monodirectional tissue p systems with

438 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 51, NO. 1, JANUARY 2021

Monodirectional Tissue P Systems With PromotersBosheng Song , Xiangxiang Zeng , Senior Member, IEEE,

Min Jiang , Senior Member, IEEE, and Mario J. Pérez-Jiménez

Abstract—Tissue P systems with promoters provide non-deterministic parallel bioinspired devices that evolve by theinterchange of objects between regions, determined by the exis-tence of some special objects called promoters. However, incellular biology, the movement of molecules across a membraneis transported from high to low concentration. Inspired by thisbiological fact, in this article, an interesting type of tissue Psystems, called monodirectional tissue P systems with promoters,where communication happens between two regions only in onedirection, is considered. Results show that finite sets of numbersare produced by such P systems with one cell, using any lengthof symport rules or with any number of cells, using a maximallength 1 of symport rules, and working in the maximally paral-lel mode. Monodirectional tissue P systems are Turing universalwith two cells, a maximal length 2, and at most one promoterfor each symport rule, and working in the maximally parallelmode or with three cells, a maximal length 1, and at most onepromoter for each symport rule, and working in the flat maxi-mally parallel mode. We also prove that monodirectional tissue Psystems with two cells, a maximal length 1, and at most one pro-moter for each symport rule (under certain restrictive conditions)working in the flat maximally parallel mode characterizes regu-lar sets of natural numbers. Besides, the computational efficiencyof monodirectional tissue P systems with promoters is analyzedwhen cell division rules are incorporated. Different uniform solu-tions to the Boolean satisfiability problem (SAT problem) areprovided. These results show that with the restrictive conditionof “monodirectionality,” monodirectional tissue P systems withpromoters are still computationally powerful. With the power-ful computational power, developing membrane algorithms formonodirectional tissue P systems with promoters is potentiallyexploitable.

Index Terms—Bioinspired computing, membrane computing,monodirectional tissue P system, NP-complete problem tissue-likenetwork, universality.

Manuscript received November 21, 2019; revised March 2, 2020 and April20, 2020; accepted June 14, 2020. Date of publication July 10, 2020; dateof current version December 22, 2020. This work was supported in part bythe National Natural Science Foundation of China under Grant 61972138and Grant 61602192, in part by the Fundamental Research Funds for theCentral Universities under Grant 531118010355, in part by the NaturalScience Foundation of Hunan Province of China under Grant 2020JJ4215,in part by the Research Project under Grant TIN2017-89842-P, cofinanced byMinisterio de Economía, Industria y Competitividad (MINECO) of Spain,through the Agencia Estatal de Investigación, and in part by the FondoEuropeo de Desarrollo Regional (FEDER) of the European Union. This arti-cle was recommended by Associate Editor L. Cheng. (Corresponding author:Xiangxiang Zeng.)

Bosheng Song and Xiangxiang Zeng are with the College of InformationScience and Engineering, Hunan University, Changsha 410082, China (e-mail:[email protected]; [email protected]).

Min Jiang is with the School of Information Science and Engineering,Xiamen University, Xiamen 361005, China (e-mail: [email protected]).

Mario J. Pérez-Jiménez is with the Department of Computer Science andArtificial Intelligence, Universidad de Sevilla, 41012 Sevilla, Spain (e-mail:[email protected]).

Digital Object Identifier 10.1109/TCYB.2020.3003060

I. INTRODUCTION

B IOINSPIRED computing focuses on the designs anddevelopments of computer algorithms and models

based on biological mechanisms and living phenom-ena, which includes quantum computing [27]; DNAcomputing [33], [63], [67]; membrane computing [42]; etc.In this article, we focus on the research area of membranecomputing, which is an active bioinspired field initiated byPaun [42], who discussed computing models motivated by thebehavior and structure of cells, understanding the processesthat take place in the compartments as computations. All thecomputing devices considered in this paradigm are called Psystems [42]. Since the initial model proposes in this area, var-ious P system models have been presented and investigated inthe aspects of computer science [1], [22], [52]; mathematics[7], [28]; and biochemistry [15], [23]. A brief summary ofmembrane computing can be found in monographs [43], [45],while a review of applications of this field can be found in [14]and [21].

An essential and important component of P systems is mem-brane structures that can be classified into two categories:1) hierarchical arrangements of membranes corresponding totrees (cell-like P systems [42]) and 2) nets of membranes (neu-rons) corresponding to arbitrary graphs (neural-like [26], [46],[56], [58], [61], [62], [64] or tissue-like P systems [31]). Thebasic models studied in this work are tissue-like P systems,which are motivated by the structure of tissue and the way ofcommunicating substances moving from one region to anotherregion. In a tissue P system, cells can communicate with othercells directly through channels and with the environment viasymport/antiport rules [41]. A rule is called symport if objectsgo the same direction; and a rule is called antiport if someobjects go the opposite directions at the same time betweentwo regions.

Inspired by various biological phenomena of cells, varioustissue P systems were constructed and investigated, most ofthem are turned out to be Turing complete [3], [4], [20], [40].Besides, from mathematical, biological, and computationalpoint of view, it is natural to incorporate mitosis (correspond-ing to computational rules “cell division”) [44] or membranefission (corresponding to computational rules “cell separa-tion”) [36] into tissue P systems, giving them the capacity togenerate an exponential space of computational units in lin-ear time. Thanks to this mechanism, different NP-completeproblems, for instance, 3-coloring [17]; vertex cover [18];subset sum [16], [47], [55]; and satisfiability problem (SATproblem) [36], [44], [57], can be efficiently solved by meansof tissue P systems with cell separation or cell division. We

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SONG et al.: MONODIRECTIONAL TISSUE P SYSTEMS WITH PROMOTERS 439

TABLE IRESULTS BETWEEN TISSUE P SYSTEMS WITH PROMOTERS AND MONODIRECTIONAL TISSUE P SYSTEMS WITH PROMOTERS, WHERE prokREPRESENTS AT MOST k PROMOTERS ASSOCIATED WITH EACH RULE, symt1 REPRESENTS SYMPORT RULES OF LENGTH AT MOST t1, antit2

REPRESENTS ANTIPORT RULES OF LENGTH AT MOST t2, ∗ REPRESENTS UNBOUNDED ON THE PARAMETER, AND ? REPRESENTS UNKNOWN RESULT

remark that under the hypothesis P �= NP, NP-complete prob-lems cannot be solved by P systems without cell division inpolynomial time [48].

Tissue P systems with promoters, motivated by the factthat biological reactions may occur in the presence of cer-tain chemicals, were raised in [8] and [50]. In [8] and [50],tissue P systems with promoters were investigated in theway of application; that is, such models were used in imageprocessing. Moreover, tissue P systems with promoters asgenerating devices of numbers were studied in [53], whichwas shown that such P systems are Turing complete whendifferent lengths of communication rules are combined (thelength of a rule is defined by all numbers of objects in sucha rule).

Inspired by the biological fact that the movement ofmolecules across a membrane is transported from high tolow concentration, the notion of “monodirectionality” was firstproposed in cell-like P systems (more precisely, P systemswith active membranes [29], readers can refer [6], [19], [22],and [39] for more details about P systems with active mem-branes), where for two given regions, communication happensonly in one direction and never in the opposite direction, thatis, for two given regions, either object send-in rules or objectsend-out rules can be used.

The motivation of this work aims at building a bridgebetween tissue P systems with promoters and a varietyof applications that involve information representation andinformation processing, thereby extending tissue P systemswith promoters to serve as a class of suitable and attractivemodels for these applications. Motivated by the monodirec-tional nature in cellular biology, a novel type of tissue Psystems with promoters, called monodirectional tissue Psystems with promoters, is introduced, where communicationhappens between two regions only in one direction, and neverin the opposite direction (hence, antiport rules are forbiddento be used systems).

Many applications require a monodirectional mech-anism, including information acquisition device forpower systems [30], [59]; some controllers for mobilerobots [9], [60]; etc. In this way, the computational modelsare usually required to have the monodirectional nature inthe sense that the transmission of information in devices canonly be one way.

The computational power of monodirectional tissue Psystems with promoters is examined as number generators. Asa result, finite sets of numbers are produced by such P systemswith one cell, using any length of symport rules or with anynumber of cells, using a maximal length 1 of symport rules,and working in the maximally parallel mode. Monodirectionaltissue P systems are Turing universal with two cells, a max-imal length 2, and at most one promoter for each symportrule, and working in the maximally parallel mode or withthree cells, a maximal length 1, and at most one promoterfor each symport rule, and working in the flat maximally par-allel mode (see Table I). We also prove that monodirectionaltissue P systems with two cells, a maximal length 1, and atmost one promoter for each symport rule (under certain restric-tive conditions) working in the flat maximally parallel modecharacterizes regular sets of natural numbers.

The computational efficiency of monodirectional tissue Psystems with promoters is also investigated by introducing celldivision rules. It is proved that the SAT problem is solvedby such systems with a maximal length 1 and at most twopromoters for each symport rule or with a maximal length 2and at most one promoter for each symport rule, working inthe flat maximally parallel mode (see Table I).

The main contributions of this article are summarized asfollows.

1) A novel type of tissue P systems with promoters,called monodirectional tissue P systems with pro-moters, is developed by introducing the notion ofmonodirectionality into tissue P systems with promot-ers. More precisely, such P systems have a networkarchitecture with the capability of complex topologyrepresentation. Moreover, the achieved monodirectionaltissue P systems with promoters are proved to be Turinguniversal. These results manifest that a Turing uni-versal monodirectional paradigm of tissue P systemswith promoters is theoretically possible and potentiallyexploitable.

2) A monodirectionality control strategy is introduced intotissue P systems with promoters to control the applica-tion of communication rules, thus making monodirec-tional tissue P systems with promoters be more suitablefor some applications which require a monodirectionalmechanism.



3) By employing network architecture as a model struc-ture and monodirectionality as information processing,monodirectional tissue P systems with promoters areattractive to some real-world problems, which involvemonodirectional nature and require networking modelprecisely.

4) Furthermore, by incorporating cell division into monodi-rectional tissue P systems with promoters, theinformation nature in cells can be replicated, thus mak-ing monodirectional tissue P systems with promoters bea more powerful modeling tool to develop various mem-brane algorithms, which makes training such systemspresumable and enhances its potential for practicalapplications.

The remainder of this article is organized as follows.Section II presents some fundamental conceptions of lan-guage and automata theory and the notion of monodirectionaltissue P systems with promoters and cell division. The com-putational power of the proposed P systems is investigatedin Section IV. In Section V, computational efficiency of theproposed P systems is presented. Finally, conclusions andsome future works are given in Section VI.

II. PRELIMINARIES AND MODEL DESCRIPTION

In this section, several fundamental conceptions from lan-guage and automata theory are recalled [51]. Also, the notionof (recognizer) monodirectional tissue P systems with promot-ers and cell division is introduced.

We define an alphabet (denoted by �) to be any nonemptyfinite set of abstract symbols. The set of strings that isconcatenated by any number of symbols is denoted by �∗.�+ = �∗\{λ} is the set that excludes the empty string (if astring does not have any symbols at all, it is called an emptystring, denoted by λ). The number of symbols in a string u isthe length of u, which is denoted by |u|.

Let � be an alphabet, and a multiset M is two tuples(�, f ) such that f is a function from � to N (the set ofnatural numbers). M(�) [respectively, M+(�)] representsthe set of all multisets (respectively, nonempty multisets).If � = {a1, . . . , ak}, then the multiset M = (�, f ) canbe denoted by {af (a1)

1 , . . . , af (ak)

k }. If we have two multisetsM1 = (�, f1) and M2 = (�, f2), then M1 +M2 is the unionof M1 and M2, which is defined by (�, f1(x) + f2(x)) suchthat each x ∈ �.

The family of finite sets of natural numbers is denotedby NFIN, the family of regular sets of natural numbers isdenoted by NREG, and we denote by NRE the family ofrecursively enumerable sets of natural numbers recognized byTuring machines.

In order to characterize NRE, the notion of programmachines is used. Readers can refer [45] for more detailsabout program machines. It is known that program machinesand Turing machines are equivalent; that is, both of themcharacterize NRE [32].

Next, we give the notion of monodirectional tissue Psystems with promoters and cell division (the readers are sug-gested to refer to [53] for more information about tissue Psystems with promoters).

Definition 1: A monodirectional tissue P system with pro-moters and cell division of degree q ≥ 1 has the followingconstruction:

� = (�, E,M1, . . . ,Mq,R, iout

)

where1) � is a finite set of alphabet of objects;2) E is a set of alphabet of objects initially placed in the

environment, such that E ⊆ �;3) Mi, 1 ≤ i ≤ q, are multisets of objects initially placed

in q cells;4) R is a set of division rules and symport rules with the

following restriction.a) Symport Rules: Either rules of form (pro|i, u/λ, j)

or rules of form (pro|i, λ/u, j) for two givenregions exist in the system such that 0 ≤ i �= j ≤ q,pro ∈ M(�), and u ∈ M+(�).

b) Division Rules: [ a ] i → [ b ] i[ c ] i such that i ∈{1, . . . , q}, i �= iout, a, b, c ∈ �.

5) iout ∈ {0, 1, . . . , q} is an output region.Note that if a system does not contain cell division rules,

then it is simply called a monodirectional tissue P system withpromoters.

Rules in monodirectional tissue P systems with promoters(and cell division) are applied in the maximally parallel mode(a maximum degree of each rule is used in parallel) [42] or inthe flat maximally parallel mode (for each computation step, amaximal applicable set of rules is selected and applied exactlyonce for each rule in this set) [5], [35], [54].

A configuration of monodirectional tissue P systems withpromoters at an instant t is defined by a tuple (N1, . . . , Nq, Ne),where Ni (1 ≤ i ≤ q) are multisets of objects over � and Ne

is a multiset of objects over �\E .The notions of computation and halting computation are

described in [42]; in particular, a configuration is a haltingconfiguration if no rule of the system is applicable to it. Onlya computation reaching a halting configuration gives a result,which is encoded by the multiset of specified objects present inthe output region iout (i.e., some objects present in the outputregion iout may not be counted).

A division rule [ a ] i → [ b ] i[ c ] i is applicable if and onlyif object a occurs in cell i, and cell i is not the output cell.When applying such a rule, cell i is divided into two cells withthe same label: object a specified in the cell i is replaced byobject b and object c in the newly generated cells, respectively;and all objects in the original cell, different from the objecttriggering the rule, are duplicated in the two new cells.

For the application of symport rules, one is referred to [53].Note that in monodirectional tissue P systems with promotersand cell division, the presence of the promoter objects makesit possible to use the associated rule as many times as possi-ble, without any restriction, that is, a promoter is valid for anynumber of rules (if these rules are associated with this pro-moter) in one step. Moreover, the promoters do not directlyparticipate in the rules. If a symport rule does not involvepromoters, then such rule is simply written by (i, u/λ, j).

A P system � that computes a set of naturalnumbers is denoted by N(�), and we denote by



NOtPmonm (prok, symt, max) [respectively, NOtPmon

m (prok,

symt, f max)] the family of all sets N(�) of natural numberscomputed by systems � with at most m cells, using a maximallength t and at most k promoters for each symport rule,and working in the maximally parallel mode (respectively,in the flat maximally parallel mode). If no bound on any ofparameters m, k, and t is forced, then we use symbol ∗ toreplace it.

The definition of recognizer tissue P systems with promot-ers and cell division was proposed in [53], such a model isconsidered to solve decision problems in a uniform way. Thereaders are referred to [53] for details.

We denote by PMCMTPDS(prok,symt,f max) the set of all deci-sion problems that are solved by a family of recognizermonodirectional tissue P systems with cell division, using amaximal length t and at most k promoters for each symportrule in a uniform and polynomial time, and working in the flatmaximally parallel mode.

III. TWO EXAMPLES

In order to illustrate the difference between monodirectionaltissue P systems and tissue P systems using only symport rules(here, we consider that both of these two kinds of P systemsare worked in the maximally parallel mode), the following twoexamples are presented.

Example 1: Let �1 = (�, E,M1,M2,R, 1) be a monodi-rectional tissue P system (cell 1 is the output cell), where� = {a}, M1 = M2 = {a}, and R = {r1 : (1, a/λ, 2)}.Note that rules of form (1, λ/a, 2) are not allowed. At step 1,object a is sent to cell 2 from cell 1, then no rule can beused in the system, and the computation halts. So the resultof computation is empty set.

Example 2: Let �2 = (�, E,M1,M2,R, 1) be a tis-sue P system using only symport rules (cell 1 is the outputcell), where � = {a}, M1 = M2 = {a}, and R ={r1 : (1, a/λ, 2); r2 : (1, λ/a, 2). Now, we analyze how suchP system works: in such P system, both rules r1 and r2 canbe chosen and used. Obviously, both rules r1 and r2 are usedin every step, and the computation never halts. So no result isobtained.

IV. COMPUTATIONAL POWER OF MONODIRECTIONAL

TISSUE P SYSTEMS WITH PROMOTERS

In this section, monodirectional tissue P systems withpromoters as generating devices for numbers are investigated.

A. Computational Power of Monodirectional Tissue PSystems With Promoters Working the Maximally ParallelMode

In this section, monodirectional tissue P systems with pro-moters as generating devices for natural numbers working inthe maximally parallel mode are investigated.

Theorem 1: NOtPmon1 (pro∗, sym∗, max) ⊆ NFIN.

Proof: Keep in mind that the system contains only onecell, objects move between the environment and cell by usingsymport rules in one direction. Hence in such a system, com-munication occurs in the following two cases: 1) objects in

Fig. 1. Membrane structure for the constructed monodirectional tissue Psystem with promoters, where circles represent cells, numbers on the side ofthe circles represent labels of cells, all cells are placed in the environmentwith label 0, and arrows indicate directions that objects are moved betweentwo regions.

the environment are moved to cell and 2) objects in the cellare moved to the environment. For case 1), symport rules arenot allowed because for objects in set E (the environment),each of them has any number of copies (if symport rules areallowed in this direction, the computation will never stop); andfor case 2), since there is a finite number of objects in the cell,the numbers of computation steps and reachable configurationsare finite. Therefore, finite sets of numbers are generated bythe system and the theorem holds.

Theorem 2: NOtPmon∗ (pro∗, sym1, max) ⊆ NFIN.Proof: It is clear that symport rules of form (pro∗|i, λ/a, 0)

(i �= 0, a ∈ E) are not allowed. So objects in set E cannotbe sent into any cells. Besides, there is a finite number ofobjects in the system, so the numbers of computation stepsand reachable configurations are finite. Hence, finite sets ofnumbers are generated by the system and the theorem holds.

Theorem 3: NOtPmon2 (pro1, sym2, max) = NRE.

Proof: Let M = (m, H, l0, lh, I) be a program machine. Thefollowing monodirectional tissue P system with promoters �

(see Fig. 1) is constructed to simulate M:

� = (�, E,M1,M2,R, 1)

where1) � = {l, l′, l′′, l′′′, liv, lv, lvi, lvii|l ∈ H} ∪ {ar|1 ≤ r ≤ m};2) E = {l, lv, lvi, lvii|l ∈ H} ∪ {ar|1 ≤ r ≤ m};3) M1 = {l′, l′′, l′′′, liv|l ∈ H} ∪ {l0}, M2 = ∅.

We design the following finite set R of symport rules.The number of object ar in cell 1 corresponds to the value

of register r in M. Each ADD instruction (respectively, SUBinstruction) in M can be simulated by six steps (respectively,eight steps) in �. Initially, cell 1 contains multisets of objects{l′, l′′, l′′′, liv|l ∈ H}∪{l0}, cell 2 is empty, and the environmentcontains multisets of objects {l, lv, lvi, lvii|l ∈ H}∪ {ar|1 ≤ r ≤m} (each of these objects has an arbitrary number of copies).The program machine M halts corresponding to cell 1 hasobject lh and no rule can be applied in system �. When thedesigned P system � halts, the computation result for M isthe number of object a1 deposited in cell 1 at this moment.

1) The following rules in R are constructed to simulateeach ADD instruction li of M:

r1,i :(1, lil

′i/λ, 2

), r2,i :

(2, lil

′i/λ, 0

)

r3,i :(1, λ/l′ilvi , 0

), r4,i :

(lvi |1, l′′i l′′′i /λ, 2

)

r5,i :(1, lvi /λ, 2

), r6,i :

(2, l′′i l′′′i /λ, 0

)

r7,i :(2, lvi /λ, 0

), r8,i :

(1, λ/l′′′i ar, 0

)



r9,i :(1, λ/l′′i lj, 0

), r10,i :

(1, λ/l′′i lk, 0

).

By applying rules r1,i, r2,i, and r3,i one by one, cell 1 willappear object lvi . At step 4, rules r4,i and r5,i are applied inparallel, cell 2 receives objects l′′i , l′′′i , and lvi , and at the nextstep, the environment will receive all these objects. At step 6,cell 1 receives objects l′′′i and ar by means of applying ruler8,i; meanwhile, the system nondeterministically chooses r9,i

and r10,i, and using one of these rules, the label object lj or lkis present in cell 1. Hence, during the process of simulation,the output cell increases one copy of object ar, which willsimulate the next instruction lj or lk.

1) The following rules in R are constructed to simulateeach SUB instruction li of M:

r11,i :(1, lil

′i/λ, 2

), r12,i :

(2, lil

′i/λ, 0

)

r13,i :(1, λ/l′ilvi , 0

), r14,i :

(lvi |1, l′′i ar/λ, 2

)

r15,i :(lvi |1, l′′′i livi /λ, 2

), r16,i :

(1, lvi /λ, 2

)

r17,i :(2, l′′i ar/λ, 0

), r18,i :

(2, l′′′i livi /λ, 0

)

r19,i :(2, lvi /λ, 0

), r20,i :

(l′′i |0, l′′′i lvi

i /λ, 1)

r21,i :(l′′i |1, λ/l′′′i lvii

i , 0), r22,i :

(lvii |1, λ/l′′i lj, 0

)

r23,i :(lvii |1, λ/livi , 0

), r24,i :

(1, lvi

i /λ, 2)

r25,i :(lviii |1, λ/livi lk, 0

), r26,i :

(1, lvii

i /λ, 2)

r27,i :(2, lvi

i /λ, 0), r28,i :

(2, lvii

i /λ, 0).

The system simulates an SUB instruction li as follows. Byapplying rules r11,i, r12,i, and r13,i one by one, object lvi willappear in cell 1 after three steps. At step 4, we have two casesto check whether cell 1 contains object ar or not.

1) Cell 1 contains object ar. Rules r14,i, r15,i, and r16,i areapplied in parallel, cell 2 receives objects l′′i , l′′′i , livi , lvi ,and ar, and these objects will be sent to the environmentby applying rules r17,i, r18,i, and r19,i. When the envi-ronment contains object l′′i , objects l′′′i and lvi

i are sentinto cell 1. If cell 1 contains object lvi

i , such cell willreceive objects l′′i , livi , and lj; meanwhile, cell 2 receivesobject lvi

i , and the environment will receive object lvii at

the next step. Hence, during this process, the output celldecreases one copy of object ar (corresponding to sub-tract one from register r), which will simulate the nextinstruction lj.

2) Cell 1 does not contain object ar. In this case, by apply-ing rules r15,i and r16,i in parallel, cell 2 receives objectsl′′′i , livi , and lvi , and the environment will receive theseobjects at the next step by applying rules r18,i and r19,i.At this moment, if cell 1 contains object l′′i , such cell willreceive objects l′′′i and lvii

i . When cell 1 contains objectlviii , objects livi and lk are sent into cell 1; meanwhile, cell

2 receives object lviii , and the environment will receive

object lviii at the next step. So when the simulation of

instruction li is finished, the system will start to simulateinstruction lk.

So an SUB instruction can be effectively simulated for bothcases.

The computation halts only when cell 1 contains object lh.The number of object a1 stored in cell 1 at this moment rep-resents the computation result of M. Hence, N(M) = N(�),and this concludes the proof.

B. Computational Power of Monodirectional Tissue PSystems With Promoters Working in the Flat MaximallyParallel Mode

In this section, the computational power of monodirectionaltissue P systems with promoters working in the flat maximallyparallel mode is presented.

Theorem 4: NOtPmon2 (pro1, sym1, f max) = NREG, where

two cells are labeled by 1 and 2, respectively; one cell (weassume cell 1, and cell 1 is the output cell) can receive objectsfrom the environment and cell 2 has no communication withthe environment; and objects are moved from cell 1 to cell 2.

Proof: We first prove that NOtPmon2 (pro1, sym1, f max) ⊆

NREG.Let �′ be an arbitrary monodirectional tissue P system with

a maximal length 1 and at most 1 promoter for each sym-port rule (under the above restrictive conditions). The maximalnumber of objects increased in the system �′ is described asfollows: each object initially placed in cell 1 and each objectinitially placed in the environment (at some step, these objectsare sent into cell 1) can be viewed as promoters, with the influ-ence of the promoter, some copies of objects are introducedinto the system; simultaneously, the promoter must be sent tocell 2, otherwise, the system never halts. Since the number ofdifferent objects initially placed in cell 1 and the number ofdifferent objects initially placed in the environment are finite,the numbers of computation steps and reachable configurationsare finite. We denote these configurations by Ci, 0 ≤ i ≤ L,and C0 is the initial configuration.

We construct a regular grammar G = (N, T, C0, P), whereN = {Ci|0 ≤ i ≤ L}, T is a set of terminal objects, and Pcontains the following productions.

1) Ci → aCj, for Ci, Cj ∈ N, a ∈ T such that there isa transition Ci ⇒ Cj between two configurations of �′during which the rules introducing terminal object a intothe output cell are used.

2) Ci → Cj, for Ci, Cj ∈ N such that there is a transitionCi ⇒ Cj between two configurations of �′ during whichthe output cell does not receive terminal object.

3) Ci → λ, for Ci ∈ N such that Ci is a haltingconfiguration of �′.

Note that if there are several copies of terminal objectsintroduced into the system (or output cell) at one step, thenwe can increase the number of configurations such that eachsubconfiguration introduces one copy of the terminal object.

We can check that each set of natural numbers generatedby system �′ can be generated by the regular grammar G.Hence, NOtPmon

2 (pro1, sym1, f max) ⊆ NREG.In what follows, we prove that the converse inclusion also

holds. Let G′ = (N′, T ′, S′, P′) be an arbitrary regular gram-mar with the productions of the form A → aB and A → a,where A, B ∈ N′, a ∈ T ′, S′ = A, and P′ is the set of pro-ductions. We construct a monodirectional tissue P system �′′



Fig. 2. Membrane structure of the constructed monodirectional tissue Psystem with promoters, where circles represent cells, numbers on the side ofthe circles represent labels of cells, all cells are placed in the environmentwith label 0, and arrows indicate directions that objects are moved betweentwo regions.

under the above restrictive conditions. Initially, N′ is the alpha-bet of system �′′, and object S′ ∈ N′. The set of rules in�′′ has the forms (p|1, λ/q, 0) and (1, p/λ, 2). A produc-tion of the form A → aB can be simulated by using rules(A|1, λ/a, 0), (A|1, λ/B, 0), and (1, A/λ, 2) in one step inthe mode of flat maximally parallelism. A production of theform A → a can be simulated by using rules (A|1, λ/a, 0)

and (1, A/λ, 2) in one step in mode of flat maximally par-allelism. We can check that system �′′ can generate thesets of natural numbers generated by grammar G′. Hence,NOtPmon

2 (pro1, sym1, f max) ⊇ NREG.The following corollary is easy to obtain according to

Theorem 4, here we omit the proof process.Corollary 1: NOtPmon

2 (pro1, sym1, f max) = NREG, wheretwo cells are labeled by 1 and 2, respectively; both cells (weassume cell 1 is the output cell) can receive objects from theenvironment, and objects are moved from cell 1 to cell 2.

Theorem 5: NOtPmon3 (pro1, sym1, f max) = NRE.

Proof: Let M = (m, H, l0, lh, I) be a program machine. Thefollowing monodirectional tissue P system with promoters �

(see Fig. 2) is constructed to simulate M:

� = (�, E,M1,M2,M3,R, 1)

where1) � = {l, l′, l′′, l′′′, liv, lv|l ∈ H} ∪ {ar|1 ≤ r ≤ m} ∪ {d};2) E = {l, lv|l ∈ H} ∪ {ar|1 ≤ r ≤ m};3) M1 = {l′, l′′, l′′′|l ∈ H} ∪ {l0}, M2 = {livi |l ∈ H} ∪ {d},

M3 = ∅.We design the following finite set R of symport rules.

The number of object ar in cell 1 corresponds to the valueof register r in M. Each ADD instruction (respectively, SUBinstruction) in M can be simulated by four steps (respectively,eight steps) in �. Initially, cell 1 contains multisets of objects{l′, l′′, l′′′|l ∈ H} ∪ {l0}, cell 2 contains multisets of objects{livi |l ∈ H}∪{d}, cell 3 is empty, and the environment containsmultisets of objects {l, lv|l ∈ H} ∪ {ar|1 ≤ r ≤ m} (each ofthese objects has an arbitrary number of copies). The programmachine M halts corresponding to cell 1 has object lh and norule can be applied in system �. When the designed P system� halts, the computation result for M is the number of objecta1 deposited in cell 1 at this moment.

1) The following rules in R are constructed to simulateeach ADD instruction li of M:

r1,i : (1, li/λ, 2), r2,i :(li|2, livi /λ, 0

)

r3,i : (2, li/λ, 0), r4,i :(livi |0, ar/λ, 1

)

r5,i :(0, livi /λ, 1

), r6,i :

(1, livi /λ, 2

)

r7,i :(livi |1, λ/lj, 0

), r8,i :

(livi |1, λ/lk, 0

).

At step 1, cell 2 receives object li by applying rule r1,i.When cell 2 contains object li, the environment will obtainobject livi ; meanwhile, the environment receives object li. Whenthe environment contains object livi , cell 1 will receive objectlivi and one copy of object ar (as flat maximal parallelism).Next, object livi is sent back to cell 2; meanwhile, the systemnondeterministically chooses r7,i and r8,i, and using one ofthese rules, and cell 1 will receive only one copy of labelobject lj or lk as using rules in the flat maximally parallelmode. Hence, during the process, the output cell increases onecopy of object ar, which will simulate the next instruction ljor lk.

1) The following rules in R are constructed to simulateeach SUB instruction li of M:

r9,i : (1, li/λ, 2), r10,i :(li|2, livi /λ, 0

)

r11,i : (2, li/λ, 0), r12,i :(1, λ/livi , 0

)

r13,i :(livi |1, l′i/λ, 2

), r14,i :

(livi |1, λ/lvi , 0

)

r15,i :(1, livi /λ, 2

), r16,i :

(l′i|2, λ/ar, 1

)

r17,i :(2, l′i/λ, 0

), r18,i :

(1, lvi /λ, 2

)

r19 : (ar|2, d/λ, 0), r20,i :(lvi |2, λ/l′′i , 1

)

r21,i :(lvi |2, λ/l′′′i , 1

), r22,i :

(1, λ/l′i, 0

)

r23,i :(2, lvi /λ, 0

), r24,i :

(l′′i |2, ar/λ, 0

)

r25,i :(ar|2, l′′i /λ, 3

), r26,i :

(ar|2, l′′′i /λ, 0

)

r27 : (1, λ/d, 0), r28,i :(d|2, l′′i /λ, 0

)

r29,i :(d|2, l′′′i /λ, 3

), r30,i :

(1, λ/l′′′i , 0

)

r31,i :(1, λ/l′′i , 3

), r32 : (1, d/λ, 2)

r33,i :(l′′′i |0, lj/λ, 1

), r34,i :

(1, λ/l′′i , 0

)

r35,i :(1, λ/l′′′i , 3

), r36,i:

(l′′i |0, lk/λ, 1

).

By the application of rules r9,i, r10,i, r11,i, and r12,i, theenvironment will receive object livi from cell 2, this object willthen be sent to cell 1. When cell 1 contains object livi , cell 2will receive object l′i from cell 1, and cell 1 receives only onecopy of object lvi ; meanwhile, cell 2 receives object livi . Next,we have two cases to check whether cell 1 contains object ar

or not.1) Cell 1 contains object ar. Rules r16,i, r17,i, and r18,i

are applied in parallel, object lvi is sent to cell 2, andthe environment will receive this object later; besides,the environment and cell 1 will receive object l′i in turn;and cell 2 receives one copy of object ar. When cell 2contains object ar, the environment, cell 1, and cell 2will receive object d in turn. If cell 2 contains objectlvi , cell 2 obtains objects l′′i and l′′′i . Next, the environ-ment will receive object ar; if cell 2 contains object ar,object l′′i is sent to cell 3, and the environment receives



object l′′′i . At step 8, objects l′′i and l′′′i are sent to cell 1and the environment, respectively; meanwhile, when theenvironment contains object l′′′i , cell 1 will receive onecopy of object lj. Hence during the process, the outputcell decreases one copy of object ar (corresponding tosubtract one from register r), which will simulate thenext instruction lj.

2) Cell 1 does not contain object ar. Only rules r17,i andr18,i are applied in parallel at step 5, cell 2 receivesobject lvi , and the environment will receive object lvi ;in parallel, the environment receives object l′i, and thencell 1 will obtain this object. If cell 2 contains objectlvi , this cell will obtain objects l′′i and l′′′i , and the envi-ronment and cell 3 will receive this object, respectively,(when object d still presents in cell 2). Next, cell 1 willobtain objects l′′i and l′′′i ; meanwhile, when the environ-ment contains object l′′i , cell 1 will receive object lk. Sowhen the simulation is finished, the system is passed tosimulate the next instruction lk.

So an SUB instruction can be effectively simulated for bothcases.

The computation halts only when cell 1 contains object lh.The number of object a1 stored in cell 1 at this moment rep-resents the computational result of M. Hence, N(M) = N(�),and this concludes the proof.

V. SOLVING SAT PROBLEM BY MONODIRECTIONAL

TISSUE P SYSTEMS WITH PROMOTERS AND CELL

DIVISION

In this section, the SAT problem is efficiently solved bymonodirectional tissue P systems with cell division by using amaximal length 1 and at most two promoters for each symportrule or using a maximal length 2 and at most one promoterfor each symport rule, working in the flat maximally parallelmode.

The propositional SAT problem is defined as follows:determining whether there exists an assignment to variablesthat satisfies a given propositional formula or not. The SATproblem was proved to be NP-complete problem [24].

Consider a propositional formula ϕ = C1 ∧ · · · ∧ Cm suchthat Ci = yi,1 ∨ · · · ∨ yi,pi , pi ≥ 1, yi,j ∈ {xk,¬xk|1 ≤ k ≤ n},1 ≤ i ≤ m, 1 ≤ j ≤ pi, and ∨ and ∧ represent or and and,respectively.

Theorem 6: SAT∈ PMCMTPDS(pro2,sym1,f max).Proof: A uniform solution to the propositional SAT problem is

given by a family of recognizer monodirectional tissue P systemswith promoters and cell division � = {�(t)|t ∈ N}, where eachsystem �(t) (t = 〈n, m〉= ((n + m)(n + m + 1)/2)+ n) withthe input multiset cod(ϕ) will process all Boolean formulas ϕ,which have m clauses and n variables.

The recognizer monodirectional tissue P system with pro-moters and cell division is constructed as follows:

�(〈n, m〉) = (�, E,M1,M2,M3,R, iin, iout)

where1) � = � ∪ E ∪ {a1, p,yes,no};2) � = {xi,j, xi,j|1 ≤ i ≤ n, 1 ≤ j ≤ m};

3) E = {ai|2 ≤ i ≤ n} ∪ {bj, cj|1 ≤ j ≤ m} ∪ {βi|0 ≤ i ≤2n + m + 3} ∪ {bm+1};

4) M1 = {a1}, M2 = {p, β0,yes, no}, M3 = ∅;5) iin = 1 and iout = 0 are the input cell and the output

region, respectively;6) The finite set of symport rules and division rules in R

is constructed as follows:

r1,i : [ ai ]1 → [ ti ]1[ fi ]1, 1 ≤ i ≤ n

r2,i,j :(tixi,j|1, λ/cj, 0

), 1 ≤ i ≤ n, 1 ≤ j ≤ m

r3,i,j :(fixi,j|1, λ/cj, 0

), 1 ≤ i ≤ n, 1 ≤ j ≤ m

r4,i : (ti|1, λ/ai+1, 0), 1 ≤ i ≤ n

r5,i : (fi|1, λ/ai+1, 0), 1 ≤ i ≤ n

r6,i : (1, ti/λ, 2), 1 ≤ i ≤ n

r7,i : (1, fi/λ, 2), 1 ≤ i ≤ n

r8 : (an+1|1, λ/b1, 0)

r9 : (1, an+1/λ, 2)

r10,j :(bjcj|1, λ/bj+1, 0

), 1 ≤ j ≤ m

r11,j :(1, bj/λ, 2

), 1 ≤ j ≤ m

r12 : (1, bm+1/λ, 2)

r13 : (bm+1|2,yes/λ, 3)

r14 : (bm+1|2, p/λ, 3)

r15 : (3,yes/λ, 0)

r16,i : (βi|2, λ/βi+1, 0), 0 ≤ i ≤ 2n + m + 2

r17,i : (2, βi/λ, 3), 0 ≤ i ≤ 2n + m + 2

r18 : (pβ2n+m+3|2,no/λ, 3)

r19 : (3, no/λ, 0).

The P system �(〈n, m〉) that solved the SAT problem con-sists of three stages: 1) the generation stage; 2) the checkingstage; and 3) the output stage. We remark that during the com-putational process, a counter object β is used for countingthe computation steps (by using rule r16,i); that is, for eachcomputation step, the subscript of β is increased by 1.

Generation Stage: In this stage, by applying division rules,all truth assignments for the formula ϕ(x1, . . . , xn) will be pro-duced (see Fig. 3). Meanwhile, the system checks the value ofall clauses by the corresponding truth assignment. This stageconsists of n iterations, and two steps are consumed for eachiteration. So this stage takes 2n steps.

At step i = 1 for 1 ≤ i ≤ n iterations, by using a rule ofr1,i, two cells with the same label are obtained from dividinga cell with label 1, where objects ti and fi are distributed ineach of these two cells, respectively.

At step i = 2 for 1 ≤ i ≤ n iterations, rulesr2,i,j, r3,i,j, r4,i, r5,i, r6,i, and r7,i are used in parallel. Whenobjects ti and xi,j (respectively, fi and xi,j) appear in cell 1,objects ai+1 and cj are introduced into that cell 1; meanwhile,by using rules r6,i and r7,i, objects ti and fi in cells 1 are sentto cell 2. The effect of rules r6,i and r7,i is to ensure thatobjects ti and fi appear in cell 1 only in one step, so only onecopy of object ai+1 and one copy of object cj are introducedinto a cell due to the flat maximal parallelism.



Fig. 3. Membrane structure of the constructed monodirectional tissue Psystem with promoters and cell division at the moment when the generationstage completes, where circles represent cells, numbers on the side of thecircles represent labels of cells, all cells are placed in the environment withlabel 0, and arrows indicate directions that objects are moved between tworegions.

Checking Stage: The system starts to check whether or notϕ assesses true by some truth assignments. Specifically, if allobjects c1, . . . , cm exist in cell 1, then it means in that cell thetruth assignment satisfies all clauses, so ϕ assesses to TRUE;if every cell 1 does not contain all the objects c1, . . . , cm, thenit means in each cell with label 1, the truth assignment doesnot satisfy at least one clause, so ϕ assesses to FALSE.

This stage begins at step 2n + 1, which costs m + 1 steps.At step 2n+1, when cell 1 contains object an+1, cell 1 will

obtain object b1; meanwhile, cell 2 receives object an+1 fromeach cell 1 by using rule r9.

At step 2n + 1 + j (1 ≤ j ≤ m), the system checks whetherobject cj exists in each cell 1. With the appearance of objects bj

and cj in cell 1, this cell will receive object bj+1; meanwhile,cell 2 receives object bj. We remark that if cell 1 presentsobject bj+1, it means clauses C1, . . . , Cj are true according tothe truth assignment assigned in that cell 1.

Output Stage: In the output stage, the computation result issent to the output region (i.e., the environment).

If the object bm+1 appears in cell 1 at step 2n+m+2, thenit means the Boolean formula evaluates to TRUE. Specifically,at step 2n + m + 2, by applying rule r12, cell 2 receives objectbm+1. When cell 2 contains object bm+1, cell 3 receives objectsp,yes, and the environment will obtain object yes at step2n + m + 4. The system halts and the computation result isaffirmative.

If every cell 1 does not contain object bm+1, then it meansthe Boolean formula evaluates to FALSE. Specifically, at step2n + m + 2, rules r16 and r17 are used in parallel, and objectβ2n+m+2 is sent into cell 2, which will be evolved to β2n+m+3at the next step. When cell 2 contains objects p, β2n+m+3,cell 3 and the environment will obtain object no in turn. Thesystem halts and the computation result is negative.

In what follows, we give the resources for constructing thesystem: 1) the size of the alphabet: 2nm + 3n + 3m + 8 ∈O(nm); 2) the number of cells initially needed in the system:3 ∈ O(1); 3) the number of objects initially needed in thesystem: 5 ∈ O(1); 4) the total number of rules in the system:2nm + 9n + 4m + 14 ∈ O(nm); and 5) the maximum lengthof a rule in the system (promoters are not included): 1 ∈

O(1). Therefore, a Turing machine exists that can constructthe system �(〈n, m〉) in polynomial time.

The designed P system �(〈n, m〉) halts at step 2n + m + 4(the last step) for the output yes or at step 2n + m + 5 (thelast step) for the output no. Thus, the system �(〈n, m〉) ispolynomially bounded concerning the numbers of clauses andvariables for the formula ϕ.

Therefore, the family of P systems � offers an efficientsolution to the SAT problem.

Theorem 7: SAT ∈ PMCMTPDS(pro1,sym2,f max).Proof: We design a family of recognizer monodirectional

tissue P systems with promoters and cell division � ={�(t)|t ∈ N} solving the SAT problem. Let �(t) (t = 〈n, m〉 =((n + m)(n + m + 1)/2) + n) be such a tissue P system [asso-ciated with input cod(ϕ)], which will deal with all Booleanformulas ϕ with m clauses and n variables.

We design the recognizer monodirectional tissue P systemwith promoters and cell division as follows:

�(〈n, m〉) = (�, E,M1,M2,M3,M4,M5,R, iin, iout)

where1) � = � ∪ E ∪ {bi, di|1 ≤ i ≤ n} ∪ {b′

i, hi, h′i|1 ≤ i ≤ m} ∪

{gi|2 ≤ i ≤ n + 1} ∪ {a1, a′n+1, b0, d0, q, z1,yes,no};

2) � = {xi,j, xi,j|1 ≤ i ≤ n, 1 ≤ j ≤ m};3) E = {ai|2 ≤ i ≤ n + 1} ∪ {cj, cj, pj, p′

j|1 ≤ j ≤ m} ∪{βi|0 ≤ i ≤ 7n + 4m + 7};

4) M1 = {b0, b1, . . . , bn, b′1, . . . , b′

m}, M2 = {β0, a1, g2,

. . . , gn+1}, M3 = {d0, d1, . . . , dn, q,yes,no}, M4 ={z1, a′

n+1, h1, . . . , hm, h′1, . . . , h′

m}, and M5 = ∅;5) iin = 1 and iout = 0 are the input cell and the output

region, respectively;6) the finite set of symport rules and division rules in R

is constructed as follows, the computation process con-sists of the generation stage, the checking stage, andthe output stage, and an overview of the computation ispresented.

Generation Stage:

r1 : [ z1 ]4 → [ z′2 ]4[ z′′

2 ]4

r2,i : [ z′i ]4 → [ z′

i+1 ]4[ z′′i+1 ]4, 2 ≤ i ≤ n

r3,i : [ z′′i ]4 → [ z′

i+1 ]4[ z′′i+1 ]4, 2 ≤ i ≤ n

r4 :(1, λ/z′

n+1, 4)

r5 :(z′

n+1|1, λ/a1, 2)

r6,i : [ ai ]1 → [ ti ]1[ fi ]1, 1 ≤ i ≤ n

r7,i,j :(ti|1, xi,j/λ, 0

), 1 ≤ i ≤ n, 1 ≤ j ≤ m

r8,i,j :(fi|1, xi,j/λ, 0

), 1 ≤ i ≤ n, 1 ≤ j ≤ m

r9,i : (ti|1, bi/λ, 0), 1 ≤ i ≤ n

r10,i : (fi|1, bi/λ, 0), 1 ≤ i ≤ n

r11,i,j :(0, xi,jcj/λ, 2

), 1 ≤ i ≤ n, 1 ≤ j ≤ m

r12,i,j :(0, xi,jcj/λ, 2

), 1 ≤ i ≤ n, 1 ≤ j ≤ m

r13,i : (0, biai+1/λ, 4), 1 ≤ i ≤ n

r14,i,j :(ti|1, λ/cj, 2

), 1 ≤ i ≤ n, 1 ≤ j ≤ m

r15,i,j :(fi|1, λ/cj, 2

), 1 ≤ i ≤ n, 1 ≤ j ≤ m

r16,i : (4, ai/λ, 3), 2 ≤ i ≤ n + 1



Fig. 4. Membrane structure of the constructed monodirectional tissue P system with promoters and cell division at the moment when the generation phasecompletes, where circles represent cells, numbers on the side of the circles represent labels of cells, all cells are placed in the environment with label 0, andarrows indicate directions that objects are moved between two regions.

r17,i : (ai|3, λ/gi, 2), 2 ≤ i ≤ n + 1

r18,i : (ai|3, di−1/λ, 0), 2 ≤ i ≤ n + 1

r19,i : (di|0, λ/ti, 1), 1 ≤ i ≤ n

r20,i : (di|0, λ/fi, 1), 1 ≤ i ≤ n

r21,i : (gi|3, ai/λ, 1), 2 ≤ i ≤ n + 1.

In the generation stage, the system works as follows. Atthe first n steps, cells with label 4 are divided by using rulesr1, r2,i, and r3,i, and 2n cells with label 4 are produced. Cellswith label 4 play an auxiliary role, which is used for checkingthe clauses that are satisfied. At the next two steps, by thesequential application of rules r4 and r5, object a1 will besent into cell 1 (the auxiliary object z′

n+1 is sent to cell 1,which is used as a promoter).

In the following steps, the system generates all truth assign-ments for the variables x1, . . . , xn and checks the clauses thatare satisfiable. This process consists n iterations, and six stepsare costed for each iteration, so it takes 6n steps for this stage.

At step i = 1 for 1 ≤ i ≤ n iterations, by using rule r6,i, twocopies of cell 1 are produced from dividing cell 1, and eachof the generated cell obtains objects ti and fi, respectively.

At step i = 2 for 1 ≤ i ≤ n iterations, when cell 1 containsobject ti (respectively, fi), the environment receives objects xi,j

and bi (respectively, xi,j and bi) by applying rules r7,i,j and r9,i

(respectively, r8,i,j and r10,i) in parallel.At step i = 3 for 1 ≤ i ≤ n iterations, cell 2 receives

xi,j, cj (respectively, xi,j, cj) by applying rule r11,i,j (respec-tively, r12,i,j); meanwhile, cell 4 receives objects bi and ai+1by using rule r13,i. Note that the number of cell 4 is morethan cell 1 at this moment, one copy of object bi and onecopy of object ai+1 enter one cell with label 4 because of theflat maximal parallelism.

At step i = 4 for 1 ≤ i ≤ n iterations, as using rules inmode of flat maximally parallelism, rule r14,i,j (respectively,

r15,i,j) is applied, cell 1 that it contains object ti (respectively,fi) receives an object cj (respectively, cj). Meanwhile, cell 3receives object ai by using rule r16,i.

At step i = 5 for 1 ≤ i ≤ n iterations, if object ai appearsin cell 3, such cell will receive object gi from cell 2, theenvironment receives object di−1 from cell 3.

At step i = 6 for 1 ≤ i ≤ n iterations, the environmentreceives objects ti and fi; meanwhile, cell 1 receives object ai.

In general, the generation stage costs 7n + 2 steps (seeFig. 4).

Checking Stage:

r22 : (1, an+1c1/λ, 0)

r23 : (1, an+1c1/λ, 0)

r24,j :(1, pjcj+1/λ, 0

), 1 ≤ j ≤ m − 1

r25,j :(1, pjcj+1/λ, 0

), 1 ≤ j ≤ m − 1

r26 :(an+1|1, b′

1/λ, 0)

r27 :(an+1|1, λ/a′

n+1, 4)

r28,j :(

pj|1, b′j+1/λ, 0

), 1 ≤ j ≤ m − 1

r29,j :(

h′j|1, λ/p′

j, 4), 1 ≤ j ≤ m − 1

r30,j :(

1, h′j/λ, 0

), 1 ≤ j ≤ m − 1

r31 :(an+1|1, a′

n+1/λ, 0)

r32,j :(

0, b′j/λ, 4

), 1 ≤ j ≤ m

r33,j :(

pj|1, p′j/λ, 0

), 1 ≤ j ≤ m − 1

r34,j :(

b′j|4, λ/pjp

′j, 0

), 1 ≤ j ≤ m

r35,j :(

4, b′jhj/λ, 5

), 1 ≤ j ≤ m

r36 :(a′

n+1|1, λ/p1h′1, 4

)



r37 :(h1|5, λ/a′

n+1, 1)

r38,j :(

p′j|1, λ/pj+1h′

j+1, 4), 1 ≤ j ≤ m − 1

r39,j :(

hj+1|5, λ/p′j, 1

), 1 ≤ j ≤ m − 1.

The system starts to check whether or not ϕ assesses trueby some truth assignments. Specifically, if the set of objects{c1, c1, . . . , cm, cm} with the subscript from 1 to m exists ina cell 1, then it means all clauses of the formula are satisfiedin that cell, so the formula evaluates to TRUE; if every cell1 does not contain the set of objects {c1, c1, . . . , cm, cm} withthe subscript from 1 to m, then it means in each cell withlabel 1, at least one clause of the formula is not satisfied inthat cell, so the formula evaluates to FALSE.

The checking stage consists of m iterations, and four stepsare needed for each iteration, thus this stage costs 4 m steps.

At step 1 of the first iteration, rules r26, r27, and r22 (orr23) are used, if cell 1 contains an object c1 or c1, then theenvironment receives objects an+1, b′

1, and c1 (or c1) fromcell 1; meanwhile, cell 1 receives object a′

n+1 from cell 4.Note that the system has the same number of cell 4 and cell1, in cell 4 each copy of object a′

i+1 enters one cell 1 becauseof the flat maximal parallelism.

At step 2 of the first iteration, if object an+1 still places in acell 1 (it means object c1 or c1 does not appear in that cell 1),the environment receives object a′

n+1 by applying rule r31. Byusing rule r32,1, each cell 4 will obtain an object b′

1 becauseof the flat maximal parallelism.

At step 3 of the first iteration, when cell 4 contains objectb′

1, such cell will receive one copy of object p1 and one copyof object p′

1 from the environment; by using rule r35,j, cell 5receives objects b′

1 and h1 from cell 4.At step 4 of the first iteration, if cell 1 still contains object

a′n+1 (it means that cell 1 has object c1 or c1), rule r36 is used,

and cell 1 receives objects p1 and h′1 from cell 4. Meanwhile,

cell 5 receives object a′n+1 from cell 1 by using rule r37.

At step i = 1 for 2 ≤ i ≤ n iterations, if cell 1 contains anobject cj or cj, then rule r24,j or r25,j is applied, the environ-ment receives objects pj−1, h′

j−1, b′j, and cj (or cj) from cell

1; meanwhile, when cell 1 contains object h′j−1, rule r29,j−1 is

used, cell 1 receives object p′j−1 from cell 4.

At step i = 2 for 2 ≤ i ≤ n iterations, if object pj−1 stillplaces in a cell 1 (it means object cj or cj does not appearin that cell with label 1), rule r33,j−1 is used, the environmentreceives object p′

j−1. Meanwhile, a cell with label 4 will obtainan object b′

j due to the flat maximal parallelism.At step i = 3 for 2 ≤ i ≤ n iterations, rules r34,j and r35,j

are used, objects pj and p′j are sent to the cell 4 that contains

object b′j, and objects b′

j and hj in cell 4 are sent to cell 5.At step i = 4 for 2 ≤ i ≤ n iterations, if cell 1 still contains

object p′j−1 (it means that cell 1 has object cj or cj), rule r38,j−1

is used, cell 1 receives objects pj and h′j from cell 4, which

contains object p′j−1. Meanwhile, cell 5 receives object p′

j−1from cell 1 by using rule r39,j−1.

Output Stage:

r40,i : (βi|2, λ/βi+1, 0), 0 ≤ i ≤ 7n + 4m + 6

r41,i : (2, βi/λ, 3), 0 ≤ i ≤ 7n + 4m + 6

r42 : (1, b0pm/λ, 0)

r43 : (b0|0, λ/d0, 3)

r44 : (0, d0pm/λ, 2)

r45 : (2, pm/λ, 3)

r46 : (pm|3,yesq/λ, 0)

r47 : (q|3, β7n+4m+6no/λ, 0).

From step 1 to step 7n+4m+7, a counter object β is usedfor counting the computation steps (by using rules r40,i andr41,i); that is, for each computation step, the subscript of β isincreased by one.

If the object pm appears in cell 1 at step 7n+4m+3, then itmeans the Boolean formula evaluates to TRUE. Specifically,by the sequential application of rules r42, r43, r44, and r45,object pm will appear in cell 3. When cell 3 contains objectpm, rule r46 is used, the environment receives objects q, yes.The P system halts and the computation result is affirmative.

If every cell 1 does not have object pm, then it means theBoolean formula evaluates to FALSE. Specifically, from step7n + 4m + 3 to step 7n + 4m + 7, only rules r40,i and r41,i areused in parallel, object β7n+4m+6 will be present in cell 3. Atstep 7n + 4m + 8, if object q still presents in cell 3, rule r47 isused, the environment receives object no. The P system haltsand the computation result is negative.

In what follows, we give the necessary resources for con-structing the system: 1) the size of alphabet: 2nm + 11n +11m + 16 ∈ O(nm); 2) the number of cells initially needed inthe system: 5 ∈ O(1); 3) the number of objects initially neededin the system: 3n + 3m + 9 ∈ O(n + m); 4) the total numberof rules in the system: 6nm + 26n + 19m + 20 ∈ O(nm); and5) the maximum length of a rule in the system (promoters isnot included): 2 ∈ O(1). Therefore, a Turing machine existsthat can construct the system �(〈n, m〉) in polynomial time.

The designed P system �(〈n, m〉) halts and the output isyes at step 7n+4m+7 (the last step) or no at step 7n+4m+8(the last step). Thus, the system �(〈n, m〉) is polynomiallybounded concerning the numbers of clauses and variables forthe formula ϕ.

Therefore, the family of P systems � offers an efficientsolution to the SAT problem.

VI. CONCLUSION

On the one hand, the hierarchical architecture of cell-likeP systems with symport/antiport rules lacks the capability ofcomplex topology representation; on the other hand, thesekinds of cell-like P systems lack the capability of processingmonodirectional information. As a result, cell-like P systemswith symport/antiport rules are not suitable for a mass ofapplications which involve monodirectional information rep-resentation and complex topology representation, and how toextend cell-like P systems to make them suitable and attrac-tive to some of these applications becomes a crucial issue. Inthis work, motivated by the monodirectional nature in cellu-lar biology, a novel type of tissue P systems with promotersis proposed, called monodirectional tissue P systems withpromoters. Specifically, monodirectional tissue P systems areTuring universal with two cells, a maximal length 2, and at



most one promoter for each symport rule, and working inthe maximally parallel mode or with three cells, a maximallength 1 and at most one promoter for each symport rule,and working in the flat maximally parallel mode. We havealso proved that monodirectional tissue P systems with twocells, a maximal length 1, and at most one promoter for eachsymport rule (under certain restrictive conditions) working inthe flat maximally parallel mode characterize regular sets ofnatural numbers. Besides, two efficient solutions to the SATproblem have been provided by such systems working in theflat maximally parallel mode (an obvious open problem isto investigate the computational efficiency of monodirectionaltissue P systems with promoters working in the maximallyparallel mode).

The advantages of proposing monodirectional tissue Psystems with promoters are summarized as follows.

1) Monodirectional tissue P systems with promoters aredeveloped by combining the monodirectional featuresand network architecture of tissue P systems, therebybeing more suitable and attractive to various applicationswhich refer to complex topology representation.

2) A monodirectionality control strategy is introduced intotissue P systems with promoters to control the applica-tion of communication rules, thus making monodirec-tional tissue P systems with promoters be more suitablefor some applications which require a monodirectionalmechanism.

3) By employing cell division into monodirectional tissue Psystems with promoters, the information nature in cellscan be replicated, thus making monodirectional tissue Psystems with promoters be a more powerful modelingtool to develop various membrane algorithms, whichmakes training such systems presumable and enhancesits potential for practical applications.

Communication for two given regions considered in thiswork is monodirectional; however, for a single cell, commu-nication may be bidirectional; that is, objects can both enterand exit a cell, for instance, cell 1 or cell 2 in Theorem 3.It is interesting to study the computational power of tissue Psystems with promoters in the sense that communication ismonodirectional for each region in a computation. (For eachregion, objects either enter this region or exit this region duringa computation.)

Membrane fission is a process by which a biological cellis split into two new ones in the manner that the contentof the initial cell is separated and distributed to the newcells [34], [49]. Consequently, the computational efficiency ofmonodirectional tissue P systems with promoters and cellseparation deserves to be investigated.

Several features for rule application inspired by biologicalor mathematical facts are introduced in membrane computing,such as asynchronism: any number of applicable rules in asystem may be used in each computational step [22]; minimalparallelism: at least one rule must be used in a membrane ifthere exists at least one applicable rule in such membrane [13];time freeness: the same result is generated by a system whichis not related to the times that rules are costed [2], [11],[55]; and sequentiality: exactly one rule is applied in one

derivation step [25]. It is interesting to consider the com-putational power of monodirectional tissue P systems withpromoters by using rules in the asynchronous mode, or inthe minimal parallel mode, or in the time-free mode, or in thesequential mode.

In [10], an interesting variant of spiking neural P systems,called spiking neural P systems with scheduled synapses,was proposed, where a duration or schedule was added toeach synapse, and each synapse was available only in theduration of its schedule. Consequently, it deserves to inves-tigate monodirectional tissue P systems with promoters andwith scheduled rules, where duration was added to eachrule, and each rule was available only in the duration ofits schedule.

Monodirectional tissue P systems with promoters have anetwork architecture with a stronger capability of complextopology representation than the hierarchical architecture ofcell-like P systems. Consequently, the prospect of monodi-rectional tissue P systems with promoters for applicationsis optimistic and promising, and the practical applicationsof monodirectional tissue P systems with promoters arewell worth investigating, for example, information acquisitiondevices for power systems or other real-world problems (e.g.,membrane-inspired evolutionary algorithms for multiobjectiveoptimization [12], [65], [66]) that need a networking modeland symbolic computing techniques.

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Bosheng Song received the Ph.D. degree in con-trol science and engineering from the HuazhongUniversity of Science and Technology, Wuhan,China, in 2015.

He spent 18 months working with the ResearchGroup on Natural Computing, University of Seville,Seville, Spain, from 2013 to 2015. He wasa Postdoctoral Researcher with the School ofAutomation, Huazhong University of Science andTechnology, from 2016 to 2019. He is currently anAssociate Professor with the College of Information

Science and Engineering, Hunan University, Changsha, China. His currentresearch interests include membrane computing and bioinformatics.

Xiangxiang Zeng (Senior Member, IEEE) receivedthe B.S. degree in automation from HunanUniversity, Changsha, China, in 2005, and the Ph.D.degree in system engineering from the HuazhongUniversity of Science and Technology, Wuhan,China, in 2011.

In 2019, he was with the Department of ComputerScience, Xiamen University, Xiamen, China. He isa Yuelu Distinguished Professor with the Collegeof Information Science and Engineering, HunanUniversity. His main research interests include mem-

brane computing, neural computing, and bioinformatics.

Min Jiang (Senior Member, IEEE) received theB.Sc. and Ph.D. degrees in computer science fromWuhan University, Wuhan, China, in 2001 and 2007,respectively.

From 2005 to 2007, he was a visiting stu-dent with the Department of Computer Science,City University of Hong Kong, Hong Kong.He is currently a Professor with the Schoolof Information Science and Engineering, XiamenUniversity, Xiamen, China. His current researchinterests mainly include machine learning andcomputational intelligence.

Mario J. Pérez-Jiménez received the B.Sc.degree in mathematics from Barcelona University,Barcelona, Spain, in 1971, and the Ph.D. degree inmathematics from the University of Seville, Seville,Spain, in 1992.

He is currently a Full Professor with theDepartment of Computer Science and ArtificialIntelligence, University of Seville, where he is theHead of the Research Group on Natural Computing.He has published 17 books in computer scienceand mathematics, and over 300 scientific papers in

international journals (collaborating with researchers worldwide). His mainresearch interests include theory of computation, computational complexitytheory, natural computing (DNA computing and membrane computing), bioin-formatics and computational modeling for systems biology, and populationdynamics.

Dr. Pérez-Jiménez is currently a Numerary Member of the AcademiaEuropaea (Academy of Europe).


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