Minimal cooperation as a way to achieve the efficiency in celllike membrane systems
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Abstract
Cooperation is doubtless a relevant ingredient on rewriting rules based computing models. This paper provides an overview on both classical and newest results studying how cooperation among objects influences the ability of celllike membrane systems to solve computationally hard problems in an efficient way. In this paper, two types of such membrane systems will be considered: (a) polarizationless P systems with active membranes without dissolution rules when minimal cooperation is permitted in object evolution rules; and (b) celllike P systems with symport/antiport rules of minimal length. Specifically, assuming that P is not equal to NP, several frontiers of the efficiency are obtained in these two computing frameworks, in such manner that each borderline provides a tool to tackle the P versus NP problem.
Keywords
Minimal cooperation Celllike P systems Computational complexity theory1 Introduction
Membrane Computing is a bioinspired computational paradigm based in the structure and behaviour of living cells. Created in 1998 by Păun [15], it has been a very fruitful area from the point of view of the applications. From theoretical computer science, results within the fields of formal languages [16], computability theory [4] and computational complexity theory [19], to applications in biology [6], robotics [2] and fault diagnosis [27]. An extensive introduction of the area can be found in [5, 12, 14, 17, 29, 30], so these reference can be useful for readers interested in a further reading.
Cooperation between cells is a widespread phenomenon in nature and generally has a cost for the cooperating cells. Cooperation and competition can play a role in achieving robust developmental processes in multicellular organisms [3]. In the framework of Membrane Computing, the dynamics of celllike membrane systems is captured by means of rewriting rules that allow the evolution of objects. Any object, alone or together with more objects can be transformed into other objects, can pass through a membrane or can dissolve/divide/separate the membrane where it currently resides [15]. Noncooperative systems contain rules abstracting proteins that transport only one solute across the membrane (for example, the glucose transporter in red blood cell membranes).
Membranes are both separators and channels of communication, but they are passive participants to the process. However, from biochemistry view the membranes are not at all passive, the passing of a chemical compound through a membrane is often done by some kind of interaction with the membrane itself [18]. P systems with active membranes are interesting particular cases of noncooperative systems. In the first version of these systems [18], electrical charges from the set \(\{ +,,0\}\) are associated with membranes. Besides, each rule is associated with a membrane (labelled by h) and it is applicable to a configuration at an instant t if the object corresponding to its lefthand side is in such membrane of that configuration, and the membrane has the polarization prefixed in the rule. Even though on the lefthand side of the rule there is only one object, there exists, in some sense, a kind of cooperation (object/label/polarization) in order to trigger the rule.
A particular case of cooperative system is celllike P systems with symport/antiport rules [13] where there exist communication rules length at least two (more than one object are involved in the rules). In these systems, communication is implemented by means of symport/antiport rules abstracting transmembrane transport of couples of chemical substances, in the same or in opposite directions.
As it is stated by the socalled Milano theorem [28], no computationally hard problems can be solved in polynomial time without using rules allowing the generation of an exponential workspace (expressed in terms of the number of membranes and objects) in polynomial time, unless \(\mathbf{P} = \mathbf{NP}\) holds. So, in order to achieve efficiency (the ability to provide polynomialtime solution to computationally hard problems), division rules (abstracting cell division process) and separation rules (abstracting membrane fission process) are also considered.
It is worth pointing out that dissolution and division rules for nonelementary membranes (they have membrane inside) are not necessary/relevant to get the computational efficiency in (noncooperative) P systems with active membranes. Nevertheless, if electrical charges are forbidden/removed, then dissolution rules start to play a relevant role in these systems from a computational complexity point of view. Only problems in class P can be efficiently solved by means of families of polarizationless P systems with active membranes which make no use of dissolution rules, even in the case that division rules for nonelementary membranes are allowed [8]. The situation is similar for noncooperative celllike P systems with symport/antiport rules where only a single object is involved in any rule (so, only symport rules with length 1 are permitted).
The main goal of this work is to present recent results concerning the efficiency of both computing frameworks: polarizationless P systems with active membranes and celllike P systems with symport/antiport rules when minimal cooperation is considered. The term “minimal cooperation” is used in the following sense: in the first case, the lefthand side of each rule has at most a couple of objects. In the second case, at most two objects are involved in each communication (symport/antiport) rules.
The paper is organized as follows: first, concepts as presumed efficiency and techniques to tackle the P vs. NP problem are explained. Section 3 is devoted to introduce the two frameworks we are going to work on. In the next section, three techniques to prove the nonefficiency of certain models will be provided. In Sect. 5, the results obtained in the two frameworks will be exposed, as well as enough citations to give the reader an extensive bibliography to consult. Finally, some conclusions and open questions are stated.
2 Efficiency of computing models
Let us recall that P (respectively, NP) denotes the class of all decision problems which can be solved by a deterministic (resp. nondeterministic) Turing machine working in polynomial time with respect to the size of the input. Problems in class P are called tractable problems. NPcomplete problems are the hardest problems within the class NP. It is unknown if they are nontractable problems but it is widely believed that NPcomplete problems are intractable.
A computing model with the ability to provide polynomialtime solutions to intractable problems (respectively, NPcomplete problems) is called an efficient (respectively, presumably efficient) computing model. In a nonefficient computing model, only problems in class P can be solved in polynomial time.
In what follows, we assume that the reader is already familiar with the basic notions and terminology of P systems. For more details, see [15].
3 The computational frameworks
In this section, the computational frameworks of membrane systems considered throughout this paper are introduced.
3.1 Polarizationless P system with active membranes
A polarizationless P system with active membranes \((\varGamma , \mu , {{\mathcal {M}}}_1, \dots , {{\mathcal {M}}}_q, {{{\mathcal {R}}}}, i_{out})\) of degree \(q \ge 1\), can be viewed as a set of q membranes, injectively labelled by elements of \(H=\{1, \dots , q\}\), arranged in a hierarchical structure \(\mu \) given by a rooted tree, such that: (a) \({\mathcal M}_1,\dots , {{\mathcal {M}}}_q\) represent the finite multisets of objects (elements of the working alphabet \(\varGamma \)) initially placed in the q membranes of the system; (b) \({\mathcal {R}}\) is a finite set of rules over \(\varGamma \) associated with the labels including division rules inspired from mitosis process; (c) \(i_{out}\in H \cup \{ env \}\) represents a distinguished zone which will encode the output of the system. We use the term zonei to refer to membrane i in the case \(i \in H\) and to refer to the environment in the case \(i=env\). The leaves of \(\mu \) are called elementary membranes; any other membrane is said to be nonelementary. Semantics of polarizationless P systems with active membranes are defined as it is usual in P systems with active membranes (see [9, 18] for details).

Separation rules for elementary membranes \([ \, a \, ]_h \rightarrow [ \, \varGamma _0 \, ]_h \; [ \, \varGamma _1 \, ]_h\), for \(h \in H{\setminus } \{ i_{out}\}\), \(a \in \varGamma \) and h is not the label of the root of \(\mu \).

Separation rules for nonelementary membranes \([ \, [ \, \, ]_{h_0} [ \, \, ]_{h_1} \, ]_{h} \rightarrow [ \ \varGamma _0 \ [ \, \, ]_{h_0} \, ]_{h} \ [ \ \varGamma _1 \ [ \, \, ]_{h_1} \, ]_{h}\), where \(h \in H {\setminus } \{i_{out}\}\) is not the label of the root of \(\mu \), \(h_0 \in H_0\) and \(h_1 \in H_1\).
A separation rule \([ \, [ \, \, ]_{h_0} [ \, \, ]_{h_1} \, ]_{h} \rightarrow [ \ \varGamma _0 \ [ \, \, ]_{h_0} \, ]_{h} \ [ \ \varGamma _1 \ [ \, \, ]_{h_1} \, ]_{h}\) is applicable to a configuration \(\mathcal{C}_t\) at an instant t, if there exists a membrane labelled by h in \({{\mathcal {C}}}_t\) such that it contains a membrane labelled by \(h_0\) and another membrane labelled by \(h_1\). When applying such a separation rule to a membrane labelled by h in \({{\mathcal {C}}}_t\), that membrane is separated into two membranes with the same label, in such a way that the contents (objects and inner membranes) are distributed as follows: The first membrane receives the objects from \(\varGamma _0\) and all inner membranes whose label belong to set \(H_0\); while the second membrane receives the objects from \(\varGamma _1\) and all inner membranes whose label belongs to set \(H_1\).
3.1.1 Minimal cooperation in object evolution rules
Next, minimal cooperation in object evolution rules is introduced in the framework of polarizationless P systems with active membranes. In general, the term “minimal cooperation” associated with membrane systems is used when there is at least one rule in the system whose lefthand side consists of exactly two symbols.
Definition 1

Primary minimal cooperation (pmc) In this case, object evolution rules are of the form \([ \, u \rightarrow v \, ]_h\), where \(h \in H\) and u, v are multisets over the working alphabet \(\varGamma \) such that \(1 \le u, v \le 2\), but at least one such rule verifies \(u=2\).

Bounded minimal cooperation (bmc) In this case, object evolution rules are of the form \([ \, u \rightarrow v \, ]_h\), where \(h \in H\) and u, v are multisets over the working alphabet \(\varGamma \) such that \(1 \le v \le u \le 2\), but at least one such rule verifies \(u=2\).

Minimal cooperation and minimal production (mcmp) In this case, object evolution rules are of the form \([ \, a \rightarrow b \, ]_h , \ [ \, a \, b \rightarrow c \, ]_h\), where \(h \in H\) and a, b, c are symbols that belong to the working alphabet \(\varGamma \), but at least one such rule is of the second type.

If \(\alpha =pmc\) (resp. \(\alpha =bmc\) or \(\alpha =mcmp\)) then primary minimal cooperation (resp. bounded minimal cooperation or minimal cooperation and minimal production) in object evolution rules are permitted. If \(\alpha =+e\) then only noncooperative rules are permitted as object evolution rules.

If \(\beta =+c\) (resp. \(\beta =c\)), then communication rules are permitted (resp. forbidden).

If \(\gamma =+d\) (resp. \(\gamma =d\)), then dissolution rules are permitted (resp. forbidden).

If \(\delta =+n\) (resp. \(\delta =n\)), then division/separation rules for elementary and nonelementary membranes are permitted (resp. only division/separation rules for elementary membranes are permitted).
3.2 Celllike P systems with symport/antiport rules
Celllike P systems with symport/antiport rules were introduced in [13] aiming to abstract the biological phenomenon of transmembrane transport of couples of chemical substances, in the same or in opposite directions. A P system with symport/antiport rules \(\Pi =(\varGamma , {\mathcal {E}}, \mu , {{\mathcal {M}}}_1,\dots , {{\mathcal {M}}}_q, {\mathcal {R}}_1, \ldots , {\mathcal {R}}_q, i_{out})\) of degree \(q \ge 1\) can be viewed as a set of q membranes, labelled by \(1, \dots , q\), arranged in a hierarchical structure \(\mu \) given by a rooted tree whose root is called the skin membrane, such that: (a) \({{\mathcal {M}}}_1,\dots , {{\mathcal {M}}}_q\) represent the finite multisets of objects initially placed in the q membranes of the system; (b) \({\mathcal {E}}\) is the set of objects initially located in the environment of the system, all of them available in an arbitrary number of copies; (c) \({\mathcal {R}}_1, \ldots , {\mathcal {R}}_q\) are finite sets of communication rules over \(\varGamma \) (\({\mathcal {R}}_i\) is associated with the membrane i of \(\mu \)); and (d) \(i_{out}\) represents a distinguished zone which will encode the output of the system. We use the term zonei (\(0 \le i \le q\)) to refer to membrane i in the case \(1 \le i \le q\) and to refer to the environment in the case \(i=0\). The length of rule (u, out) or (u, in) (resp. (u, out; v, in)) is defined as u (resp. \(u+v\)). In these models, division rules and separation rules are introduced in a similar way that in polarizationless P systems with active membranes, both in syntax and semantics.
For each natural number \(k \ge 1\), we denote by \(\mathcal{CDC}(k)\) (respectively, \(\mathcal{CSC}(k)\)) the class of P systems with symport/antiport rules and division rules (respectively, separation rules) such that the length of the communication rules is at most k. When division or separation also for nonelementary membranes is permitted, then we denote \(\mathcal{CD}_{ne}{{\mathcal {C}}}(k)\) and \(\mathcal{CS}_{ne}{{\mathcal {C}}}(k)\), respectively. If the set \({{{\mathcal {E}}}}\) associated with the environment is the empty set, then we denote \(\widehat{\mathcal{CDC}}(k)\), \(\widehat{\mathcal{CSC}}(k)\), \(\widehat{\mathcal{CD}_{ne}{{\mathcal {C}}}}(k)\) and \(\widehat{\mathcal{CS}_{ne}{{\mathcal {C}}}}(k)\), respectively.
4 Techniques to prove nonefficiency

Dependency graph A directed graph (dependency graph) \(G_{\Pi }\) associated with a membrane system \(\Pi \) is considered in such a way that there exists an accepting computation of \(\Pi \) if and only if there exists a path between two distinguished nodes in the dependency graph associated with it.

Algorithmic technique A deterministic algorithm \(\mathcal {A}\) working in polynomial time that receives as input a membrane system \(\Pi \) and an input multiset m of \(\Pi \) is considered in such a manner that algorithm \(\mathcal {A}\) reproduces the behaviour of a single computation of the system \(\Pi \) with input multiset m.

Simulation technique A membrane system \(\Pi '\) simulating (in an efficient way) a given membrane system \(\Pi \) is constructed, in such a way that: (a) \(\Pi '\) is obtained from \(\Pi \) by removing some syntactic/semantic ingredients; and (b) there exists an efficient and injective correspondence between accepting computations of \(\Pi '\) and accepting computations of \(\Pi \).
5 Nonefficiency and presumed efficiency of membrane systems
The role of minimal cooperation is studied from the computational complexity point of view. In the case of polarizationless P systems with active membranes, object evolution rules whose lefthand side consists of a couple of objects are considered. In the case of celllike P systems with symport–antiport rules, minimal cooperation that is expressed by the property of the length of communication (symport/antiport) rules equals to two.
5.1 Polarizationless P systems with active membranes
P systems with active membranes and electrical charges are (noncooperative) presumably efficient computing models even though dissolution rules and division for nonelementary membranes are forbidden. However, if polarizations are removed the dissolution rules play a crucial role from complexity point of view. In fact, using the dependency graph technique it has been shown that families of noncooperative polarizationless P systems with active membranes which make no use of dissolution rules can only solve tractable problems in an efficient way [8, 21].
Theorem 1
\(\mathbf{P}=\mathbf{PMC}_{\mathcal{DAM}^0(+e,+c,d,+n)}=\mathbf{PMC}_{\mathcal{SAM}^0(+e,+c,d,+n)}\).
Nevertheless, when dissolution rules are allowed in that framework, the situation is quite different. If division rules for nonelementary membranes are also permitted, then PSPACEcomplete problems can be solved in polynomial time [1].
Theorem 2
\(\mathtt{QSAT} \in \mathbf{PMC}_{\mathcal{DAM}^0(+e,+c,+d,+n)}\).
It is an open question what happens if division rules only for elementary membranes are permitted in polarizationless P systems with active membranes and with dissolution rules (the socalled Păun’s conjecture).
Open question:\(\mathbf{PMC}_{\mathcal{DAM}^0(+e,+c,+d,n)} = \mathbf{P}\) ?
The presumed efficiency of polarizationless P systems with active membranes and membrane division can be reached by considering bounded minimal cooperation, even in the case that dissolution rules and division rules for nonelementary membranes are forbidden [22].
Theorem 3
\(\mathtt{SAT} \in \mathbf{PMC}_{\mathcal{DAM}^0(bmc,+c,d,n)}\).
However, if separation rules are used instead of division rules in polarizationless P systems with active membranes which make use of bounded minimal cooperation then, using the algorithmic technique, the limitation on the efficiency of these systems has been shown [24].
Theorem 4
\(\mathbf{PMC}_{\mathcal{SAM}^0(bmc,+c,d,+n)}=\mathbf{P}\).
Families of polarizationless P systems with active membranes and membrane separation which use primary minimal cooperation in object evolution rules can solve NPcomplete problems in polynomial time, even in the case that dissolution rules and division rules for nonelementary membranes are forbidden [21].
Theorem 5
\(\mathtt{SAT} \in \mathbf{PMC}_{\mathcal{SAM}^0(pmc,+c,d,n)}\).
The result of Theorem 3 has been improved in the sense that minimal cooperation in object evolution rules producing only a single object suffices to reach the efficiency of polarizationless P systems with active membranes [25].
Theorem 6
\(\mathtt{SAT} \in \mathbf{PMC}_{\mathcal{DAM}^0(mcmp,+c,d,n)}\).
By considering the corresponding “counting version” of membrane systems from \(\mathcal{DAM}^0(mcmp,+c,d,n)\), the previous result has been extended to the counting problem \(\# SAT\) [23], which is a wellknown \(\#\)Pcomplete problem [11].
Bearing in mind that minimal cooperation with minimal production in object evolution rules is a particular case of bounded minimal cooperation, we deduce the following result:
Theorem 7
\(\mathbf{PMC}_{\mathcal{SAM}^0(mcmp,+c,d,+n)}=\mathbf{P}\).
Non efficiency  Presumed efficiency  Frontier 

\(\mathcal {DAM}^0(+e, +c, d, +n)\)  \(\mathcal {DAM}^0(+e, +c, +d, +n)\)  Dissolution rules 
\(\mathcal {DAM}^0(+e, +c, d, n)\)  \(\mathcal {DAM}^0(bmc, +c, +d, n)\)  Bounded minimal cooperation 
\(\mathcal {SAM}^0(bmc, +c, d, +n)\)  \(\mathcal {SAM}^0(pmc, +c, +d, +n)\)  Kind of minimal cooperation 
\(\mathcal {SAM}^0(mcmp, +c, d, n)\)  \(\mathcal {DAM}^0(mcmp, +c, +d, n)\)  Kind of rules 
Let us recall that each such frontier provides a tool to tackle the P versus NP problem. For instance, in the framework of polarizationless P systems with active membranes and membrane division (\(\mathcal {DAM}^0(+e, +c, ??, +n)\)), passing from forbidding the use of dissolution rules to permitting them amounts to passing from nonefficiency to presumed efficiency.
5.2 Celllike P systems with symport/antiport rules
From the complexity point of view, noncooperative celllike P systems with symport/antiport rules and polarizationless P systems with active membranes and without dissolution rules have some similarities. In particular, the dependency graph technique used to prove Theorem 1 can be adapted to show that such noncooperative systems are not efficient unless \(\mathbf{P}=\mathbf{NP}\) holds.
Theorem 8
\(\mathbf{P} = \mathbf{PMC}_{\mathcal{CD}_{ne}{{\mathcal {C}}}(1)} = \mathbf{PMC}_{\mathcal{CS}_{ne}{{\mathcal {C}}}(1)}\).
However, if minimal cooperation in communication rules is considered, then computational efficiency can be reached when membrane division rules are used [26].
Theorem 9
\(\mathtt{HAMCYCLE} \in \mathbf{PMC}_{\mathcal{CDC}(2)}\).
Concerning membrane separation rules, if the length of communication rules is at most two then, using the algorithmic technique, it can be shown that separation rules are not enough to provide polynomialtime solutions to computationally hard problems, assuming that \(\mathbf{P} \ne \mathbf{NP}\). Nevertheless, P systems with symport/antiport rules with length at most three and with membrane separation are computationally efficient [10].
Theorem 10
\(\mathbf{P} = \mathbf{PMC}_{\mathcal{CSC}(2)}\; and \;\mathtt{SAT} \in \mathbf{PMC}_{\mathcal{CSC}(3)}\).
In [20], a polynomialtime solution for the QSAT problem, which is a wellknown PSPACEcomplete problem [7], has been given by means of a family of celllike P systems with communication rules with length at most three which make use of division rules for nonelementary membranes.
Theorem 11
\(\mathtt{QSAT} \in \mathbf{PMC}_{\mathcal{CD}_{ne}{{\mathcal {C}}}(3)}\).
Next, the role of environment associated with P systems with symport/antiport rules is studied from the complexity point of view. Let us recall that in these systems environment provides an arbitrary large amount of objects at the initial configuration.
First, it has been shown that the role of the environment is not relevant in P systems with symport/antiport rules and membrane division using simulation technique.
Theorem 12
For each \(k \ge 1\), we have \(\mathbf{PMC}_{\widehat{\mathcal{CDC}}(k)}=\mathbf{PMC}_{\mathcal{CDC}(k)}\).
Second, it has been shown that P systems with symport/antiport rules without environment and with membrane separation can only solve problems in class P in polynomial time [10] using algorithmic technique.
Theorem 13
For each\(k \ge 1\)we have\(\mathbf{P}=\mathbf{PMC}_{\widehat{\mathcal{CSC}}(k)}\).
Hence, in this kind of P systems the role of environment is important:
\(\mathbf{P}=\mathbf{PMC}_{\widehat{\mathcal{CSC}}(3)}\) and \(\mathtt{SAT} \in \mathbf{PMC}_{\mathcal{CSC}(3)}\).
Non efficiency  Presumed efficiency  Frontier 

\(\mathcal {CDC}(1)\)  \(\mathcal {CDC}(2)\)  Length of rules 
\(\mathcal {CSC}(2)\)  \(\mathcal {CSC}(3)\)  Length of rules 
\(\mathcal {CSC}(2)\)  \(\mathcal {CDC}(2)\)(2)  Length of rules 
\(\widehat{{{\mathcal {C}}}{{\mathcal {S}}}{{\mathcal {C}}}(2)}\)  \(\widehat{{{\mathcal {C}}}{{\mathcal {D}}}{{\mathcal {C}}}(2)}\)  Kind of rules 
\(\widehat{\mathcal {CDC}}\)  \(\mathcal {CSC}\)  Environment 
6 Conclusions
In this work, an overview of the results related to the computational efficiency of membrane systems which make use of (minimal) cooperation is presented. Informally speaking, one could say that it is not surprising that in most cases models using cooperation prove to be more powerful than their noncooperative counterparts, by setting an analogy with the existing gap between contextfree and contextsensitive rules in the Chomsky hierarchy. This paper investigates the impact of (minimal) cooperation in the framework of celllike membrane systems. That is, the goal is to restrict the cooperation to its limit, while minimizing the rest of ingredients of the model, in such a way that borderlines of efficiency (expressed in terms of the degree of cooperation) are unveiled. More precisely, the paper focuses on two case studies: allowing some “minimal” degree of cooperation in object evolution rules for polarizationless P systems with active membranes, and reducing symport/antiport rules to “minimal” length for celllike P systems.

To reach the efficiency, can the use of division rules for nonelementary membranes be avoided?

What about the efficiency of polarizationless P systems with active membranes and separation rules which incorporates minimal cooperation with minimal production in communication rules?

Is there a family of systems from \(\mathcal{DAM}^0(mcmp,+c,d,+n)\) (or from \(\mathcal{SAM}^0(mcmp,+c,d,+n)\)) providing a polynomialtime solution to the QSAT problem?

\(\mathtt{QSAT} \in \mathbf{PMC}_{\mathcal{CD}_{ne}{{\mathcal {C}}}(2)}\) ?

\(\mathtt{QSAT} \in \mathbf{PMC}_{\mathcal{CS}_{ne}{{\mathcal {C}}}(3)}\) ?
Notes
Acknowledgements
This work is supported by the research project TIN201789842P, cofinanced by Ministerio de Economía, Industria y Competitividad (MINECO) of Spain, through the Agencia Estatal de Investigación (AEI), and by Fondo Europeo de Desarrollo Regional (FEDER) of the European Union. The authors also acknowledge the Grants No 61320106005 of the National Natural Science Foundation of China.
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