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On the computational efficiency of tissue P systems with evolutional symport/antiport rules
... In theoretical studies, many kinds of TPs based on various biological motivations have been constructed and developed to expand the types of MC [9,10]. The analysis of these extended systems on computing power and computational efficiency is an essential point in theoretical works [11,12]. TPs with evolutional symport/antiport are presented to rewrite and modify objects in the communication process [13,14]. ...
... Specifically, mean squared error (MSE) [67], as the fitness function of comparative clustering approaches, is introduced in the comparison experiments, which is determined by (12) in the following: ...
... Therefore, all these comparison results validate the clustering efficiency of the proposed ECPSO-MTP. Specifically, mean squared error (MSE) [67], as the fitness function of comparative clustering approaches, is introduced in the comparison experiments, which is determined by (12) in the following: ...
In order to establish a highly efficient P system for resolving clustering problems and overcome the computation incompleteness and implementation difficulty of P systems, an attractive clustering membrane system, integrated with enhanced particle swarm optimization (PSO) based on environmental factors and crossover operators and a distributed parallel computing model of monodirectional tissue-like P systems (MTP), is constructed and proposed, which is simply named ECPSO-MTP. In the proposed ECPSO-MTP, two kinds of evolution rules for objects are defined and introduced to rewrite and modify the velocity of objects in different elementary membranes. The velocity updating model uses environmental factors based on partitioning information and randomly replaces global best to improve the clustering performance of ECPSO-MTP. The crossover operator for the position of objects is based on given objects and other objects with crossover probability and is accomplished through the hybridization of the global best of elementary membranes to reject randomness. The membrane structure of ECPSO-MTP is abstracted as a network structure, and the information exchange and resource sharing between different elementary membranes are accomplished by evolutional symport rules with promoters for objects of MTP, including forward and backward communication rules. The evolution and communication mechanisms in ECPSO-MTP are executed repeatedly through iteration. At last, comparison experiments, which are conducted on eight benchmark clustering datasets from artificial datasets and the UCI Machine Learning Repository and eight image segmentation datasets from BSDS500, demonstrate the effectiveness of the proposed ECPSO-MTP.
... In theoretical studies, many kinds of TPs based on various biological motivations have been constructed and developed to expand the types of MC [9,10]. The analysis of these extended systems on computing power and computational efficiency is an essential point in theoretical works [11,12]. TPs with evolutional symport/antiport are presented to rewrite and modify objects in the communication process [13,14]. ...
... Specifically, mean squared error (MSE) [67], as the fitness function of comparative clustering approaches, is introduced in the comparison experiments, which is determined by (12) in the following: ...
Citation: Wang, L.; Liu, X.; Qu, J.; Zhao, Y.; Gao, L.; Ren, Q. An Extended Membrane System with Monodirectional Tissue-like P Systems and Enhanced Particle Swarm Optimization for Data Clustering. Appl. Sci. 2023, 13, 7755.
Nature-inspired computing is a type of human-designed computing motivated by nature, which is based on the employ of paradigms, mechanisms, and principles underlying natural systems. In this article, a versatile and vigorous bio-inspired branch of natural computing, named membrane computing is discussed. This computing paradigm is aroused by the internal membrane function and the structure of biological cells. We first introduce some basic concepts and formalisms of membrane computing, and then some basic types or variants of P systems (also named membrane systems ) are presented. The state-of-the-art computability theory and a pioneering computational complexity theory are presented with P system frameworks and numerous solutions to hard computational problems (especially NP -complete problems) via P systems with membrane division are reported. Finally, a number of applications and open problems of P systems are briefly described.
Tissue
$P$
systems with promoters provide nondeterministic parallel bioinspired devices that evolve by the interchange of objects between regions, determined by the existence of some special objects called
promoters
. However, in cellular biology, the movement of molecules across a membrane is transported from high to low concentration. Inspired by this biological fact, in this article, an interesting type of tissue
$P$
systems, called monodirectional
tissue
$P$
systems with promoters
, where communication happens between two regions only in one direction, is considered. Results show that finite sets of numbers are produced by such
$P$
systems with one cell, using any length of symport rules or with any number of cells, using a maximal length 1 of symport rules, and working in the maximally parallel mode. Monodirectional tissue
$P$
systems are Turing universal with two cells, a maximal length 2, and at most one promoter for each symport rule, and working in the maximally parallel mode or with three cells, a maximal length 1, and at most one promoter for each symport rule, and working in the flat maximally parallel mode. We also prove that monodirectional tissue
$P$
systems with two cells, a maximal length 1, and at most one promoter for each symport rule (under certain restrictive conditions) working in the flat maximally parallel mode characterizes regular sets of natural numbers. Besides, the computational efficiency of monodirectional tissue
$P$
systems with promoters is analyzed when cell division rules are incorporated. Different uniform solutions to the Boolean satisfiability problem (SAT problem) are provided. These results show that with the restrictive condition of “monodirectionality,” monodirectional tissue
$P$
systems with promoters are still computationally powerful. With the powerful computational power, developing membrane algorithms for monodirectional tissue
$P$
systems with promoters is potentially exploitable.
In the field of membrane computing, a great attention is traditionally paid to the results demonstrating a theoretical possibility to solve NP-complete problems in polynomial time by means of various models of P systems. A bit less common is the fact that almost all models of P systems with this capability are actually stronger: some of them are able to solve PSPACE-complete problems in polynomial time, while others have been recently shown to characterize the class \(\mathbf {P^{\#P}}\) (which is conjectured to be strictly included in PSPACE). A large part of these results has appeared quite recently. In this survey, we focus on strong models of membrane systems which have the potential to solve hard problems belonging to classes containing NP. These include P systems with active membranes, P systems with proteins on membranes and tissue P systems, as well as P systems with symport/antiport and membrane division and, finally, spiking neural P systems. We provide a survey of computational complexity results of these membrane models, pointing out some features providing them with their computational strength. We also mention a sequence of open problems related to these topics.
In P systems with active membranes, the question of understanding the power of non-confluence within a polynomial time bound is still an open problem. It is known that, for shallow P systems, that is, with only one level of nesting, non-confluence allows them to solve conjecturally harder problems than confluent P systems, thus reaching \(\mathbf{PSPACE }\). Here we show that \(\mathbf{PSPACE }\) is not only a bound, but actually an exact characterization. Therefore, the power endowed by non-confluence to shallow P systems is equal to the power gained by confluent P systems when non-elementary membrane division and polynomial depth are allowed, thus suggesting a connection between the roles of non-confluence and nesting depth.
We investigate the influence that the flow of information in membrane systems has on their computational complexity. In particular, we analyse the behaviour of P systems with active membranes where communication only happens from a membrane towards its parent, and never in the opposite direction. We prove that these “monodirectional P systems” are, when working in polynomial time and under standard complexity-theoretic assumptions, much less powerful than unrestricted ones: indeed, they characterise classes of problems defined by polynomial-time Turing machines with \({\mathbf{NP}}\) oracles, rather than the whole class \({\mathbf{PSPACE}}\) of problems solvable in polynomial space.
In spite of the fact that many ways of using the evolution rules in a P system were already investigated, there is still a case, which we call the flat maximal parallelism, which appeared in several papers, but which deserves a more careful attention: in each step, in each membrane, a maximal set of applicable rules is chosen and each rule in the set is applied exactly once. In this work, flat maximal parallelism is studied for non-cooperating P systems with promoters. Specifically, we prove that non-cooperating P systems with at most one promoter associated with any rule, working in the flat maximally parallel way, are Turing universal (the Turing universality of such P systems is open if they work in the maximally parallel way). Moreover, a uniform solution to the SAT problem is provided by using non-cooperating P systems with promoters and membrane division, working in the flat maximal parallel way.
For many models of P systems and tissue P systems, the main behavior of a specific system can be simulated by a corresponding system with only one membrane or cell, respectively; this effective construction is called flattening. In this paper we describe the main procedure of flattening for specific variants of static (tissue) P systems as well as for classes of dynamic (tissue) P systems with a bounded number of possible membrane structures or a bounded number of cells during any computation.
In the framework of recognizer cell–like membrane systems it is well known that the construction of exponential number of objects in polynomial time is not enough to efficiently solve NP–complete problems. Nonetheless, it may be sufficient to create an exponential number of membranes in polynomial time.
In this paper, we study the computational efficiency of recognizer tissue P systems with communication (symport/antiport) rules and division rules. Some results have been already obtained in this direction: (a) using communication rules and making no use of division rules, only tractable problems can be efficiently solved; (b) using communication rules with length three and division rules, NP–complete problems can be efficiently solved. In this paper, we show that the length of communication rules plays a relevant role from the efficiency point of view for this kind of P systems.
The “classical” model of P systems was introduced by Gheorghe Păun in 1998; this model with symbol objects was shown by Gh. Păun [J. Comput. Syst. Sci. 61, 108-143 (2000; Zbl 0956.68055)] to be computationally universal provided that catalysts and priorities of rules are used. We now show by reduction via register machines that the priorities may be omitted from the model without loss of computational power. As a consequence, several universality results for P systems in Gh. Păun’s book [Membrane computing: An introduction. Berlin: Springer (2002; Zbl 1034.68037)] are improved.
We introduce a new variant of membrane systems where the rules are directly assigned to membranes and, moreover, every membrane carries an energy value that can be changed during a computation by objects passing through the membrane. The result of a successful computation is considered to be the distribution of energy values carried b y the membranes. We show that for sys- tems working in the sequential mode with a kind of priority relation on the rules we already obtain universal computational power. When omitting the priority relation, we obtain a characterization of the family of Parikh sets of languages generated by context-free matrix grammars. On the other hand, when using the maximally parallel mode, we do not need a priority relation to obtain com- putational completeness. Finally, we introduce the corresponding model of tissue P systems with energy assigned to the membrane of each cell and objects moving from one cell to another one in the environment as well as being able to change the energy of a cell when entering or leaving the
Recognizer P systems with active membranes have proven to be very efficient computing devices, being able to solve NP-complete decision problems in a polynomial time. However such solutions usually exploit many powerful features, such as electrical charges (polarizations) associated to membranes, evolution rules, communication rules, and strong or weak forms of division rules. In this paper we contribute to the study of the computational power of polarizationless recognizer P systems with active membranes. Precisely, we show that such systems are able to solve in polynomial time the NP-complete decision problem 3-SAT by using only dissolution rules and a form of strong division for non-elementary membranes, working in the maximallly parallel way.
Monodirectional tissue P systems with promoters are natural inspired parallel computing paradigms, where only symport rules are permitted, and with the restriction of “monodirectionality”, objects for two given regions are transferred in one direction. In this article, a novel kind of P systems, monodirectional evolutional symport tissue P systems with promoters (MESTP P systems) is raised, where objects may be revised during the movement between two regions. The computational theory of MESTP P systems that rules are employed in a flat maximally parallel pattern is investigated. We prove that finite natural number sets are created by MESTP P systems applying one cell, at most 1 promoter and all evolutional symport rules having a maximal length 2 or with arbitrary number of cells, promoters and all evolutional symport rules having a maximal length 2. MESTP P systems are Turing universal when two cells, at most 1 promoter and all evolutional symport rules having a maximal length 2 are employed. In addition, with the help of cell division mechanism, monodirectional evolutional symport tissue P systems with promoters and cell division (MESTPD P systems) are employed to solve
NP
-complete (the
SAT
) problem, where system uses at most 1 promoter and all evolutional symport rules having a maximal length 3. These results show that MESTP(D) P systems are still computationally powerful even if monodirectionality control mechanism is imposed, thereby developing membrane algorithms for MESTP(D) P systems is theoretically possible as well as potentially exploitable.
We analyse the computational efficiency of tissue P systems, a biologically-inspired computing device modelling the communication between cells. In particular, we focus on tissue P systems with fission rules (cell division and/or cell separation), where the number of cells can increase exponentially during the computation. We prove that the complexity class characterised by these devices in polynomial time is exactly , the class of problems solved by polynomial-time Turing machines with oracles for counting problems.
We prove that non-confluent (i.e., strongly nondeterministic) P systems with active membranes working in polynomial time are able to simulate polynomial-space nondeterministic Turing machines, and thus to solve all \({\mathbf{PSPACE }}\) problems. Unlike the confluent case, this result holds for shallow P systems. In particular, depth 1 (i.e., only one membrane nesting level and using elementary membrane division only) already suffices, and neither dissolution nor send-in communication rules are needed.
Tissue P systems with symport/antiport rules are a class of distributed parallel computing models inspired by the cell intercommunication in tissues, where objects are never modified in the process of communication, just changing their place within the system. In this work, a variant of tissue P systems, called tissue P systems with evolutional symport/antiport rules is introduced, where objects are moved from one region to another region and may be evolved during this process. The computational power of such P systems is studied. Specifically, it is proved that such P systems with one cell and using evolutional symport rules of length at most 3 or using evolutional antiport rules of length at most 4 are Turing universal (only the family of all finite sets of positive integers can be generated by such P systems if standard symport/antiport rules are used). Moreover, cell division rules are considered in tissue P systems with evolutional symport/antiport rules, and a limit on the efficiency of such P systems is provided with evolutional communication rules of length at most 2. The computational efficiency of this kind of models is shown when using evolutional communication rules of length at most 4.
This paper represents a significant advance on the issue of testing for implementations specified by P systems with transformation and communicating rules. Using the X-machine framework and the concept of cover automaton, it devises a testing approach for such systems, that, under well defined conditions, it ensures that the implementation conforms to the specification. It also investigates the issue of identifiability for P systems, that is an essential prerequisite for testing implementations based on such specifications and establishes a fundamental set of properties for identifiable P systems.
The decision problems solved in polynomial time by P systems with elementary active membranes are known to include the class \(\mathbf{P}^{\# \mathbf{P}}\). This consists of all the problems solved by polynomial-time deterministic Turing machines with polynomial-time counting oracles. In this paper we prove the reverse inclusion by simulating P systems with this kind of machines: this proves that the two complexity classes coincide, finally solving an open problem by Păun on the power of elementary division. The equivalence holds for both uniform and semi-uniform families of P systems, with or without membrane dissolution rules. Furthermore, the inclusion in \(\mathbf{P}^{\# \mathbf{P}}\) also holds for the P systems involved in the P conjecture (with elementary division and dissolution but no charges), which improves the previously known upper bound \(\mathbf{PSPACE}\).
With mathematical motivation, we consider a combination of basic features of multiset-rewriting P systems and of spiking neural P systems, that is, we consider cell-like P systems with spiking rules in their membranes (hence dealing with only one kind of objects, the spikes). The universality of these systems as number generating devices is proved for the two usual ways to define the output (internally or externally) and for various restrictions on the spiking rules. Several research topics are also pointed out.
Adaptive infinite impulse response (IIR) filter has been preferably used in modeling real-world systems because of its reduced number of coefficients and better performance over finite impulse response (FIR) filter. However, it is still a challenging problem how to design an optimal IIR filter due to its nonlinear and multimodal error surface. This paper introduces membrane computing to design an optimal IIR filter and proposes a novel design method that employs a tissue-like membrane system with ring-shaped topology structure. A modification of the different evolution mechanism was developed as evolution rules for objects according to the special membrane structure. Under the control of the object's evolution-communication mechanism, the tissue-like membrane system can effectively find the global minima for an IIR filter design problem. The proposed method was evaluated over several benchmark IIR systems and was compared with several state-of-the-art evolutionary algorithms. The experimental results show that the proposed method has better identification performance compared with other algorithms.
We continue the investigation of the power of the computability models introduced in [12] under the name of transition super-cell systems. We compare these systems with classic mechanisms in formal language theory, context-free and matrix grammars, E0 and ET0 systems, interpreted as generating mechanisms of number relations (we take the Parikh image of the usual language generated by these mechanisms rather than the language). Several open problems are also formulated.
Membrane computing, known as P systems or membrane systems, is a novel class of distributed and parallel computing models. This paper proposes a membrane clustering algorithm using hybrid evolutionary mechanisms to address data clustering problem. It uses a tissue P system consisting of three cells to find the optimal cluster centers for a data set to be clustered. Its object is used to express candidate cluster centers, and the three cells use three different evolutionary mechanisms: genetic operators, velocity-position model and differential evolution mechanism. Particularly, the velocity-position model and differential evolution mechanism used in the process are the improved versions proposed in this paper according to the special membrane structure and communication mechanism. The hybrid evolutionary mechanisms can enhance the diversity of objects in the system and improve the convergence performance. Under the control of the hybrid evolutionary mechanisms and communication mechanism, the membrane clustering algorithm can determine a good partition for a data set. The proposed membrane clustering algorithm is evaluated on three artificial data sets and five real-life data sets and compared with k-means and several evolutionary clustering algorithms. Statistical significance tests have been performed to establish the superiority of the proposed membrane clustering algorithm. © Copyright 2015, Institute of Software, the Chinese Academy of Sciences. All right reserved.
Polynomial-time P systems with active membranes characterise PSPACE by exploiting membranes nested to a polynomial depth, which may be subject to membrane division rules. When only elementary (leaf) membrane division rules are allowed, the computing power decreases to P-PP = P-#P, the class of problems solvable in polynomial time by deterministic Turing machines equipped with oracles for counting (or majority) problems. In this paper we investigate a variant of intermediate power, limiting membrane nesting (hence membrane division) to constant depth, and we prove that the resulting P systems can solve all problems in the counting hierarchy CH, which is located between P-PP and PSPACE. In particular, for each integer k >= 0 we provide a lower bound to the computing power of P systems of depth k.
The paper focuses on the relation between biological and computational information processing. Several simple biological operations, as the exchange of molecules via cellular membrane or the cellular growth and separation, are abstracted into a mathematical model called membrane system (P system). This paper studies tissue P systems where a fixed interaction graph defines the communication between various types of cells. Polynomially uniform families of tissue P systems with the operation of cell separation were recently studied. Their computational power in polynomial time was shown to range between P and NP boolean OR co - NP, characterizing borderlines between tractability and intractability by the length of rules controlling the interchange of objects. Here, we show that the computational power of these uniform families is limited by the class PSPACE, which is the class characterizing the power of classical parallel computing models, such as PRAM or the alternating Turing machine.
We introduce a weak uniformity condition for families of P systems, DLOGTIME uniformity, inspired by Boolean circuit complexity. We then prove that DLOGTIME-uniform families of P systems with active membranes working in logarithmic space (not counting their input) can simulate logarithmic-space deterministic Turing machines.
P systems are parallel molecular computing models based on processing multisets of objects in cell-like membrane structures. Various variants were already shown to be computationally universal, equal in power to Turing machines. In this paper one proposes a class of P systems whose membranes are the main active components, in the sense that they directly mediate the evolution and the communication of objects. Moreover, the membranes can multiply themselves by division. We prove that this variant is not only computationally universal, but also able to solve NP-complete problems in polynomial (actually, linear) time. We exemplify this assertion with the well-known SAT problem.
Spiking neural P systems (SN P systems) are a new class of computing models inspired by the neurophysiological behavior of biological spiking neurons. In order to make SN P systems capable of representing and processing fuzzy and uncertain knowledge, we propose a new class of spiking neural P systems in this paper called weighted fuzzy spiking neural P systems (WFSNP systems). New elements, including fuzzy truth value, certain factor, weighted fuzzy logic, output weight, threshold, new firing rule, and two types of neurons, are added to the original definition of SN P systems. This allows WFSN P systems to adequately characterize the features of weighted fuzzy production rules in a fuzzy rulebased system. Furthermore, a weighted fuzzy backward reasoning algorithm, based onWFSN P systems, is developed, which can accomplish dynamic fuzzy reasoning of a rule-based system more flexibly and intelligently. In addition, we compare the proposed WFSN P systems with other knowledge representation methods, such as fuzzy production rule, conceptual graph, and Petri nets, to demonstrate the features and advantages of the proposed techniques.
We prove that arbitrary single-tape Turing machines can be simulated by uniform families of P systems with active membranes with a cubic slowdown and quadratic space overhead. This result is the culmination of a series of previous partial results, finally establishing the equivalence (up to a polynomial) of many space complexity classes defined in terms of P systems and Turing machines. The equivalence we obtained also allows a number of classic computational complexity theorems, such as Savitchʼs theorem and the space hierarchy theorem, to be directly translated into statements about membrane systems.
We study the P
versus
NP problem through membrane systems. Language accepting P systems are introduced as a framework allowing us to obtain a characterization
of the P ≠ NP relation by the polynomial time unsolvability of an NP–complete problem by means of a P system.
Membrane systems, also called P systems, are biologically inspired theoretical models of distributed and parallel computing. This paper presents a new class of tissue-like P systems with cell separation, a feature which allows the generation of new workspace. We study the efficiency of the class of P systems and draw a conclusion that only tractable problems can be efficiently solved by using cell separation and communication rules with the length of at most 1. We further present an efficient (uniform) solution to SAT by using cell separation and communication rules with length at most 6. We conclude that a borderline between efficiency and non-efficiency exists in terms of the length of communication rules (assuming ). We discuss future research topics and open problems.
We consider tissue-like P systems with states associated with the links (we call them synapses) between cells, controlling the passage of objects across the links. We investigate the computing power of such devices for the case of using—in a sequential manner—antiport rules of small weights. Systems with two cells are proved to be universal when having arbitrarily many states and minimal antiport rules, or one state and antiport rules of weight two. Also the systems with arbitrarily many cells, three states, and minimal antiport rules are universal. In contrast, the systems with one cell and any number of states and rules of any weight only compute Parikh sets of matrix languages (generated by matrix grammars without appearance checking); characterizations of Parikh images of matrix languages are obtained for such one-cell systems with antiport rules of a reduced weight.
In this paper we present a theory of computational complexity in the framework of membrane computing. Polynomial complexity
classes in recognizer membrane systems and capturing the classical deterministic and non-deterministic modes of computation, are introduced. In this context, a characterization of the relation
P =
NP is described.
Starting from the way the inter-cellular communication takes place by means of protein channels and also from the standard
knowledge about neuron functioning, we propose a computing model called a tissue P system, which processes symbols in a multiset rewriting sense, in a net of cells similar to a neural net. Each cell has a finite
state memory, processes multisets of symbol-impulses, and can send impulses (“excitations”) to the neighboring cells. Such
cell nets are shown to be rather powerful: they can simulate a Turing machine even when using a small number of cells, each
of them having a small number of states. Moreover, in the case when each cell works in the maximal manner and it can excite
all the cells to which it can send impulses, then one can easily solve the Hamiltonian Path Problem in linear time. A new
characterization of the Parikh images of ET0L languages are also obtained in this framework.
A recently introduced variant of P-systems considers membranes which can multiply by division. These systems use two types of division: division for elementary membranes (i.e. membranes not containing other membranes inside) and division for non-elementary membranes. In two recent papers it is shown how to solve the satisfiability problem and the Hamiltonian path problem (two well known NP-complete problems) in linear time with respect to the input length, using both types of division. We show in this paper that P-systems with only division for elementary membranes suffice to solve these two problems in linear time. Is it possible to solve NP complete problems in polynomial time using P-systems without membrane division? We show, moreover, that (if P=NP) deterministic P-systems without membrane division are not able to solve NP-complete problems in polynomial time.
In the attempt to have a framework where the computation is done by communication only, we consider the biological phenomenon of trans-membrane transport of couples of chemicals (one say symport when two chemicals pass together through a membrane, in the same direction, and antiport when two chemicals pass simultaneously through a mem- brane, in opposite directions). Surprisingly enough, membrane systems without changing (evolving) the used objects and with the communica- tion based on rules of this type are computationally complete, and this result is achieved even for pairs of communicated objects (as encountered in biology). Five membranes are used; the number of membranes is re- duced to two if more than two chemicals may collaborate when passing through membranes.
Current P systems which solve NP–complete numerical problems represent the instances of the problems in unary notation. However, in classical complexity theory, based upon Turing machines, switching from binary to unary encoded instances generally corresponds to simplify the problem. In this paper we show that, when working with P systems, we can assume without loss of generality that instances are expressed in binary notation. More precisely, we propose a simple method to encode binary numbers using multisets, and a family of P systems which transforms such multisets into the usual unary notation. Such a family could thus be composed with the unary P systems currently proposed in the literature to obtain (uniform) families of P systems which solve NP–complete numerical problems with instances encoded in binary notation.
We introduce also a framework which can be used to design uniform families of P systems which solve NP–complete problems (both numerical and non-numerical) working directly on binary encoded instances, i.e., without first transforming them to unary notation. We illustrate our framework by designing a family of P systems which solves the 3-SAT problem. Next, we discuss the modifications needed to obtain a family of P systems which solves the PARTITION numerical problem.
We prove that recognizer P systems with active membranes using polynomial space characterize the complexity class PSPACE. This result holds for both confluent and nonconfluent systems, and independently of the use of membrane division rules.
We describe a solution to the SAT problem via non-confluent P systems with active membranes, without using membrane division rules. Furthermore, we provide an algorithm for simulating such devices on a nondeterministic Turing machine with a polynomial slowdown. Together, these results prove that the complexity class of problems solvable non-confluently and in polynomial time by this kind of P system is exactly the class NP.
We consider recognizer P systems having three polarizations associated to the membranes, and we show that they are able to
solve the PSPACE-complete problem Quantified 3SAT when working in polynomial space and exponential time. The solution is uniform (all the instances of a fixed size are solved
by the same P system) and uses only communication rules: evolution rules, as well as membrane division and dissolution rules,
are not used. Our result shows that, as it happens with Turing machines, this model of P systems can solve in exponential
time and polynomial space problems that cannot be solved in polynomial time, unless P = SPACE.
On three variants of P systems with string-objects
- Ferretti