The computational power of monodirectional tissue P systems with symport rules

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Abstract

Tissue P systems with symport/antiport rules are parallel bio-inspired devices that evolve by communicating objects between two regions in both directions. In this paper, a type of tissue P systems, called monodirectional tissue P systems with symport rules (MTS P systems) is considered, where communication happens between two regions only in one direction. Results show that MTS P systems are Turing universal with three cells and using a maximal length 2 of symport rules working in a maximally parallel way. Besides, the Turing universality of MTS P systems is also achieved by using two cells, and a maximal length 1 of symport rules working in a synchronized way by imposing a flat maximal parallelism among rules in the set. These results show that with the restrictive condition of “monodirectionality”, MTS P systems are still computationally powerful. Therefore, developing membrane algorithms for MTS P systems is potentially exploitable.

Introduction

Membrane computing is a nature inspired computational paradigm introduced in 1998 by Gheorghe Păun, which studies computing devices inspired by the functioning and structure of biological cells [39], [41], [50]. All the computing models investigated in the framework of membrane computing are called P systems [37]. Since the models of P systems proposed in [37], many variants of P systems have been proposed and investigated from the aspect of computer science [1], [21], mathematical [4], [26], [31], [49] and biological motivations [9], [13], [22]. Except for the theoretical results obtained in membrane computing, P systems have also been used to solve various real-world problems [12], [20], [61], such as ecology and system biology [22], [23], image and signal processing [14], [43], [60], machine learning [44], [62].

An essential and important component of P systems is a membrane structure, which has two main categories: a hierarchical (cell-like) arrangement of membranes [37] and a net of membranes (tissue-like) [29] or neurons (neural-like) [25], [45], [42], [59]. In this work, we focus on tissue-like P systems with symport/antiport rules [3], [5], [35], [36], where a multiset of objects moved in one direction between two regions is called a symport rule; while an antiport rule means that two multisets of objects are moved in two opposite directions between two regions.

Many variants of tissue P systems have been proposed, and the computational power of such kinds of systems has also been investigated, which turned out to be Turing universal, for instance, tissue P systems with channel states: a channel associated with a state between two regions is used to control communication [18]; tissue P systems with promoters: any numbers of promoters may be associated with a rule, and the application of rules is regulated by promoters [33]; tissue P systems with cell division: cell division rules are introduced into tissue P systems [40]. If cell division is introduced into tissue P systems, then many NP-complete problems can be solved in polynomial time, such as the subset sum problem [15], [46], [53], the 3-Coloring problem [16], the vertex cover problem [17], the SAT problem [11], [34], [40], [56].

Inspired from the biological fact that substances are transported from the higher concentration side to the lower concentration side, in [8], unidirectional evolution-communication P systems were proposed, it was proved that such P systems using evolution rules and communication rules are Turing universal [8]. Besides, in [27], a close variant of P systems, called monodirectional P systems with active membranes was proposed, where communication happens between two regions only in one direction (readers can refer [7], [38] for more details). The computational complexity of monodirectional P systems with active membranes has been investigated in [27], where it is shown that such P systems working in polynomial time and under standard complexity-theoretic assumptions characterize classes of problems defined by polynomial-time Turing machines with NP oracles. So, a current line of research is to investigate the computational power of P systems with the restrictive condition of “monodirectionality”. In [54], [55], monodirectional tissue P systems with promoters or with channel states were investigated, and the computational power of these kinds of P systems was studied.

The motivation of this work aims at building a bridge between tissue P systems and the potential application prospects that involve information processing, thereby extending tissue P systems to serve as a class of suitable and attractive models for these applications. Motivated by the monodirectional nature in cellular biology, a novel type of tissue P systems, called monodirectional tissue P systems with symport rules (MTS P systems), is introduced, where for two given regions, communication happens only in one direction, and never in the opposite direction, hence antiport rules are forbidden to use in such systems.

Actually, many applications require a monodirectional mechanism, but it was not explicitly and systematically investigated separately, such as information acquisition device for power systems [28], [58]. Consequently, MTS P systems are much more attractive to deal with some real-world problems which involve monodirectional mechanism.

The computational power of MTS P systems is investigated. Results show that finite sets of vectors of non-negative integers are produced by MTS P systems with one cell and using any length of symport rules or with any number of cells and using a maximal length 1 of symport rules. Besides, we prove that MTS P systems are Turing universal with three cells and using a maximal length 2 of symport rules, and working in a maximally parallel way. The Turing universality of MTS P systems is also achieved by using two cells, and a maximal length 1 of symport rules working in a synchronized way by imposing a flat maximal parallelism among rules in the set; in such working mode, MTS P systems with two cells and using a maximal length 1 of symport rules (under certain restrictive conditions) can compute at least all Parikh sets of matrix languages.

Contributions of the present work are summarized as follows.

  • (a)

    A novel type of tissue P systems, called monodirectional tissue P systems with symport rules (MTS P systems), is developed by introducing the notion of monodirectionality into tissue P systems. More precisely, MTS P systems have a network architecture with the capability of complex topology representation, and are proved to be Turing universal. These results manifest that a Turing universal monodirectional paradigm of MTS P systems is theoretically possible and potentially applicable.

  • (b)

    A new strategy of using rules called synchronous by imposing a flat maximal parallelism among rules in the set, is proposed, which provides a useful tool in proving various properties. The new strategy of using rules broaden current research topic in membrane computing, which helps to find more realistic P systems from a biological point of view.

  • (c)

    A monodirectionality control strategy is introduced into tissue P systems to control the application of communication among cells, thus making MTS P systems be more suitable for some applications which require a monodirectional mechanism.

  • (d)

    By employing network architecture as model structure and monodirectionality as information processing, MTS P systems are attractive to some real-world problems which involve monodirectional nature and require networking model precisely.

The remainder of this paper is structured as follows. In section 2, some fundamental concepts from formal language are introduced. In section 3, the details of MTS P systems are described. Two examples are given in section 4. In section 5, the results about the computational power of MTS P systems are demonstrated. Finally, conclusions and future work are presented in section 6.

Section snippets

Preliminaries

In this section, some fundamental concepts from formal language and automata theory are recalled. For further information about this aspect, one can refer to [48].

We define an alphabet (denoted by Γ) to be any nonempty finite set of abstract symbols. The set of strings obtained by concatenating any number of symbols is denoted by Γ. Γ+=Γ{λ} is the set that excludes the empty string (if a string does not have any symbols at all, it is called empty string, denoted by λ). The number of

Model description

We give the definition of monodirectional tissue P systems with symport rules.

Definition 2

A monodirectional tissue P system with symport rules (MTS P system) of degree q1 is a tupleΠ=(Γ,T,E,M1,,Mq,R,iout), where

  • Γ is an alphabet, and every element in such alphabet is named an object;

  • TΓ is an alphabet of terminal objects;

  • E is an alphabet of objects initially placed in the environment such that EΓ;

  • Mi, 1iq, are multisets of objects initially placed in the q cells;

  • R is a finite set of symport rules with

Two examples

In this section, two examples are given, Example 1 illustrates the difference between MTS P systems and tissue P systems using only symport rules (rules in these two kinds of P systems are used in a maximally parallel mode). Example 2 is used to illustrate how an MTS P system works in two different ways of applying rules.

Example 1

Let Π1=(Γ,T,E,M1,M2,R,2) be an MTS P system (cell 2 is the output cell), where Γ=T={a,b,c}, M1=, M2={a,b}, the environment contains an arbitrary number of copies of object c, R

Systems working in a maximally parallel mode

In this subsection, the computational power of MTS P systems is investigated, where rules in MTS P systems are applied in a maximally parallel mode [37].

Theorem 5.1

PsOtP1mon(sym,max)PsFIN.

Proof

As the system contains only one cell, communication can occur only between the cell and the environment in one direction. There are two cases: (1) objects are moved from the environment to cell; (2) objects are moved from cell to the environment. For case (1), symport rules are not allowed because each object in set E

Conclusions and further works

The advantages of MTS P systems are summarized in the following: (1) MTS P systems are developed by combining the monodirectional features and a network architecture of tissue P systems, thereby being more suitable and attractive to various applications which refer to complex topology representation; (2) a monodirectionality control strategy is introduced into tissue P systems to control the application of communication rules, thus making MTS P systems be more suitable for some applications

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The work was supported by National Natural Science Foundation of China (61972138, 61872309), the Fundamental Research Funds for the Central Universities (531118010355), Hunan Provincial Natural Science Foundation of China (2020JJ4215), and the Key Research and Development Program of Changsha (kq2004016).

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