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
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 [
Inspired by the biological phenomenon of trans-membrane transport of couples of chemicals, communication P systems with symport/antiport rules were proposed in [
Since the notion of tissue P systems was proposed, numerous research topics have been arisen [
Tissue P systems have been used to find polynomial time solutions to
Computational complexity theory in the framework of tissue P systems was introduced in [
Tissue-like P systems with evolutional symport/antiport rules (TESA P systems, for short) were proposed in [
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
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 variant of tissue P systems with evolutional symport/antiport rules, called 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 A new simulator
Let us start this section by recalling some notions from formal language theory used in this work; the reader can find details in [
An alphabet
A
A tissue P system with evolutional symport/antiport rules and cell separation of degree evolutional communication rules: separation rules:
A tissue P system with evolutional symport/antiport rules and cell separation of degree
A
An evolutional symport rule
A separation rule
The rules of a TESAS P system are applied in a maximally parallel manner, and we have the restriction that when a cell
A transition from a configuration
A natural framework to solve decision problems is to use recognizer P systems; one can refer to [
A recognizer tissue P system with evolutional symport/antiport rules and cell separation of degree the tuple for each multiset
It is worth pointing out that, in any recognizer TESAS P systems, all computations halt. Then, any symport rule of the type
For each ordered pair of natural numbers
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 [
A decision problem the family there exists a polynomial encoding
The set of all decision problems that can be solved by recognizer TESAS P systems with evolutional communication rules of length at most
In this subsection, we use tissue P systems with cell separation and evolutional communication rules of length at most
The proof uses a similar technique as in [ We denote the label of a cell as a pair If a separation rule is applied to a cell with label
Note that if communication rules occur in two cells, then the labels of these two cells do not change.
A configuration at an instant
Let
Let
Let
Let
Let
Let
Let
Let
Let
If
Next, we show that TESAS P systems from
If
Let for each the number of created cells along the computation
(1) Let us notice that
For each
(2) By induction on
(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
The algorithm
A transition of a recognizer tissue P system
It is easy to check that Algorithm
Algorithm
One has
Because
The algorithm Every computation of The output of the algorithm
Hence,
For each
Indeed, it suffices to notice that, at any transition step, the application of each rule consumes at least one object from
For each
In [
The
One has
We provide a polynomial time uniform solution to the
For each
We consider a propositional formula
Let
An instance of the
The generation phase takes
In the initial configuration, we have objects
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 place in cell 3.
At the first step of the
Note that at the first step of the 1st loop involving cells 1 and 2, only rules
At the second step of the
Note that at the second step of the 1st loop involving cells 1 and 2, only rules
At the third step of the
Note that at the third step of the 1st loop involving cells 1 and 2, only rules
At the fourth step of the
Note that at the fourth step of the 1st loop involving cells 1 and 2, only rules
At step
At step
At step
At the first
At the first step of the
At the second step of the
At the third step of the
If the input formula
If the input formula
From the computation process, we can check that rules of a system size of the alphabet: initial number of cells: initial number of objects: number of rules: maximum length of a rule (the total number of objects involved in a rule):
Hence, there exists a deterministic Turing machine that builds the system
The tissue P system
According to Definition
One has
The theorem holds because the
For each natural number
It suffices to notice that for each natural number
Figure
Summary of the results and open problems.
The solution to the
As described in Algorithm
New model type, @model<tsec> Communication rules with two cells in the right hand of the rule, as:
Input tables to introduce the propositional formulas. The data introduced in these tables are then converted into parameters for the model that can be used to generate the corresponding Use of existing SAT plugin to provide end users with a more natural way of introducing the formulas that are then converted to the format of the previous tables. Definition of the output showing in different formats if the formula is satisfiable or not, generated from the objects of the computation.
The computational complexity of tissue P systems with cell separation was first investigated in [
In this work, membrane fission as a mechanism to generate an exponential workspace (expressed in terms of number of membranes and number of objects) has been considered instead of membrane division, in the framework of tissue P systems with evolutional symport/antiport rules. The computational efficiency of this kind of tissue P systems has been investigated. In this context, the main contributions of the paper are the following: (a) only problems in class
As future works, we propose the following: In Section In the framework of tissue P systems, the environment is a singular region since the objects initially placed in it have an arbitrary large number of copies. Tissue P systems with cell separation and without environment (the alphabet of the environment is empty) were considered in [ Besides much investigated maximal parallelism, several ways of using rules were also considered in membrane computing, such as flat maximal parallelism [ In [
The solution to the
In what follows, we will develop a new simulator MeCoSim to check the correctness of the solution to the
Taking the existing P-Lingua syntax for P systems introduced in [
The P-Lingua syntax for these classical types of communication rules was the following: [ T / [ E /
Thus, the syntax tried to mimic the theoretical description of the computing model. This kind of rules might also be present in P systems in
@model<tsec>
The rest of the file will then define the main elements describing the P system, typically consisting of
These elements are described in detail in the following subsections. The sets of rules will include the new elements introduced by this work, and other subsections, without significant changes with respect to other existing models, are included here to make the work self-contained.
@mu = [ [ ]’1 [ ]’2 [ ]’3 ]’0;
As it can be seen, an external structure labelled by
@ms1 = a, b, c; / @ms2 = d, e, f; /
The alphabet of the environment
It would be written in P-Lingua as @ms(0) = E @ms(0) += al /
Note that while a simple set can be assigned by the symbol
No explicit definition of
@ms(1) = A
The grammar designed in P-Lingua for
Note that the suffix
Let us illustrate this with the solution for the
MeCoSim input table.
In the given solution, the input cell is 3, so the input multiset def define_input ( @ms(3) += xb x
As it can be seen, the assignment of objects to the input cell iterates over each pair
MeCoSim SAT plugin.
Regarding the output cell, again there is no explicit definition in P-Lingua, but the whole system will follow a computation; when a halting configuration is reached, one can interpret the output in many different ways. However, another mechanism in MeCoSim allows the customization of outputs to focus on the regions, objects, or results expected by the user. Thus, for SAT problem, a P system designer can pay attention to every object appearing inside each cell or in the environment, as shown in Figure
MeCoSim output for P systems designer.
MeCoSim output for end user.
In the proof of Theorem Initialization For each computation step, while some rules are applicable: Selection of rules Execution of rules
The
The
As a result of this selection phase, a set of rules will have been selected, verifying that every membrane with applicable rules has selected exactly one.
Then, the
First of all, these tools include parsers to detect possible errors in P-Lingua files that could be translation errors or due to inaccuracies from the solution, maybe not meeting some features required by the computing model. Throughout the debugging process, we are informed about the rules that are being generated for the system, so that we can check that the expected sets of rules are actually available for the computation of the system, as shown in Figure
Debugging process.
In addition, parsing tab will show at the end the initial configuration of the system, thus allowing the P systems designer to contrast if the expected multisets were produced.
Once the solution has been proved correct after the debugging process, we can be interested in checking that the system evolves according to our manual traces; to do so, we can follow the computation step by step, informing about the rules applied for each step and the objects contained inside each region of every configuration, as shown in Figure
Step by step simulation process.
In addition, several visual aids are available to ease the checking of information concerning alphabets, structure, or multisets inside each region. These aids can be visualized in any moment of the computation. An example of this kind of viewers is given in Figure
An example of simulation information in a computation.
Additionally, along with the checking of the specific solution, the development of this type of tools shows itself as a good way of revisiting the general algorithms proposed for the model, paying attention to subtle details, given the need of reproducing every detail specified in the formal definition of the model, so these development tasks emerge as enriching complementary tasks for the robustness of the computing models and their corresponding theoretical definitions.
The authors declare that there are no conflicts of interest regarding the publication of this article.
The work of Linqiang Pan and Bosheng Song was supported by National Natural Science Foundation of China (61602192, 61772214, 61320106005, and 61033003), China Postdoctoral Science Foundation (2016M600592, 2017T100554), and the Innovation Scientists and Technicians Troop Construction Projects of Henan Province (154200510012).