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Complexity
Volume 2018 (2018), Article ID 3745210, 21 pages
https://doi.org/10.1155/2018/3745210
Research Article

The Computational Complexity of Tissue P Systems with Evolutional Symport/Antiport Rules

1Key Laboratory of Image Information Processing and Intelligent Control of Education Ministry of China, School of Automation, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
2School of Electric and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China
3Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain

Correspondence should be addressed to Bosheng Song; nc.ude.tsuh@gnosgnehsob

Received 29 May 2017; Accepted 18 December 2017; Published 23 April 2018

Academic Editor: Sigurdur F. Hafstein

Copyright © 2018 Linqiang Pan 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.

Abstract

Tissue P systems with evolutional communication (symport/antiport) rules are computational models inspired by biochemical systems consisting of multiple individuals living and cooperating in a certain environment, where objects can be modified when moving from one region to another region. In this work, cell separation, inspired from membrane fission process, is introduced in the framework of tissue P systems with evolutional communication rules. The computational complexity of this kind of P systems is investigated. It is proved that only problems in class P can be efficiently solved by tissue P systems with cell separation with evolutional communication rules of length at most , for each natural number . In the case where that length is upper bounded by , a polynomial time solution to the SAT problem is provided, hence, assuming that a new boundary between tractability and NP-hardness on the basis of the length of evolutional communication rules is provided. Finally, a new simulator for tissue P systems with evolutional communication rules is designed and is used to check the correctness of the solution to the SAT problem.

1. Introduction

A cell is the basic unit of biological organization that constitutes all living organisms. There are many different types of biological cells, which have different specialized functions that maintain an organism working properly. Inspired by the structure and functioning of living cells, Păun proposed a computing paradigm in 2000 [1], called membrane computing, which has become an active research area (for other research domains of natural computing, one can refer to [2, 3]). A large number of theoretical models were proposed [46] and they have been used to solve various real problems [710]. These distributed and parallel computation models investigated in membrane computing are called P systems. In general, there exist two main families of P systems: cell-like P systems [1], which have a hierarchical arrangement of membranes described by a rooted tree (corresponding to cell-like membrane structure), and tissue-like P systems [11] or neural-like P systems [12], which have a net of cells or neurons described by a directed graph. An overview of membrane computing can be found in [13]. The present work deals with tissue-like P systems.

Inspired by the biological phenomenon of trans-membrane transport of couples of chemicals, communication P systems with symport/antiport rules were proposed in [14], where symport rules move objects (without evolution) between two regions in one direction, and antiport rules move objects (without evolution) between two regions in opposite directions. Later, symport/antiport rules were considered in tissue-like P systems [11], where cells are placed in nodes of a directed graph, and an arc between two nodes corresponds to a communication channel between cells placed in these nodes.

Since the notion of tissue P systems was proposed, numerous research topics have been arisen [1518], and various ingredients (energy, catalyst, mitosis, etc.) from other computational models were considered in the context of tissue P systems. In [19], tissue P systems with channel states controlling the communication between two regions were proposed, and several Turing universality results were achieved, where the systems work in a maximally parallel way with sequential behavior on channels. In [20], a general model of tissue P systems with channel states that allow us to model hybrid cooperating grammar systems was considered, where the results were established for strings and arrays.

Tissue P systems have been used to find polynomial time solutions to NP-complete problems. In [21], membrane division rules used in P systems with active membranes have been introduced into tissue P systems yielding tissue P systems with cell division, and a polynomial time uniform solution to the SAT problem was shown. Since then, tissue P systems with cell division were also considered to solve other NP-complete problems: bin packing [22], subset sum [23], vertex cover [24], and so on. Cell division rules have a replication functioning; that is, two new created cells by a division rule have exactly the same objects except for at most a pair of different objects. Inspired from membrane fission process, cell separation rules are another way to obtain an exponential workspace in polynomial time, but they do not have the duplication functioning; that is, when a cell is separated, the objects in the cell are distributed in each of the newly created cells. Tissue P systems with cell separation have also been used to solve NP-complete problems in polynomial time; one can refer to [25, 26] for these investigations.

Computational complexity theory in the framework of tissue P systems was introduced in [21] and it has been studied in [2730]. It was shown that in the framework of tissue P systems with cell division, only tractable problems can be efficiently solved by using communication rules with length at most one [31] (the length of such a rule is the total number of objects involved in it), but a uniform polynomial time solution to the HAM-CYCLE problem by a family of such P systems using communication rules with length at most two has been given [30]. On the other hand, in the framework of tissue P systems with cell separation, by using communication rules with length at most two, only tractable problems can be efficiently solved, but the SAT problem can be solved by this kind of P systems using communication rules with length at most three [26]. Moreover, frontiers between efficiency and nonefficiency in terms of the length of symport/antiport rules in the framework of cell-like P systems have been investigated in [32].

Tissue-like P systems with evolutional symport/antiport rules (TESA P systems, for short) were proposed in [33], where objects are moved from one region to another region and may be evolved during this process. In [33], the computational efficiency of TESA P systems with cell division (TESAD P systems, for short) was investigated. It is shown that a limit on the efficiency of TESAD P systems is provided with evolutional communication rules of length at most 2. However, when using evolutional communication rules of length at most 4, the SAT problem can be solved by TESAD P systems. However, it is still an open problem as formulated in [33] related to the role of evolutional communication rules in tissue P systems with cell separation from a computational complexity point of view.

During those computational complexity studies for new variants of P systems, the solutions designed for NP-complete problems are frequently difficult to follow, and requerying makes sure that the evolution of the systems is exactly as expected. In this context, the aid of computer tools to assist in both the design and verification tasks may be crucial, producing much more reliable solutions. In this sense, the development of P-Lingua [3436] implied a significant progress. This open source framework includes a standard language aiming to specify the elements of different types of P systems using a notation very close to the researchers in membrane computing community. Besides, it contains simulation engines for a number of P system types. On top of that, MeCoSim [37, 38] provides an additional layer of abstraction with a visual application where researchers can explore at a higher level the evolution of their solutions based on P systems.

In this work, we investigate tissue P systems with evolutional symport/antiport rules and cell separation (TESAS P systems, for short) from a computational complexity point of view.

Contributions of the present work are summarized as follows:(a)A variant of tissue P systems with evolutional symport/antiport rules, called tissue P systems with evolutional symport/antiport rules and cell separation (TESAS P systems, for short), and the corresponding recognizer version are proposed. In TESAS P systems, the length of an evolutional symport/antiport rule is defined as an ordered pair whose first component is the total number of objects involved in the left hand side (LHS) of the rule and the second component is the total number of objects involved in the right hand side (RHS) of the rule; that is, . The set of all recognizer TESAS P systems with evolutional communication rules of length at most is denoted by TSEC.(b)The computation efficiency of recognizer TESAS P systems is investigated in terms of the length of evolutional communication rules. It is shown that only tractable problems can be efficiently solved by families of systems from TSEC or from TSEC, for each natural number . We further show that the SAT problem can be solved in polynomial time by a family of systems from TSEC, hence, assuming that , a new boundary between tractability and NP-hardness on the basis of the length of evolutional communication rules, is provided.(c)A new simulator MeCoSim is designed in order to check the correctness of the solution to the SAT problem. By using the software MeCoSim, we can analyse the designed P systems, then run the simulation, and obtain the computation results.

2. Tissue P Systems with Evolutional Symport/Antiport Rules and Cell Separation

Let us start this section by recalling some notions from formal language theory used in this work; the reader can find details in [39].

An alphabet is a nonempty set. Any sequence of elements from is called a string over . The length of , denoted by , is the number of occurrences in of symbols from .

A multiset over an alphabet is a function from to the set of natural numbers . The multiplicity of a symbol in the multiset is . The support of is the set of symbols such that . A multiset is finite and its support is a finite set. The set of all finite multisets over is denoted by , and by we denote the set of all nonempty finite multisets over an alphabet , and the empty multiset is denoted by . The cardinal of a finite multiset , denoted by , is the sum of all multiplicities of elements in the support of . If and are multisets over , then we define the union of and , denoted by , as follows: , for each .

Definition 1. A tissue P system with evolutional symport/antiport rules and cell separation of degree is a tuple where (i) and are finite alphabets such that ;(ii), are nonempty sets such that and ;(iii) are multisets over ;(iv) is a finite set of rules of the following forms:(1)evolutional communication rules:(a), where , , , (evolutional symport rules);(b), where , , , (evolutional antiport rules);(2)separation rules:(a), where , , ;(v).

A tissue P system with evolutional symport/antiport rules and cell separation of degree , , can be viewed as a set of cells, labelled by such that (a) represent the multisets of objects initially placed in the cells of the system; (b) is the set of objects initially located in the environment of the system, all of them available in an arbitrary number of copies; and (c) represents a distinguished region which will encode the output of the system. We use the term region () to refer to cell in the case and to refer to the environment in the case .

A configuration at any instant of a TESAS P system is described by multisets of objects in each cell and the multiset of objects over in the environment at that moment. The initial configuration of is .

An evolutional symport rule is enabled at a configuration at an instant if there is a region from which contains multiset . By applying an evolutional symport rule, the multiset of objects in region from is consumed and the multiset of objects is generated in region from . An evolutional antiport rule is enabled at a configuration at an instant if there is a region from which contains multiset of objects and a region from which contains multiset of objects . By applying an evolutional antiport rule, (a) the multiset of objects in region from and the multiset of objects in region from are consumed; (b) the multiset of objects is generated in region from configuration ; and (c) the multiset of objects is generated in region from configuration . The length of an evolutional symport/antiport rule is an ordered pair of natural numbers: .

A separation rule is enabled at a configuration at an instant if there is a cell from which contains object and . By applying a separation rule to a such a cell , (a) object is consumed from such cell; (b) two new cells with label are generated at configuration ; (c) in the original cell , the objects from are placed in one of the new cells, while the other objects from are placed in another one.

The rules of a TESAS P system are applied in a maximally parallel manner, and we have the restriction that when a cell is separated at one transition step, no other rules can be applied for that cell at that step.

A transition from a configuration to another configuration is obtained by applying rules in a maximally parallel manner following the previous remarks. A computation of the system is a (finite or infinite) sequence of transitions starting from the initial configuration, where any term of the sequence other than the first is obtained from the previous configuration in one transition step. If the sequence is finite (called halting computation) then the last term of the sequence is a halting configuration, that is, a configuration where no rule is applicable to it. A computation gives a result only when an halting configuration is reached, and that result is encoded by the multiset of objects present in the output region .

A natural framework to solve decision problems is to use recognizer P systems; one can refer to [21, 29] for further details.

Definition 2. A recognizer tissue P system with evolutional symport/antiport rules and cell separation of degree is a tuple where(i)the tuple is a TESAS P system of degree , where strictly contains an (input) alphabet and two distinguished objects yes, no, and () are multisets over ;(ii) is the input cell and is the label of the environment;(iii)for each multiset over the input alphabet , any computation of the system with input starts from the configuration of the form and always halts and either object yes or object no (but not both) must appear in the environment at the last step.

It is worth pointing out that, in any recognizer TESAS P systems, all computations halt. Then, any symport rule of the type must verify the following condition: multiset contains some object from .

For each ordered pair of natural numbers greater than or equal to 1, the class of recognizer TESAS P systems with cell separation and with evolutional communication rules of length at most is denoted by . This means that the LHS (resp., RHS) of any evolutional communication rule in a system from involves at most objects (resp., objects).

Next, we define the concept of solving a problem in a uniform way and in polynomial time by a family of recognizer TESAS P systems (see [40] for details).

Definition 3. A decision problem is solvable in a uniform way and in polynomial time by a family of recognizer TESAS P systems if the following conditions hold:(i)the family is polynomially uniform by Turing machines;(ii)there exists a polynomial encoding of in such that (a) for each instance , is a natural number and is an input multiset of the system ; (b) for each , is a finite set; and (c) the family is polynomially bounded, sound, and complete with regard to .

The set of all decision problems that can be solved by recognizer TESAS P systems with evolutional communication rules of length at most in a uniform way and polynomial time is denoted by .

3. The Computational Complexity of Tissue P Systems with Evolutional Symport/Antiport Rules

3.1. The Limitation on the Efficiency of

In this subsection, we use tissue P systems with cell separation and evolutional communication rules of length at most to provide a new characterization of the classical complexity class .

The proof uses a similar technique as in [32]. Firstly, some representations of TESAS P systems from are given. By (resp., ) we denote the set of communication rules (resp., separation rules) of . We will fix a total order in and a total order in . Because several cells with the same label are generated by using separation rules, in order to identify the different cells with the same label, the following recursive definition is used to modify the labels of the new generated cells:(i)We denote the label of a cell as a pair , where and is a binary string.(ii)If a separation rule is applied to a cell with label , then the new created two cells will be labelled by and , respectively. We mention that, for the system during any computation, we consider a lexicographical order over the set of labels of cells.

Note that if communication rules occur in two cells, then the labels of these two cells do not change.

A configuration at an instant of a tissue P system from is described by the multisets of objects over contained in each cell and the multiset of objects over in the environment. Hence, a configuration of can be described as follows:

Let , , be an evolutional symport rule of and . We denote by the multiset of objects and the corresponding the multiset of objects . In a similar way, and are defined when is of the form .

Let , , be an evolutional symport rule of and . We denote by the multiset of objects , and the corresponding is . In a similar way, and are defined when is of the form .

Let , , be an evolutional symport rule of and . We denote by the multiset of objects and we denote by the corresponding the multiset of objects . In a similar way, and are defined when is of the form .

Let , , be an evolutional symport rule of and . We denote by the multiset of objects The corresponding is . In a similar way, and are defined when is of the form .

Let , , be an evolutional antiport rule of and . We denote by the multiset of objects and the corresponding the multiset of objects . In a similar way, and are defined when is of the form .

Let , , be an evolutional antiport rule of and . We denote by the multiset of objects , and the corresponding is .

Let , , be an evolutional antiport rule of and . We denote by the multiset of objects and we denote by the corresponding the multiset of objects

Let , , be an evolutional antiport rule of and . We denote by the multiset of objects and we denote by the corresponding the multiset of objects .

Let , , be an evolutional antiport rule of and . We denote by the multiset of objects The corresponding is .

If is a configuration of , then the multiset obtained by replacing in every occurrence of by is denoted by . Moreover, we denote by (resp., ) a multiset of labelled objects added to (resp., removed from) the configuration .

Next, we show that TESAS P systems from can only solve tractable problems.

If of () is a halting computation, then we denote by the length of . For each , the multiset of objects over contained in all cells labelled by at configuration is denoted by . We denote by the multiset of objects over contained in the environment at configuration . Finally, the finite multiset is denoted by .

Lemma 4. Let be a recognizer tissue P system of degree from . Let and let be a computation of . Then, one has(1), and for each , , ;(2)for each , , ;(3)the number of created cells along the computation by the application of cell separation rules is bounded by .

Proof. (1) Let us notice that Let be a recognizer tissue P system from and let be the set of rules associated with , which contains the following types of evolutional communication rules:(a), () or ();(b), () or ();(c), ;(d), ;(e), ;(f), () or ();(g), () or (). For each , , in the transition from configuration to configuration , by using any rules of types from (a) to (g), at least one object from is consumed and at most one object is produced in . Hence, in any transition step the number of objects in the system is not increased.
(2) By induction on , let us start analyzing the basic case . The result is trivial because of . By induction hypothesis, let us suppose the result holds for , . Then, Hence, the result is also true for .
(3) According to the fact that the application of a cell separation rule consumes an object and produces two new cells, result (3) can be obtained from (2) easily.

Next, a deterministic algorithm working in polynomial time is presented, which receives as input a P system from and an input multiset of , in such manner that algorithm reproduces the behaviour of a computation of . If the system is confluent, then the algorithm will provide the same answer of . We give Pseudocode 1 of the algorithm to describe the simulation process.

Pseudocode 1

The algorithm receives a recognizer tissue P systemfrom and an input multiset . Let . Let any computation of perform at most ( is a natural number) transition steps. Hence, from Lemma 4, the number of cells in the system along any computation is bounded by .

A transition of a recognizer tissue P system is performed in two phases: selection phase and execution phase (see Algorithms 1 and 2). For the detailed information of a transition of such P system, one can refer to [32].

Algorithm 1: Selection phase.
Algorithm 2: Execution phase.

It is easy to check that Algorithm 1 is deterministic and the running time of this algorithm is polynomial in the size of because the number of cycles of the first main loop for is of order ; the number of cycles of the second main loop for is of order ; and the number of cycles of the third main loop for is of order .

Algorithm 2 is deterministic and the running time of this algorithm is polynomial in the size of because the number of cycles of the first main loop for is of order ; the number of cycles of the second main loop for is of order ; and the number of cycles of the third main loop for is of order .

Theorem 5. One has .

Proof. Because is closed under polynomial time reduction and nonempty, hence . In what follows, we show that . Let and let be a family of recognizer tissue P systems from TSEC solving according to Definition 3. Let be a polynomial encoding associated with that solution. If is an instance of the problem , then will be processed by the system . We consider the deterministic algorithm as shown in Algorithm 3.
The algorithm receives an instance of the decision problem , working in a polynomial time. The following three assertions are equivalent:(i); that is, the answer of problem to instance is affirmative.(ii)Every computation of is an accepting computation.(iii)The output of the algorithm with input is Yes. Hence, .

Algorithm 3

Corollary 6. For each , , one has .

Proof. Indeed, it suffices to notice that, at any transition step, the application of each rule consumes at least one object from and produces at most one object from . Thus, along any computation of the system, the total number in it is not increased.

Theorem 7. For each , , one has .

Proof. In [31], it was shown that the class of decision problems solvable in polynomial time by means of families of tissue P systems with cell division and symport rules with length 1 is equal to class P. Bearing in mind that systems from are noncooperative ones, the dependency graph technique used in the cited paper can be used to obtain the result, in a similar way.

3.2. An Efficient Solution to the SAT Problem by P Systems in

The SAT problem is a well-known NP-complete problem; here we give an efficient solution to the SAT problem by a family of tissue P systems with evolutional communication rules of length at most .

Theorem 8. One has SAT  .

Proof. We provide a polynomial time uniform solution to the SAT problem [41] by a family of recognizer tissue P systems from . Each system () can process all Boolean formula in conjunctive normal form with variables and clauses.
For each , we consider the recognizer tissue P system whereand the set of rules consists of the following rules:We consider a propositional formula , which contains () clauses, , for , and , for , .
Let be a polynomial encoding of instances from SAT in , where and . So the propositional formula will be processed by the system .
An instance of the SAT problem is solved by the system , which can be separated into three phases: generation phase, checking phase, and output phase.
Generation Phase. In this phase, by using separation rules in cell with label 1, all truth assignments for the variables associated with the Boolean formula will be generated. When this phase completes, there are copies of cell with label 1 such that each of them encodes a different truth assignment of variables .
The generation phase takes steps, which has two parallel processes. On the one hand, loops are executed, and each loop takes four steps. When the loops are completed, three additional steps are executed. On the other hand, there is a counter object in cell 3 that evolves from to , and objects , () are produced in cell 3 after steps at this phase.
In the initial configuration, we have objects , in cell 1, objects , , , in cell 2, and objects , , , in cell 3.
In what follows, we first analyze the computation process that takes place in cells 1 and 2; then we explain the computation process that takes