^0\to {\left[{\beta }_{nm+m+3}\right|no]}_1^0$.
  • $r_{12}\equiv {\left[{\beta }_{nm+m+3}\right|yes]}_1^0\to yes{\left[{\beta }_{nm+m+3}\right|]}_1^0$.
  • $r_{13}\equiv {\left[{\beta }_{nm+m+3}\right|no]}_1^0\to no{\left[{\beta }_{nm+m+3}\right|]}_1^0$.

Generation stage.

The system produces $2^n$ truth assignments for n variables in the generation stage, $\Pi (⟨n,m⟩)$ checks each truth assignment to decide whether all clauses are true; the generation stage takes $nm+n$ steps, which contain n iterations (each iteration has $m+1$ steps). Next, we analyze each iteration as follows.

In the first step of the i-th iteration, the system employs rule $r_{1,i}$ for each membrane 1 divided into two membranes without changing its label, and each protein $p_{i,1}$ is replaced by $p_{i,1}^{+}$ and $p_{i,1}^{-}$, respectively.

In the next $m-1$ steps of the i-th iteration, in membrane 1 containing protein $p_{i,j}^{+}$ (resp., protein $p_{i,j}^{-}$), one of the rules in $r_{2,i,j}$ (resp., $r_{3,i,j}$) can be employed. If membrane 1 contains protein $p_{i,j}^{+}$ and object $d_{i,j}$ (resp., $d_{i,j}^{\prime }$ or $d_{i,j}^{″}$), with the influence of protein $p_{i,1}^{+}$, object $d_{i,j}$ is modified to $c_j$, and the protein is modified to $p_{i,j+1}^{+}$. If membrane 1 contains protein $p_{i,j}^{-}$ and object $d_{i,j}^{\prime }$ (resp., $d_{i,j}$ or $d_{i,j}^{″}$), with the influence of protein $p_{i,j}^{-}$, object $d_{i,j}^{\prime }$ is modified to $c_j$, and the protein is modified to $p_{i,j+1}^{-}$.

In step $m+1$ of the i-th iteration, if membrane 1 contains protein $p_{i,m}^{+}$ and object $d_{i,m}$ (resp., $d_{i,m}^{\prime }$ or $d_{i,m}^{″}$), with the influence of protein $p_{i,m}^{+}$, one of the rules in $r_{4,i,j}$ is used, object $d_{i,m}$ is modified to $c_m$, and the protein is modified to $p_{i+1,1}$. If membrane 1 contains protein $p_{i,m}^{-}$ and object $d_{i,m}^{\prime }$ (resp., $d_{i,m}$ or $d_{i,m}^{″}$), with the influence of protein $p_{i,m}^{-}$, one of the rules in $r_{5,i,j}$ is used, object $d_{i,m}^{\prime }$ is modified to $c_m$, and the protein is modified to $p_{i+1,1}$.

Checking stage.

The checking stage takes m steps. In the checking stage, each assignment in each membrane 1 is checked for whether all clauses are satisfied. More specifically, when at least one membrane 1 includes all objects $c_1,\cdots ,c_m$, then it indicates that all clauses in $\phi$ are satisfied for that assignment in that membrane; consequently, the computation result of $\phi$ is TRUE; when no membrane 1 includes all objects $c_1,\cdots ,c_m$, then it indicates that in each membrane 1, the Boolean formula $\phi$ cannot be satisfied; consequently, the computation result of $\phi$ is FALSE.

At the j-th ($1\le j\le m$) step of the checking stage, rule $r_{6,j}$ is employed, and protein $p_{n+1,j}$ is modified to $p_{n+1,j+1}$. Note that when object $c_j$ ($1\le j\le m$) does not appear in a membrane 1, rule $r_{6,j}$ cannot be employed in this step.

Output stage.

The system sends the correct result to the environment in this stage. Rules $r_{9,i}$ are used for counting the computation steps except when rules $r_{1,i}$ are used.

If there exists a membrane 1 that has protein $p_{n+1,m+1}$ ($\phi$ is satisfied), at step $nm+n+m+1$, rule $r_7$ is employed, object m is sent into membrane 1, and it is modified to e. At step $nm+n+m+2$, rule $r_8$ is employed, and object e is sent to the environment. At step $nm+n+m+3$, the system employs rule $r_{10}$, and object e is sent to membrane 1, which is modified to yes in this process. At step $nm+n+m+4$, rule $r_{12}$ is applied, object yes is sent to the environment, and the system stops. Consequently, the computation result of $\phi$ is positive.

If protein $p_{n+1,m+1}$ does not appear in any membrane 1 ($\phi$ is not satisfied), then at step $nm+n+m+1$ and step $nm+n+m+2$, only rules $r_{9,i}$ are applied. At step $nm+n+m+3$, rule $r_{11}$ is applied, object m is sent to membrane 1, and it is modified to no. At step $nm+n+m+4$, rule $r_{13}$ is applied, object no is sent to the environment, and the system stops. Consequently, the computation result of $\phi$ is negative.

To design the PAM P system, the necessary resources are counted as follows: (1) the total number of objects is $4nm+n+2m+7$; (2) the total number of proteins is $4nm+3n+4m+6$; (3) the initial number of membranes is 1; (3) the total number of objects at initial configuration is $nm+1$; (4) the total number of rules used in the system is $7nm+n+2m+8$; and (5) the length of a rule (not counting proteins) is no more than 2. Consequently, a PAM P system $\Pi (⟨n,m⟩)$ is constructible in polynomial time by a Turing machine.

Object yes (resp., no) as the result of the PAM P system $\Pi (⟨n,m⟩)$ is transferred to the environment in the last step $nm+n+m+4$. Consequently, the PAM P system $\Pi (⟨n,m⟩)$ is polynomially bounded concerning the sizes of clauses and variants for $\phi$.

Consequently, the family of PAM P systems $\Pi$ offers a uniform result of a Boolean satisfiability problem.    □

5. Conclusions

In this paper, a novel class of P systems whose name is P systems with proteins on active membranes is introduced. The computational power and computational efficiency of PAM P systems are discussed. We showed that PAM P systems achieve Turing universality when the systems use two membranes, one protein and one polarization. In addition, PAM P systems with the help of membrane division rules are able to make the SAT problem solvable.

We showed conclusions that PAM P systems (using rules of types (3) and (4)) have Turing universality and (using rules of types (2), (3), (4) and (6)) can give the result of the SAT problem. A clear remaining problem is discussing whether the numbers of the types of rules of PAM P systems are optimal in order to achieve universality and to give the result of the SAT problem.

Many variants of P systems are able to make the QSAT problem solvable [4,5,32]. A clear remaining problem is how to make the QSAT problem solvable in PAM P systems.