Computational power of cell separation in tissue P systems

Information Sciences xxx (2014) xxx–xxx

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Information Sciences

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Computational power of cell separation in tissue P systems

http://dx.doi.org/10.1016/j.ins.2014.04.0310020-0255/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author at: Research Institute of the IT4Innovations Centre of Excellence, Faculty of Philosophy and Science, Silesian University i74601 Opava, Czech Republic. Tel.: +420 553 684 364.

E-mail addresses: [email protected] (P. Sosík), [email protected] (L. Cienciala).

Please cite this article in press as: P. Sosík, L. Cienciala, Computational power of cell separation in tissue P systems, Inform. Sci.http://dx.doi.org/10.1016/j.ins.2014.04.031

Petr Sosík a,b,⇑, Ludek Cienciala a

a Research Institute of the IT4Innovations Centre of Excellence, Faculty of Philosophy and Science, Silesian University in Opava, 74601 Opava, Czech Republicb Departamento de Inteligencia Artificial, Facultad de Informática, Universidad Politécnica de Madrid, Campus de Montegancedo s/n, Boadilla del Monte,28660 Madrid, Spain

a r t i c l e i n f o

Article history:Received 4 September 2013Received in revised form 12 March 2014Accepted 10 April 2014Available online xxxx

Keywords:Membrane computingTissue P systemCell separationComputational complexityPSPACE

a b s t r a c t

The paper focuses on the relation between biological and computational informationprocessing. Several simple biological operations, as the exchange of molecules via cellularmembrane or the cellular growth and separation, are abstracted into a mathematical modelcalled membrane system (P system). This paper studies tissue P systems where a fixedinteraction graph defines the communication between various types of cells.

Polynomially uniform families of tissue P systems with the operation of cell separationwere recently studied. Their computational power in polynomial time was shown to rangebetween P and NP [ co� NP, characterizing borderlines between tractability and intracta-bility by the length of rules controlling the interchange of objects. Here, we show that thecomputational power of these uniform families is limited by the class PSPACE, which is theclass characterizing the power of classical parallel computing models, such as PRAM or thealternating Turing machine.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

The importance of studying biological processes from the point of view of information is now a widely respected fact,resulting in the fast emergence of several scientific disciplines on the borderlines of biology, information and computer sci-ence. On the one hand, various kinds of mathematical models and tools are employed to understand intricate processes inbiological systems. On the other hand, many bio-inspired computing models have emerged, allowing to solve computationalproblems using strategies inspired by nature. Membrane system is a cell-inspired mathematical model which has been inten-sively studied for 15 years now. The main components of membrane systems (also called P systems) are (i) a hierarchicalsystem of membranes encompassing regions with discrete objects and (ii) rules allowing transport and transformation ofobjects; in some models also dissolution/division of membranes, etc. An overview of several variants of P systems with manyexamples can be found, e.g., in the monograph [20]. P systems have been applied not only to model biological processes,such as signalling pathways [6], membrane processes [23], FAS mediated apoptosis [3,7], quorum sensing [22] and others,but also to find ‘‘soft’’ solutions to a range of difficult problems, often in combination with other soft-computing techniques,such as swarm optimization [26], evolutionary search [5,29] and differential evolution [25].

Tissue P systems are a specific variant of P systems introduced in [11,12]. A biological motivation of the model is theintercellular communication and cooperation between tissue cells by the interchange of signaling molecules. The original

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2 P. Sosík, L. Cienciala / Information Sciences xxx (2014) xxx–xxx

motivation was (1) to construct a distributed cell-inspired computing model based on the interaction of simple discrete-state components and (2) to characterize the information-processing potential of cell-to-cell interaction in tissue-like envi-ronment. However, the model can also provide a theoretical background for engineered cell-based systems harnessing thepower of biological information processing, a recent hot topic in synthetic biology.

Unlike the hierarchical arrangement of membranes which is usual in P systems, the underlying structure of tissue P sys-tems is defined as a fixed interaction graph with membranes/cells in its nodes. The communication among cells alongsidethe edges of the virtual graph is based on symport/antiport rules [19], transporting a certain number of objects across mem-branes. This number is called the length of the rule. From the original definitions of tissue P systems [11,12], several researchlines have been developed (see, for example, [1,2,4,8,9]).

The variant of tissue P systems studied here is extended by the operation of cell separation introduced in [13] and calledtissue P systems with cell separation [14]. The biological inspiration is the following: new cells are produced by cell separationin tissues in a natural way. The computational efficiency of the model was investigated and the following results have beenpreviously obtained: (a) only tractable problems can be efficiently solved when the length of communication rules is at most2 [15] and (b) an efficient (uniform) solution to the SAT problem by using communication rules with length at least 8 (and, ofcourse, separation rules) [14]. The latter result was recently improved and the borderline between tractability and intracta-bility was found between the length of rules 2 and 3 [16]. The tissue P systems with cell separation solving intractable prob-lems like the partition problem or the independent set problem in polynomial time were introduced in [27] or [28].

In this paper we demonstrate an upper bound of the power of tissue P systems with cell separation. We show that theclass of problems solvable by uniform families of these systems in polynomial time is contained in the class PSPACE. Thisclass characterizes the power of many classical models of parallel computing machines, such as PRAM or the alternating Tur-ing machine, relating, thus, classical and bio-inspired parallel computing devices. A preliminary version of this resultappeared in [24], with very limited proof material. Here, we give a full version of the proofs together with an extended pre-sentation of the results. The rest of the paper is organized as follows: after introducing preliminary notation, definitions oftissue P systems with cell separation are given. Next, computational complexity classes induced by tissue P systems aredefined. In Section 3, we demonstrate that any such tissue P system can be simulated with a classical computer (and, hence,also with Turing machine) in a polynomial space. The paper concludes with a short discussion.

2. Tissue P systems with cell separation

2.1. Mathematical background

A multiset m over an underlying set A is a pair ðA; f Þ where f : A! N is a mapping. If m ¼ ðA; f Þ is a multiset, then, itssupport is defined as suppðmÞ ¼ fx 2 A j f ðxÞ > 0g. The total number of elements in a multiset, including repeatedmemberships, is the cardinality of the multiset. A multiset is empty (resp. finite) if its support is the empty set (resp. a finiteset). If m ¼ ðA; f Þ is a finite multiset over A, and suppðmÞ ¼ fa1; . . . ; akg then it can also be represented by the string

af ða1Þ1 . . . af ðakÞ

k over the alphabet fa1; . . . ; akg. Nevertheless, all permutations of this string precisely identify the same multisetm. Throughout this paper, we speak about ‘‘the finite multiset m’’ where m is a string, meaning ‘‘the finite multisetrepresented by the string m’’.

If m1 ¼ ðA; f1Þ;m2 ¼ ðA; f2Þ are multisets over A, then we define the union of m1 and m2 as m1 þm2 ¼ ðA; gÞ, whereg ¼ f1 þ f2.

For any sets A and B the relative complement A n B of B in A is defined as follows:

Pleasehttp:/

A n B ¼ fx 2 A j x R Bg:

2.2. Basic definition

Although this section contains a complete definition of tissue P systems with cell separation, the reader might want toconsult a more detailed description in the monograph [20] or visit the web page http://ppage.psystems.eu. The model ofP system described here is based, besides the symport/antiport rules, on the operation of cell separation. The two cellsobtained by separation have the same labels as the original cell, and an interaction with other cells or with the environmentis blocked during the separation process.

Definition 1 [14]. A tissue P system with cell separation of degree q P 1 is a tuple

P ¼ ðC;C1;C2; E;M1; . . . ;Mq;R; ioutÞ;

where:

1. C is a finite alphabet whose elements are called objects;2. fC1;C2g is a partition of C, that is, C ¼ C1 [ C2;C1;C2 – ;, C1 \ C2 ¼ ;;

cite this article in press as: P. Sosík, L. Cienciala, Computational power of cell separation in tissue P systems, Inform. Sci. (2014),/dx.doi.org/10.1016/j.ins.2014.04.031

http://ppage.psystems.eu


P. Sosík, L. Cienciala / Information Sciences xxx (2014) xxx–xxx 3

3. E # C is a finite alphabet representing the set of objects initially in the environment of the system, and 0 is the label of theenvironment (the environment is not properly a cell of the system); let us assume that objects in the environment appear inarbitrary copies each;

4. M1; . . . ;Mq are strings over C, representing the finite multisets of objects placed in the q cells of the system at thebeginning of the computation; 1;2; . . . ; q are labels which identify the cells of the system;

5. R is a finite set of rules of the following forms:(a) Communication rules: ði;u=v ; jÞ, for i; j 2 f0;1;2; . . . ; qg; i – j;u;v 2 C�; juv j > 0; the communication rule ði;u=v ; jÞ can

be applied to two cells i and j such that u is contained in cell i and v is contained in cell j; the application of this rulemeans that the objects of the multisets represented by u and v are interchanged between the two cells.

(b) Separation rules: ½a�i ! ½C1�i½C2�i, where i 2 f1;2; . . . ; qg and a 2 C, and i – iout . In reaction with an object a, the cell i isseparated into two cells with the same label; at the same time, object a is consumed; the objects from C1 are placed inthe first cell, those from C2 are placed in the second cell; the output cell iout cannot be separated;

6. iout 2 f0;1;2; . . . ; qg is the output cell.

A communication rule ði;u=v ; jÞ is called a symport rule if u ¼ k or v ¼ k. A symport rule ði;u=k; jÞ, with i – 0; j – 0, providesa virtual arc from cell i to cell j. A communication rule ði;u=v ; jÞ is called an antiport rule if u – k and v – k. An antiport ruleði;u=v; jÞ provides two arcs: one from cell i to cell j and another one from cell j to cell i. Thus, every tissue P systems has anunderlying directed graph whose nodes are the cells of the system and the arcs are obtained from communication rules. Inthis context, the environment can be considered as a virtual node of the graph such that their connections are defined byantiport rules of the form ði;u=v ; jÞ, with i ¼ 0 or j ¼ 0. The length of the communication rule ði;u=v ; jÞ is defined as juj þ jv j.

The object a triggers rule ½a�i ! ½C1�i ½C2�i and it is consumed. Nevertheless, this rule does not produce any new object innew cells. The remaining objects in cell i are distributed into the new cells, according to sets C1 and C2. If there are n objectsin the cell i where the rule is applied, the total number of objects in the cells created is n� 1. Note that a separation rule canbe applied repeatedly, resulting in an exponential growth of the number of cells.

The rules of a system like the above one are used in the non-deterministic maximally parallel manner as customary inmembrane computing: at each step, all cells which can evolve must evolve in a maximally parallel way (at each step weapply a multiset of rules which is maximal, no further rule can be added being applicable). The label of a cell identifiesthe rules which can be applied to it precisely. There is one important restriction: when a cell is separated, the separationrule is the only one which is applied for that cell at that step; thus, the objects inside that cell do not evolve by means ofcommunication rules. However, there is no priority between rules, hence, if both communication and separation rules areapplicable to a cell, then one option (one separation rule or possibly multiple communication rules) is arbitrarily chosen.

A configuration of a tissue P system with cell separation at any instant is described by all multisets of objects over C asso-ciated with all the cells present in the system, and the multiset of objects over C� E associated with the environment at thatmoment. Bearing in mind that the objects from E have infinite copies in the environment, they are not properly changedalong the computation. The initial configuration is ðM1; . . . ;Mq; ;Þ. A configuration is a halting configuration if no rule ofthe system is applicable in it.

We say that configuration C1 yields configuration C2 in one transition step, denoted C1)PC2, if the system can pass fromC1 to C2 by applying the rules from R as specified above. A computation of P is a (finite or infinite) sequence of configurationssuch that:

1. the first term of the sequence is the initial configuration of a sequence;2. each non-initial configuration of the sequence is obtained from the previous configuration by applying rules of the system

in a maximally parallel manner with the restrictions previously mentioned; and3. if the sequence is finite (called halting computation), then the last term of the sequence is a halting configuration.

Halting computations give a result which is encoded by the objects present in the output cell iout in the haltingconfiguration.

Example 1. Let P ¼ ðC;C1;C2; ;; E;M1;M2;R;0Þ be a tissue P system with cell separation, two cells and with:

Pleasehttp:/

C ¼ fx; y; zgC1 ¼ fxgC2 ¼ fy; zgE ¼ ;M1 ¼ xy

M2 ¼ xxz

R ¼ fr1 ¼ ð1; y=x;2Þ; r2 ¼ ½y�2 ! ½C1�2½C2�2g

:

The initial configuration of the system is C0 ¼ ðxy; xxz; ;Þ. Only rule r1 is applicable in this configuration. After the firststep, the configuration of the system is C1 ¼ ðxx; xyz; ;Þ. The separation rule r2 is applicable in configuration C1. After thesecond step, the system consists of three cells in configuration C2 ¼ ðxx; x; z; ;Þ. The situation is illustrated in Fig. 1.



Fig. 1. First three configurations of the tissue P system P described in Example 1.


2.3. Recognizer tissue P systems with cell separation

Let us denote a decision problem as a pair ðIX ; hXÞ where IX is a language over a finite alphabet (whose elements are calledinstances) and hX is a total boolean function over IX . A natural correspondence between decision problems and languages overa finite alphabet can be established as follows. Given a decision problem X ¼ ðIX ; hXÞ, its associated language isLX ¼ fw 2 IX : hXðwÞ ¼ 1g. Conversely, given a language L over an alphabet R, its associated decision problem isXL ¼ ðIXL ; hXL Þ, where IXL ¼ R�, and hXL ¼ fðx;1Þ : x 2 Lg [ fðx;0Þ : x R Lg. The solvability of decision problems is definedthrough the recognition of the languages associated with them by using languages recognizer devices.

In order to study the computational efficiency of membrane systems, the notions from classical computational complexitytheory are adopted for tissue P systems, and the idea of recognizer tissue P systems is introduced in [21].

Definition 2 [14]. A recognizer tissue P system with cell separation of degree q P 1 is a tuple

Pleasehttp:/

P ¼ ðC;C1;C2;R; E;M1; . . . ;Mq;R; iin; ioutÞ

where:

1. ðC;C1;C2; E;M1; . . . ;Mq;R; ioutÞ is a tissue P system with cell separation of degree q P 1 (as defined in the previoussection).

2. The working alphabet C has two distinguished objects yes and no such that at least one copy of them is present in someinitial multisets M1, . . ., Mq, but none of them is present in E.

3. R is an (input) alphabet strictly contained in C, and E # C n R.4. M1; . . . ;Mq are strings over C n R;5. iin 2 f1; . . . ; qg is the input cell.6. The output region iout is the environment.7. All computations halt.8. If C is a computation of P, then either object yes or object no (but not both) must have been released into the environ-

ment, and only in the last step of the computation.

For each w 2 R�, the computation of the system P with input w 2 R� starts from the configuration of the formðM1;M2; . . . ;Miin þw; . . . ;Mq; ;Þ, that is, the input multiset w has been added to the contents of the input cell iin. Therefore,we have an initial configuration associated with each input multiset w (over the input alphabet R) in this kind of systems.

Given a recognizer tissue P system with cell separation, we say that a computation C is an accepting computation (respec-tively, rejecting computation) if object yes (respectively, object no) appears in the environment associated with thecorresponding halting configuration of C, and neither object yes nor no appears in the environment associated with anynon-halting configuration of C.

For each natural number k P 1, we denote by TSCðkÞ the class of recognizer tissue P systems with cell separation andcommunication rules of length at most k. We denote by TSC the class of recognizer tissue P systems with cell separationand without restriction on the length of communication rules. Obviously, TSCðkÞ# TSC for all k P 1.

2.4. Polynomial complexity classes of tissue P systems

Next, we define what does it mean to solve a decision problem in the framework of tissue P systems efficiently and in auniform way. Bearing in mind that they provide devices with a finite description, a numerable family of tissue P systems willbe necessary in order to solve a decision problem efficiently.

Definition 3 [14]. We say that a decision problem X ¼ ðIX ; hXÞ is solvable in a uniform way and polynomial time by a familyP ¼ fPðnÞjn 2 Ng of recognizer tissue P systems (with cell separation) if the following holds:




1. The family P is polynomially uniform by Turing machines, that is, there exists a deterministic Turing machine working inpolynomial time which constructs the system PðnÞ from n 2 N.

2. There exists a pair ðcod; sÞ of polynomial-time computable functions over IX such that:(a) for each instance u 2 IX ; sðuÞ is a natural number and codðuÞ is an input multiset of the system PðsðuÞÞ;(b) for each n 2 N; s�1ðnÞ is a finite set;(c) the family P is polynomially bounded with regard to ðX; cod; sÞ, that is, there exists a polynomial function p, such that

for each u 2 IX every computation of PðsðuÞÞ with input codðuÞ is halting and it performs at most pðjujÞ steps;(d) the family P is sound with regard to ðX; cod; sÞ, that is, for each u 2 IX , if there exists an accepting computation of

PðsðuÞÞ with input codðuÞ, then hXðuÞ ¼ 1;(e) the family P is complete with regard to ðX; cod; sÞ, that is, for each u 2 IX , if hXðuÞ ¼ 1, then every computation of

PðsðuÞÞ with input codðuÞ is an accepting one.

From the soundness and completeness conditions above, we deduce that every P system PðnÞ is confluent, in the followingsense: every computation of a system with the same input multiset must always give the same answer.

Let R be a class of recognizer tissue P systems. We denote by PMCR the set of all decision problems which can be solved ina uniform way and polynomial time by means of families of systems from R. The following results are known:

Theorem 1 [15]. P ¼ PMCTSCð2Þ.

Theorem 2 [16]. NP [ co�NP # PMCTSCð3Þ.

3. Simulation of tissue P systems with cell separation in polynomial space

In this section we demonstrate that any computation of a recognizer tissue P system with cell separation can be simulatedin polynomial space associated with its initial size and the number of steps. Instead of simulating a computation of a P sys-tem from its initial configuration onwards (which would require an exponential space for storing configurations), we create arecursive function which computes the content of any cell h after a given number of steps. The recursive function evaluatescontents of the cells interacting with h in a reverse time order (towards the initial configuration). Thus, we do not need tostore the content of any cell but we re-calculate it recursively whenever needed.

Simulated P systems are confluent, hence possibly non-deterministic, but the simulation will be performed in a determin-istic way: only one possible sequence of configurations of the P system is traced. This corresponds to a weak priority relationbetween rules:

(i) separation rules are always applied prior to communication rules,(ii) priority between communication rules given by the order in which they are listed,

(iii) priority between cells to which the rules are applied.

However, the confluency condition ensures that such a simulation is correct as all the computations starting from thesame initial configuration must lead to the same result.

Each cell of P is assigned a unique label in the initial configuration. But cells may be separated during computation of P,thus producing more cells with the same label. To identify cells uniquely, we add to each label a compound index. Each indexis an empty string in the initial configuration. If a cell is not separated in a computational step, digit 1 is attached to its index.If a separation rule is applied, the first resulting cell has attached digit 1 and the second cell digit 2 to its index. Hence, theindex of each cell is an n-tuple of digits from f1;2g after n steps of computation. We also apply the same indexing to theenvironment which is treated as a virtual cell that never separates. A similar idea of indexing of separating cells was usedalready in [10], in the form of extended cell labels.

Notice that some possible n-tuples may correspond to non-existing cells as cells do not need to separate at each step. Thesituation is illustrated in Fig. 2: cell h is separated in the first step, cells g1 and h2 are separated in the second step. Anexample of a reference to non-existing cell is h12.

Fig. 2. An example of indexing of cells during first two computational steps.

Please cite this article in press as: P. Sosík, L. Cienciala, Computational power of cell separation in tissue P systems, Inform. Sci. (2014),http://dx.doi.org/10.1016/j.ins.2014.04.031



Consider a confluent recognizer tissue P system with cell separation of degree q P 1, described formally as

Pleasehttp:/

P ¼ ðC;C1;C2;R; E;M1; . . . ;Mq;R; iin; ioutÞ:
For any cell of P we refer to the multiset of objects contained in it at any instant simply as its content. We construct functionContent which computes recursively the content of any cell h (with a given compound index) of P after n P 0 steps of com-putation. The function is described in an informal way but it could be processed by a Turing machine working in (asymp-totically) the same space.
Let m denote the total number of communication rules in R. A simplified scheme of the function content computingrecursively the content of a cell labeled h with index ind of P in configuration Cn;n P 1 follows:

1. for all communication rules ri ¼ ðj;u=v ; kÞ;1 6 i 6 m, repeat step 2, keeping the multiset of already applied rules invariable rulesAppliedTotal;

2. for each pair of cells labeled j and k with various compound indices repeat:(a) calculate recursively the contents of these two cells in configuration Cn�1;(b) if a separation rule is applicable to any of them, skip the rest of the cycle and start a new iteration.(c) remove from cells j and k the objects consumed by the maximum multiset of rules r1; . . . ; ri�1 in

rulesAppliedTotal;(d) calculate the maximum multiset of rules ri applicable to the remaining contents of cells j; k and add it to

rulesAppliedTotal;(e) store the maximum multiset of rules ri applicable to cell h with index ind;

3. calculate recursively the content of cell h with index ind in configuration Cn�1; if possible, apply a separation rule,otherwise apply the multisets of rules stored at step 2e to get the content of the cell in configuration Cn.

All rules at a particular step n are applied in a maximally parallel way to all cells, but we simulate this process sequen-tially, following a fixed order of rules in R and a fixed order of cells to which these rules are applied.

Observe that the recursion only includes the number of configuration n and not the number of already applied rules, althoughan application of a rule ri may depend on the contents of other (possibly all) cells which were already affected by rules r1; . . . ; ri�1

at the same step. Note that, thanks to the maximal parallelism and fixed order of rules and of pairs of cells, one can always re-cal-culate the effect of rules r1; . . . ; ri�1 already applied to a particular pairs of cells, as in the paragraphs 2b and 2c above.

A description of the function content as follows. Assume for simplicity that an input multiset of objects w is alreadyincluded in the initial multiset Miin .

function content

Input: h 2 f1; . . . ; qg – label of a celli1i2 . . . in – a compound indexn P 0 – a number of configuration

Output: the content of cell labeled h with compound index i1i2 . . . inin configuration Cn, or null if such a cell does not exist.

Variables significant for a space complexity calculation:

rulesAppliedToh, rulesAppliedTotal, rulesForCell1, rulesForCell2;(Multisets of rules with the underlying set R)contentCell1, contentCell2, contentFinal

(Multisets storing contents of cells)

if n ¼ 0 then return Mh; (return the initial multiset of cell h)

rulesAppliedTotal :¼ ;;rulesAppliedToh :¼ ;;

for each communication rule ri ¼ ðj;u=v ; kÞ;1 6 i 6 m do begin

rulesForCell1 :¼ rulesAppliedTotal;

for all j1j2 . . . jn�1 2 f1;2gn�1

do begin

contentCell1 = content(j; j1j2 . . . jn�1;n� 1);(Content of cell j with index j1j2 . . . jn�1 in the previous configuration.)

if contentCell1 = null or the cell can apply a separation rule then

skip the rest of the cycle body;

M :¼ maximum multiset of rules in rulesForCell1 applicable to cell j




with contentCell1 in order r1; . . . ; ri�1;remove M from the multiset rulesForCell1;remove the objects consumed by the rules in M in cell jfrom contentCell1;

rulesForCell2 :¼ rulesAppliedTotal;

for all k1k2 . . . kn�1 2 f1;2gn�1do begin

contentCell2 = content(k; k1k2 . . . kn�1;n� 1);(Content of cell k with index k1k2 . . . kn�1 in the previousconfiguration.)

if contentCell2 = null or the cell can apply a separation rule then

skip the rest of the cycle body;

M :¼ maximum multiset of rules in rulesForCell2 applicable to cell kwith contentCell2 in order r1; . . . ; ri�1;

remove M from the multiset rulesForCell2;remove the objects consumed by the rules in M in cell kfrom contentCell2;

(Now contentCell1 and ContentCell2 contain only the objectsunaffected by the application of rules r1; . . . ; ri�1.)

x :¼ maximum copies of rule ri ¼ ðj;u=v ; kÞ applicable to cells j; kwith contentCell1 and contentCell2, respectively;

remove x copies of u from contentCell1;

add x copies of rule ri to rulesAppliedTotal;

if one of the cells j or k with their respective indices is identicalto cell h with index i1i2 . . . in�1 then

add x copies of rule ri to rulesAppliedToh;

end cycle; (cell k with index k1k2 . . . kn�1)end cycle; (cell j with index j1j2 . . . jn�1)

end cycle; (rule ri)(Now rulesAppliedToh contains the rules applied in step n to cell h withindex i1i2 . . . in�1.)

contentFinal = content(h; i1i2 . . . in�1;n� 1);(The content of cell h with index i1i2 . . . in�1 in the previous configuration Cn�1.)

if contentFinal = null then return null and exit;

if a separation rule ½a�h ! ½C1�h½C2�h exists such that a 2 contentFinal then

if in ¼ 1 then remove a and every b R C2 from contentFinal

else remove a and every b R C1 from contentFinal;(Cell h with index i1i2 . . . in�1 separates in step n)

else

if in = 2 then return null and exit;

(The last element in of compound index corresponds to a copy of cell hseparating in step n which is not the case, hence this copy does not exist.)

else apply all the rules in rulesAppliedToh to contentFinal, i.e.,add/remove the multisets of objects entering/leaving cell h to/fromcontentFinal;

return contentFinal;

3.1. Example of the simulation

Consider the tissue P system with cell separation P described in Example 1. Let us compute the values of content (1,1,1)and content (1,2,1). Recall that the first parameter is the label of a cell, the second parameter is the compound index andthe third parameter is the number of configuration.




1. The computation of content (1,1,1) starts by initializing the values of multisets rulesAppliedToh andrulesAppliedTotal to ;. Then, for all communication rules we count the number of their parallel application. Thisnumber depends on the contents of cells in the previous configuration which are calculated recursively. We add thenumbers of applications of each rule to the multiset rulesAppliedTotal and, if the rule involves cell labeled 1 withcompound index 1, then also to the multiset rulesAppliedToh as h ¼ 1.The rule r1 ¼ ð1; y=x;2Þ – the variables cell1 and cell2 adopt the values 1 and 2, respectively. Because it is the firstexamined rule, the sets rulesForCell1 and rulesForCell2 are empty. The next step is to compute the content ofcell1 (and later cell2) in the previous configuration.

contentCell1 :¼ contentð1; k;0Þ ¼ xy,contentCell2 :¼ contentð2; k;0Þ ¼ xxz.

One copy of rule r1 is applicable to cells cell1 and cell2. We add ðr1;1Þ to both multisets rulesAppliedTotal andrulesAppliedToh since h ¼ 1 and cell 1 with compound index 1 is involved in the rule r1.As there are no more cells labeled 1 or 2 with different compound indices in the previous configuration, we proceed to thenext step. Because there is no other communication rule, we can continue with applying rules from the variablerulesAppliedToh.At this moment, the variable rulesAppliedToh contains the complete multiset of communication rules applied in step 1 tocell 1 with index 1. Now, we again compute the content of cell 1 in the previous configuration:

contentFinal: = content ð1; k;0Þ ¼ xy.

No separation rule is applicable to cell 1 and the last digit of its compound index is 1. So, we apply all rules in rules-

AppliedToh to contentFinal and we obtain the result contentð1;1;1Þ ¼ xx.2. In the case of the function content(1,2,1), the computation proceeds analogously. In the last phase, no separation rule is

applicable to the cell labeled 1. As the last digit of compound index is 2 (the cell is created by separation in present step)the result is content(1,2,1) = null.

3.2. Formal verification

Theorem 3. Let P ¼ ðC;C1;C2;R; E;M1; . . . ;Mq;R; iin; ioutÞ be a recognizer confluent tissue P system with cell separation and letC0;C1; . . .be a computation. Let h be a label of a cell in the initial configuration of P and let i1 . . . it 2 f1;2gt be a compound index,for a t 2 N.

� If cell hi1 ...it exists in configuration Ct of the system P, then the function contentðh; i1 . . . it ; tÞ returns the multiset of objects inthis cell in configuration Ct.� Otherwise, the function contentðh; i1 . . . it ; tÞ returns null.

Proof. We prove correctness of the recursive function content by induction on t.

� Let t ¼ 0. The content of a given cell in the initial configuration is returned correctly in the first step of computation.� Let the Theorem hold for some t P 0 (the induction hypothesis I.H.1). We show that it holds also for t þ 1. Consider the

function content with parameters h; i1 . . . ititþ1 and t þ 1. The following claims hold true:

1. As n ¼ t þ 1, the condition ‘‘if n ¼ 0 then. . .’’ is false.2. The algorithm sets rulesAppliedToh = rulesAppliedTotal = ;.3. In the next phase the algorithm computes the multiset of communication rules applicable in the configuration Ct . For all

communication rules ri 2 R;1 6 i 6 m, the algorithm examines how many copies of the rule are applicable in the whole Psystem and particularly to the cell hi1 ...it .We prove that after i iterations of the cycle for rules r1; r2; . . . ; ri;0 6 i 6 m, the following holds:(i) the variable rulesAppliedTotal contains the total multiset of communication rules r1; . . . ; ri applied in all cells at

step t þ 1;(ii) the variable rulesAppliedToh contains the total multiset of communication rules r1; . . . ; ri applied to the cell hi1 ...it

at step t þ 1.The proof is constructed by induction on the number of rule i.� Let i ¼ 0 – the statement certainly holds as no rule has been examined yet; the variables rulesAppliedToh and

rulesAppliedTotal contain an empty multiset.� Let the statement hold for an i;0 6 i 6 m� 1 (the induction hypothesis I.H.2). We prove that the statement also holds

for iþ 1. Let riþ1 ¼ ðj;u=v ; kÞ be the (i + 1)th communication rule in R. The algorithm proceeds as follows:




(a) The variable rulesForCell1 has assigned the total multiset of rules r1; . . . ; ri already applied at step t þ 1 which is,by the I.H.2, stored in the variable rulesAppliedTotal.

(b) Steps 3c–3j are iterated through all cells with label j and different compound indices j1 . . . jt .(c) The content of a processed cell in configuration Ct is computed recursively, with a correct result by the I.H.1. If the

obtained value is null or a separation rule is applicable to the content of the cell, no communication rules can beapplied and we proceed to the next index j01 � � � j

0t .

(d) The rules r1; . . . ; ri stored in rulesForCell1 are now scanned in this order for a maximal multiset of each of themapplicable to contentCell1. The total multiset M of applicable rules is removed from rulesForCell1 and theobjects consumed by these rules are removed from contentCell1.

(e) The variable rulesForCell2 has assigned the total multiset of rules r1; . . . ; ri already applied at step t þ 1 which is,by the I.H.2, stored in the variable rulesAppliedTotal.

(f) Next, steps 3g–3j are iterated through all cells with label k and different compound indices k1 . . . kt .(g) The same operations as in steps 3c–3d are now applied to variables rulesForCell2 and contentCell2.(h) Since the rules in R and the cells of the system are always iterated in the same order, the variables contentCell1

and contentCell2 now contain the content of cells jj1 ...jtand kk1 ...kt , respectively, after the application of rules

r1; . . . ; ri at step t þ 1.(i) At this moment the application of rule riþ1 ¼ ðj;u=v ; kÞ to this pair of cells can be simulated. The multiset with max-

imum applicable copies of this rule is added to the variable rulesAppliedTotal. Since this step is iterated over allpairs of cells labeled j; k, the first part of the statement of I.H.2 is proved.

(j) If the cell h with the compound index i1 . . . it is identical to one of the processed cells j; k, with their respective indices,we add the same multiset of rules to the variable rulesAppliedToh. Since this step is iterated over all pairs of cellslabeled j; k, the second part of the statement of I.H.2 is proved, too.

We have proved that now the variable rulesAppliedToh contains the multiset of all rules applied in step t þ 1 to cell hwith compound index i1 . . . it .4. The content of cell labeled h with index i1 . . . it is calculated recursively and stored to the variable contentFinal. The

result is correct by the I.H.1.5. If the value of contentFinal is null, cell h with compound index i1 . . . it does not exist in configuration Ct . Therefore, the

cell with index i1 . . . it itþ1 cannot exist in configuration Ctþ1 and function content correctly returns value null.6. If a separation rule a½ �h ! ½C1�h½C2�h is applicable to cell hi1 ...it , then we assign it a priority over communication rules. The

function content simulates its application and returns the result.7. If no separation rule is applicable and itþ1 ¼ 2, then the cell hi1 ...itþ1 could exist only as a result of separation at step t þ 1.

Since no such separation is possible, the cell cannot exist and the function content returns null.8. If none of the previously listed cases occurs, the result is correctly computed as an application of the multiset of rules in

rulesApplicableToh to the objects in contentFinal.

The above analysis proves that the function content returns a correct result for any combination of its valid inputparameters. �

3.3. Space complexity of the simulation

Theorem 4. A result of any computation consisting of n steps of a recognizer confluent tissue P system with cell separation can becomputed by Turing machine in polynomial space associated to n.

Proof. Consider a recognizer confluent tissue P system with cell separation

Pleasehttp:/

P ¼ ðC;C1;C2;R; E;M1; . . . ;Mq;R; iin; ioutÞ:

The function content described above evaluates the content of a particular cell (including the environment treated as a vir-tual cell) after n steps. By Definition 2, the system always halts in a configuration when the object yes or no is present in theenvironment. The result of computation of P with an input w can be therefore obtained as follows:

1. Prepare the initial configuration of P, add w to Miin .2. Subsequently compute content(iout;11 . . . 1;n) for n ¼ 0;1;2; . . .. until the presence of objects yes or no in the

environment.3. Return the corresponding result of computation.

Space complexity of computation of the function contentðh; index;nÞ is determined by the space occupied by variablesstoring multisets of objects and multisets of applicable rules. We focus first on the variables storing multisets of objectscontained in the cells. Cardinality of each such multiset is bounded by the total number of objects in the system after n steps.Denote this number by on. By definition, we have




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o0 ¼Xq

i¼1

cardðMiÞ þ jwj: ð1Þ

Note that new objects can be introduced to the system only via antiport rules from the environment. Let Rcom be the set ofcommunication rules in R, and let

c ¼ maxði;u=v;jÞ2Rcom

fjuj; jv jg; ð2Þ

be the maximum number of objects possibly imported to the system by the application of one rule. As the number of appli-cable antiport rules is bounded by the number of objects already present in the system, we obtain

on 6 c � on�1; n P 1: ð3Þ

Hence, we have on 6 o0cn, which is a value represented by d � n bits for a constant

d 6 log o0 þ log c: ð4Þ

Finally, jCjd � n bits are necessary to describe any multiset with cardinality d � n and with the underlying set C. This is also themaximum size of any variable of this type.

The situation is similar when considering the size of variables storing multisets of applicable rules. The number of copiesof any rule applicable in a configuration Cn is limited by the number on of objects in the system. Hence the space required tostore the total multiset of applicable rules in a configuration Cn is at most jRjd � n.

Finally, recall that the function content with parameter n performs recursive calls of itself with parameter n� 1. It usesthree variables storing multisets of objects and four variables with multisets of rules. For its space complexity CðnÞ we cantherefore write:

Cð0Þ ¼ log o0 ð5ÞCðnÞ 6 Cðn� 1Þ þ 3jCjd � nþ 4jRjd � n; n P 1: ð6Þ

The solution to this recurrence is

CðnÞ ¼ OððjCj þ jRjÞd � n2 þ log o0Þ: ð7Þ

Hence, with the aid of the function content described above, a conventional computer can simulate n steps of computationof the systems P in space polynomial to n, and as the space necessary for Turing machine performing the same computationis asymptotically the same, the statement follows. �

Theorem 5. PMCTSC # PSPACE

Proof. Consider a family P ¼ fPðnÞjn 2 Ng of recognizer tissue P systems with cell separation satisfying conditions of Def-inition 3, which solves in a uniform way and polynomial time a decision problem X ¼ ðIX ; hXÞ. For each instance u 2 IX , denote

PðsðuÞÞ ¼ ðC;C1;C2;R; E;M1; . . . ;Mq;R; iin; ioutÞ;

and let w ¼ codðuÞ be the corresponding input multiset. By Definition 3, paragraphs 1 and 2(a), the values ofcardðwÞ; cardðM1Þ; . . . ; cardðMqÞ, and lengths of rules in R are exponential with respect to juj (they must be constructedby a deterministic Turing machine in polynomial time). Furthermore, values of jCj and jRj are polynomial to juj. (Actually,only polynomially many of the elements of the alphabet C could appear in the rules of system PðsðuÞÞ and the rest could beignored.)

By Definition 3, paragraph 2(c), also the number of steps n of any computation of system PðsðuÞÞ is polynomial to juj. ByTheorem 4, the computation of system PðsðuÞÞ can be simulated with Turing machine in space polynomial to n using thefunction content. Its space complexity is described by the Eq. (7) containing constants d and o0. By (1)–(4), the value of bothd and log o0 is polynomial to juj and, hence, so is the space complexity of function content. Therefore, each instance u 2 IX

can be solved with a Turing machine in space polynomial to juj. �

4. Discussion

We have investigated the properties of certain cellular processes as the controlled transport of molecules through mem-branes and cell separation. In order to relate them with standard computer science methodology, they were formalized inthe framework of tissue P system with cell separation, a discrete, multiset-based mathematical model. While it was previ-ously shown that this model can characterize various complexity classes and their borderlines, as well as to solve efficientlyintractable (NP-complete) problems, here we have proved that the class PSPACE imposes an upper bound of the power of‘‘reasonably’’ restricted families of tissue P systems with cell separation. These restrictions include polynomial time compu-tations, polynomial uniformity and confluent behavior of the families of P systems and their members, respectively.




Note that the achieved upper bound by PSPACE applies trivially also to the original definition of tissue P systems withoutcell separation in [11,12]. An exact characterization of the power of tissue P systems with cell separation remains open as weonly know that the class of problems they can solve in polynomial time contains NP [ co�NP. Some recent results [17,18]with a related (but different) model of P systems with active cells suggest that it might be possible to solve in polynomialtime all the problems in PP or in the polynomial hierarchy.

Another open problem is the power of families of non-confluent (hence non-deterministic) tissue P systems with cellseparation. We cannot adopt simply our proof technique to this case just by using a non-deterministic Turing (or anotherequivalent) machine for simulation. Observe that in our recursive algorithm the same configuration of a P system is typicallyre-calculated many times during one simulation run. If the simulation was non-deterministic, we could obtain differentresults for the same configuration which would make the simulation inconsistent.

Acknowledgements

This work was supported by the European Regional Development Fund in the IT4Innovations Centre of Excellence Project(CZ.1.05/1.1.00/02.0070), and by the Ministerio de Ciencia e Innovación, Spain, under Project TIN2012-36992.

References

[1] A. Alhazov, R. Freund, M. Oswald, Tissue P systems with antiport rules and small numbers of symbols and cells, in: C. De Felice, A. Restivo (Eds.),Developments in Language Theory, Lecture Notes in Computer Science, Vol. 3572, Springer, Berlin, 2005, pp. 54–78.

[2] F. Bernardini, M. Gheorghe, Cell communication in tissue P systems and cell division in population P systems, Soft Comput. 9 (9) (2005) 640–649.[3] S. Cheruku, A. Paun, F. Romero-Campero, M. Pérez-Jiménez, O. Ibarra, Simulating FAS-induced apoptosis by using P systems, Prog. Nat. Sci. 17 (4)

(2007) 424–431.[4] R. Freund, G. Paun, M. Pérez-Jiménez, Tissue P systems with channel states, Theor. Comput. Sci. 330 (2005) 101–116.[5] C. García-Martínez, C. Lima, J. Twycross, N. Krasnogor, M. Lozano, P system model optimization by means of evolutionary based search algorithms, in:

Proceedings of the 12th Annual Conference on Genetic and Evolutionary Computation, GECCO ’10, ACM, New York, NY, USA, 2010, pp. 187–194.[6] J. Jack, A. Paun, Discrete modeling of biochemical signaling with memory enhancement, Trans. Comput. Syst. Biol. 11 (2009) 200–215.[7] J. Jack, A. Paun, A. Rodríguez-Patón, Effects of HIV-1 proteins on the Fas-mediated apoptotic signaling cascade: a computational study of latent T cell

activation CD4+, in: D.W. Corne, P. Frisco, G. Paun, G. Rozenberg, A. Salomaa (Eds.), Membrane Computing: 9th International Workshop, Lecture Notesin Computer Science, vol. 5391, Springer, 2009, pp. 246–259.

[8] S. Krishna, K. Lakshmanan, R. Rama, Tissue P systems with contextual and rewriting rules, in: G. Paun, G. Rozenberg, A. Salomaa, C. Zandron (Eds.),Membrane Computing, Lecture Notes in Computer Science, vol. 2597, Springer, Berlin, 2003, pp. 339–351.

[9] K. Lakshmanan, R. Rama, The computational efficiency of insertion deletion tissue P systems, in: K. Subramanian, K. Rangarajan, M. Mukund (Eds.),Formal Models, Languages and Applications, World Scientific, 2006, pp. 235–245.

[10] L.F. Macías-Ramos, M.J. Pérez-Jiménez, A. Riscos-Núñez, M. Rius-Font, L. Valencia-Cabrera, The efficiency of tissue P systems with cell separation relieson the environment, in: E. Csuhaj-Varjú, M. Gheorghe, G. Rozenberg, A. Salomaa, G. Vaszil (Eds.), Int. Conf. on Membrane Computing, Lecture Notes inComputer Science, vol. 7762, Springer, 2013, pp. 243–256.

[11] C. Martín Vide, J. Pazos, G. Paun, A. Rodríguez Patón, A new class of symbolic abstract neural nets: tissue P systems, in: O. Ibarra, L. Zhang (Eds.),Computing and Combinatorics, Lecture Notes in Computer Science, vol. 2387, Springer, Berlin, 2002, pp. 573–679.

[12] C. Martín Vide, J. Pazos, G. Paun, A. Rodríguez Patón, Tissue P systems, Theor. Comput. Sci. 296 (2003) 295–326.[13] L. Pan, T.-O. Ishdorj, P systems with active membranes and separation rules, J. Univ. Comput. Sci. 10 (5) (2004) 630–649.[14] L. Pan, M. Pérez-Jiménez, Computational complexity of tissue–like P systems, J. Complex. 26 (3) (2010) 296–315.[15] L. Pan, M.J. Pérez-Jiménez, A. Riscos-Núñez, M. Rius-Font, New frontiers of the efficiency in tissue P systems, in: L. Pan, G. Paun, T. Song (Eds.),

Preproceedings of Asian Conference on Membrane Computing, Huazhong University of Science and Technology, Wuhan, China, 2012, pp. 61–73.[16] M. Pérez-Jiménez, P. Sosík, Improving the efficiency of tissue P systems with cell separation, in: M. Martínez-del-Amor, G. Paun, I. Pérez-Hurtado, F.

Romero-Campero (Eds.), Proceedings of the Tenth Brainstorming Week on Membrane Computing, Fénix Editora, Sevilla, 2012, pp. 105–140.[17] A. Porreca, A. Leporati, G. Mauri, C. Zandron, Elementary active membranes have the power of counting, Int. J. Nat. Comput. Res. 2 (3) (2011) 35–48.[18] A. Porreca, A. Leporati, G. Mauri, C. Zandron, P systems with elementary active membranes: beyond NP and coNP, in: M. Gheorghe, T. Hinze, G. Paun,

G. Rozenberg, A. Salomaa (Eds.), Int. Conf. on Membrane Computing, Lecture Notes in Computer Science, vol. 6501, Springer, 2011, pp. 383–392.[19] A. Paun, G. Paun, The power of communication: P systems with symport/antiport, New Gener. Comput. 20 (3) (2002) 295–306.[20] G. Paun, G. Rozenberg, A. Salomaa (Eds.), The Oxford Handbook of Membrane Computing, Oxford University Press, Oxford, 2010.[21] G. Paun, M. Pérez-Jiménez, A. Riscos-Núñez, Tissue P systems with cell division, Int. J. Comput. Commun. Control 3 (3) (2008) 295–303.[22] F.J. Romero-Campero, M.J. Pérez-Jiménez, A model of the quorum sensing system in vibrio fischeri using P systems, Artif. Life 14 (1) (2008) 95–109.[23] S. Sedwards, T. Mazza, Cyto-Sim: a formal language model and stochastic simulator of membrane-enclosed biochemical processes, Bioinformatics 23

(20) (2007) 2800–2802.[24] P. Sosík, L. Cienciala, Tissue P systems with cell separation: upper bound by PSPACE, in: A.-H. Dediu, C. Martín-Vide, B. Truthe (Eds.), Theory and

Practice of Natural Computing, TPNC 2012, Lecture Notes in Computer Science, vol. 7505, Springer, 2012, pp. 201–215.[25] G. Zhang, J. Cheng, M. Gheorghe, Q. Meng, A hybrid approach based on differential evolution and tissue membrane systems for solving constrained

manufacturing parameter optimization problems, Appl. Soft Comput. 13 (3) (2013) 1528–1542.[26] G. Zhang, H. Rong, M. Gheorghe, J. Cheng, F. Ipate, R. Lefticaru, A particle swarm optimization based on P systems, in: S. Yue, H.-L. Wei, L. Wang, Y. Song

(Eds.), Sixth International Conference on Natural Computation, ICNC2010, IEEE, Yantai, 2010, pp. 3003–3007.[27] X. Zhang, S. Wang, Y. Niu, L. Pan, Tissue P systems with cell separation: attacking the partition problem, Sci. China Inf. Sci. 54 (2) (2011) 293–304.[28] X. Zhang, X. Zeng, B. Luo, Z. Zhang, A uniform solution to the independent set problem through tissue P systems with cell separation, Front. Comput.

Sci. 6 (4) (2012) 477–488.[29] Y. Zhang, L. Huang, A variant of P systems for optimization, Neurocomputing 72 (4–6) (2009) 1355–1360.


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Computational power of cell separation in tissue P systems