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On power of p systems using sequential andparallel rewritingShankara narayanan Krishna a & Raghavan Rama aa Department of Mathematics , Indian Institute of Technology Madras , Chennai,Tamilnadu, 600 036, IndiaPublished online: 19 Mar 2007.

To cite this article: Shankara narayanan Krishna & Raghavan Rama (2001) On power of p systems usingsequential and parallel rewriting, International Journal of Computer Mathematics, 76:3, 317-330, DOI:10.1080/00207160108805028

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ON POWER OF P SYSTEMS USING SEQUENTIAL AND PARALLEL REWRITING

SHANKARA NARAYANAN KRISHNA and RAGHAVAN RAMA*

Department of Mathematics, Indian Institute of Technology Madras, Chennai-36, Tamilnadu, India

(Received 15 December 1999)

A new class of distributed computing models inspired from biology, that of P Systems, was recently introduced by Gh. Pgun. Several variants of P Systems were already shown to be computationally universal, equal in power to Turing Machines. We investigate in this paper the power of computability of P Systems based on rewriting, with cooperation, priorities and external output. It is established that rewriting P Systems with priorities and two membranes is computationally universal, thereby making an improvement in the existing result that RE 2 RP3(Pri). We give a new model in P Systems stressing the importance of parallelism. The power of computability of such models is investigated by comparing them with classic mecha- nisms in L-Systems: TOL, EOL and ETOL Systems.

Keywork Cooperation and priorities; Computationally universal; External output; Mem- branes of variable thickness; Rewriting P Systems

C. R. Categories: F 4.3, D 3.1, F 4.2

0. INTRODUCTION

The P Systems are a class of distributed parallel computing devices of a biochemical inspiration. In this system, one considers a membrane structure consisting of several cell membranes which are hierarchially embedded in the main membrane, called the skin membrane. The membranes delimit regions, where objects are placed. The obtained construct resembles a super- cell system in which objects are placed in regions subject to evolution rules.

*Corresponding author. e-mail: ramar@acer.iitm.ernet.in

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An object can be transformed into other objects, can pass through adjacent membranes and can dissolve the membrane in which it is placed. Such a construct led to a computing device: Start from an initial configuration and evolve until a halting configuration is obtained. (The number of) The objects in a specified output membrane is the result of the computation.

P Systems can be used as a support for a computing device based on any type of objects and any type of evolution rules associated with them [l - 131. The set of objects can evolve in many ways defined by string processing rules: Rewriting (Sequential and Parallel), point mutations, insertion and deletion and so on. Transition P Systems can be interpreted as using no data structure for codifying the information; the numbers are encoded as the cardinality of multisets , hence they are represented in the base one. This can be adequate to a biochemical implementation, but it looks inefficient from a classic point of view. Moreover, in this way we can deal only with problems on numbers not with symbolic computation. That is why we look for representing information by using a data structure of a standard type, strings. Thus transformation in the form of rewriting steps has been con- sidered, as usual in formal language theory. Consequently, the evolution rules are given as rewriting rules. Strings evolving using rewriting and by splicing has been considered in [7] and it is proved that P Systems based on splicing characterize the family of recursively enumerable languages. Also it is established in [7] that rewriting P Systems with priorities and three membranes is computationally universal.

We consider here systematically the four possibilities: P Systems with or without priorities, with or without cooperation. The four hierarchies on the number of membranes used in a system are compared to languages generated by devices in the Chomsky hierarchy. A characterization of re- cursively enumerable languages is also obtained in the case of P Systems based on sequential rewriting. The proof uses the same technique as in [7], but this time, the number of membranes used is still smaller: two.

In a transition P System, a string passes through membranes as a unique entity, its symbols do not follow different itineraries, as it was possible for the objects in a multiset. In a transition P System based on parallel rewriting, a string passes through as a unique entity, and evolves parallely. That is, every symbol in the entity is rewritten whenever it is possible to do so. The rewriting rules are rules of L- systems and hence the entity evolves parallely. We consider the different possibilities of P Systems based on parallel rewrit- ing with or without priorities, with or without cooperation, etc. and compare with TOL, EOL and ETOL language families. We also observe that P Systems using parallel rewriting with cooperation is universally computable.

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P SYSTEMS 319

In the usual P Systems, besides the action of dissolving a membrane, the action of making a membrane thicker has been considered in [8]. Here, all membranes are of thickness one initially. If a rule in a membrane of thickness one introduces the symbol T , then the membrane becomes of thickness two. A membrane of thickness two does not become thicker by using further rules which introduce the symbol r , but no object can enter or exit it. If a rule which introduces 6 is used in a membrane of thickness one, then the membrane is dissolved; if the membrane had thickness two, then it returns to thickness one. If at the same step, one uses rules which introduce both r and 6 in the same membrane, then the membrane does not change its thickness. We prove that systems of variable thickness using context-free rules can characterize RE, which was conjectured in [8].

1. P SYSTEMS BASED ON SEQUENTIAL AND PARALLEL REWRITING WITH EXTERNAL OUTPUT

We refer the reader to [7,8] for basic notions, notations, and results about P Systems; here we directly introduce the class of systems we investigate.

DEFINITION 1.1 A P System based on rewriting is a language generating mechanism H = (V, T, P, WI, ~ 2 , . . . , Wm, (R1, PI), (R2, ~21, . . . , (Rm, P,)) where

(1) m 2 1; (2) V is an alphabet (The total alphabet of the system); (3) T E V (The terminal alphabet); (4) p is a membrane structure; (5) wl, . . . , w, are strings over V, describing the jinite languages over V (6) R,, R2,. . . , R, are jinite sets of developmental rules, of the following

forms:

(a) Context-free evolution rules of the form X+v(tar) where tar E {here, out, in,), XE V, v E V*.

(b) Non Context-free rules of the form u ---t v(tar) of radius greater than one, u, v E V*.

(7) pl, p2, . . . , pm are partial order relations on R1, R2, . . . , R,,

In P Systems based on sequential rewriting, we apply atmost one rule in every step; the parallelism refers to the fact that we are processing simul- taneously all available strings by all applicable rules. If several rules can be applied to a string, at several places each, we take only one rule and

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only one possibility to apply it and consider the obtained string as the next state of the object described by the string.

In P Systems based on parallel rewriting, the parallelism is in two ways: We process simultaneously all strings in all membranes by all applicable rules; also we rewrite every symbol of the string which can be rewritten. Thus several rules are applied to a string; each symbol is processed by one rule at a time; if a symbol occurs at several places of the string and if different rules can be applied to it, then at each position we nondetermin- istically choose a rule and apply it.

We denote by ERP (a, P) the family of languages generated by P Systems based on sequential rewriting with external output, a E {Pri, nPri), P E {Coo, nCoo). If non context-free rules are not used, then the system is said to be non cooperative. Also EPP((r, /3) denotes the family of languages generated by P Systems based on parallel rewriting with external output, a E {Pri, nPri),P E {Coo,nCoo). If atmost m membranes have been used, then we write ERP,(a, P) and EPP,(a, P) for the family of languages generated by P Systems based on sequential and parallel rewriting respectively.

The language generated by the system (both parallel and sequential) is defined as follows: If there are any strings inside the system purely over terminals after a complete computation, they are not listed in the language. Likewise, strings going out of the system consisting of both terminals and nonterminals are also not listed in the language. Only those strings which go out of the system and which are purely over terminals belong to the language.

Proof This result can be proved in exactly the same way as Theorem 2 in [7].

Proof The fact that CF & ERP1(Pri,nCoo) can be proved similarly as above.

To prove the strict inclusion, we consider the following example. Consider the system II = (V, T, [1]1, aBbGc, (R1, pl)), where V= {A, B, F, G, a, b, c, + }, T= {a, b, c).

R1 consists of the following rules:

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r6:F--+F r7:G+ + rg :F+ + r9 :A+ + r lo :B+ + rll: + -+A rl2 : a + a (out).

Then L(n) = {a"bncn : n > 11, which is not Context-free. Initially, we have aBbGc in the system. To get abc, we have to use G+ + first. Then the string becomes aBb+c. Now, the rule B-aA cannot be applied as, + + X has a higher priority. But then, B ---+ + has a higher priority than + +A. Hence, B - + is applied, and then + + A, and, ultimately, a -+ a(out). Now suppose we have a2" A ~ ~ " F c ~ " in membrane 1. If we propose to generate further, the rule F+ bGc has higher priority and so we get a2" ~ b ~ " + 'GC~"+ . Here, the rule A +aB has more priority than G--+ + and G+bFc and so we get a2"+'~b2"+ 'GC~"+ ' . Here either the string can go out as a2n+1b2n+1~2n+1 as discussed above in the case of abc or, again we have to apply G - bFc first to get a 2n + 1 Bb2n + 2FC2n + 2 . Now we can apply only B ---+ aB because of its higher priority than F--+ + and F+ bGc. Also, here we cannot apply B+ + as F+ F has a higher priority. So we get a2"+2~b2n+2~c2n+2 . This string can go out after applying F+ + , A + +, + + X in order in which case, we get a2n+2b2n+2~2n+2. Thus, the language generated by the system is L(II) = {a"bncn : n 2 1).

THEOREM 1.3 RE C ERPl(nPri, Coo).

Proof The computational universality can be proved by considering a type-0 grammar G in Kuroda normal form.

Proof Let G = (N, T, S, F ) be a matrix grammar in binary normal form, N = N1 U N2 u { S , + ), N1 n N2 = 4. Then the rules of F are of the following forms:

(1) (S - XA), X E N1, A E N2. (2) (X+ Y,A+x),X, Y E N1,A E N2,x E (N2UT)*. (3) (X- Y, A --+ +), X, Y E N1, A E N2, and + is a special symbol. (4) (X--+X,A--+x) ,X€ N1,A E N2,x E T*.

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There is only one matrix of type 1, matrices of type 3 correspond to ap- pearance checking rules. The symbol + is a trap symbol, once introduced, it is never removed. All derivations can terminate only by a rule of type 4. Assume that there are k matrices, numbered ml, m2, . . . , mk.

We construct the P System II = (V, T, [1L1211, WI , ~ 2 , (RI, PI), (R2, ~ 2 ) ) where V = N 1 ~ N 2 U { X i , X ~ I l ~ i I k , X ~ N l ) ~ ( A i , A ~ I 1 I i I k , A € N 2 ) ~ { i , i l l l < i < k ) .

wl = XA such that ( S + XA) is a matrix in F, wi = 4, i # 1. The rules are of the following forms:

R1 : r l : {X+ YiImi:(X+ Y,A+x) is of type 2) r2 : {X-Y,'lmi : (X-+Y,A--++) is of type 3) r3:{X-+i11mi:(X+X,A-+x) is of type4) r4:{Yi-+i1YiIY€ N1,l < i l k). r5:{Yi+if1Y:(in 2)1Y€NI, 1 < i l k ) r6:{i+Xl 1 < i 5 k) r7 : {A + Ai (in 2) 1 mi : (X --+ Y, A + x) or (X -+ A, A + x) is a matrix of

type 2 or 4) rg:{A- + Imi:(X+ Y,A-+ +) is a matrix of type 3) r, : {a + a(out) I a E T ) rlo:{+ + +) r, : {X + + I X E N1 and there does not exist any mi containing a rule for X) r12 : {A- + I A E N2 and there does not exist any mi containing a rule

for A). 113: { i l+ i f l l 5 i 5 k)

R2 : rl : {Ai+iA:IA f N2) r2:{if-ill < i 5 k)u{ iN+XI1 5 i < k) r$:{Y;+Y(inl)lY€Nl, 1 < i < k } r3: {Yi+ Y(in 1) I Y E N1) r4:{+ + +) rj : {A:-xIm;:(X+Y,A-+x) is of type 2)U

{A:--+x (in 1) I mi : (X-+X, A+x) is of type 4) U

{A:+ + I mi: (X--+Y,A++) is of type 3), 1 5 i 5 k $ : o - - + I l < j I k )

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P SYSTEMS 323

The system works as follows: Assume that in membrane 1, we have a string of the form Xw, (XE Nl, w E (N2 U T )*). In membrane 1, one simulates matrix mi as follows: First, the nonterminal from N1 is rewritten along with the index i of the matrix, and then the nonterminal from N2 is rewritten and the string goes to membrane 2, in case of a matrix of type 2 or 4. In case of an checking rule, the rule A ---t + is applied and the com- putation never halts.

In membrane 2, we have to rewrite the rule corresponding to Ai,Ai, as this rule has higher priority than r3, which is the only rule to quit the membrane. Ai is first rewritten as iAi and the rule corresponding to A; is applied if there does not exist two different indices i , j in the membrane. That is, A: is re- written if matrix mi has been correctly simulated. If there exists two differ- ent indices in the string, which means the simulation has not been correctly done, the rule rj:j++ is applied and the computation never halts. If the simulation of the matrix is done correctly, the string goes to mem- brane 1 using r3 or ri according as a matrix of type 2 or 4 has been simu- lated. So, if the simulations are done correctly, we end up in membrane one with a terminal string which belongs to L(G). This string can go out using a - a(out), a E T. Hence, the language generated by the system is same as L(G).

Note 1.1 We do not know whether this result is optimal. The problem whether a characterization of RE can be obtained with one membrane is left as an open problem.

2. P SYSTEMS BASED ON PARALLEL REWRITING WITH EXTERNAL OUTPUT

In P Systems based on parallel rewriting and external output , the objects we consider are strings, and all symbols which can be replaced in a step are replaced. Consider a string ala2.. .a, and suppose there are rules ai ---t bi (tar i) where tar i represents some membrane. If all symbols are targeted to different membranes, one membrane is chosen nondeterminis- tically and the string is moved there. Otherwise, if there are k targets occurring in the rules of ai, n,, n2,. . . , nk times each, the string moves to target i provided ni = max{nl, n2, . . . , nk} and assuming that there are no wrong rules. That is, there are no rules Ri of the form a + v with v introducing b (inj) where j is not a membrane placed immediately inside i. Always, the string moves to a target only after all possible symbols are rewritten.

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324 S. N. KRISHNA A N D R. RAMA

Example 2.1 The system II = ({a), bl, a, b), {al, bl, a, bHll1, albl, (R1, 411, R1 = {albl - ab) U {albl --+ ab(out)) U {ab -+ abab) U {ab --, abab(out)) generates the language { ( ~ b ) ~ " 1 n 2 0).

The string albl can either go out as ab or can remain in the system as abab. In the latter case, ab becomes abab. Here, if to one ab, the rule ab --+ abab(out) is applied, and if to one ab, the rule ab -4 abab is applied, then nondeterministically, the string can go out as abababab or remain in the system as abababab. Proceeding like this, the language generated is {(ab12" 1 n 2 0).

THEOREM 2.1 EOL = EPPl(nPri, nCoo).

Proof Let G = (V, T, w, P) be an EOL System. We construct the parallel rewriting P System II = (V, T, w, (R1, 4)), where R1 consists of all rules in P and in addition, the rules {a + x (out) I a ---+ x is in P).

The strings can go out of the system after each step of rewriting. Note that, in every step, every symbol is rewritten. So, the strings generated by II are precisely the same as generated by the EOL System. Conversely, let =(V, T, Lo, (R1, 4)) be a parallel rewriting system of degree one over some alphabet V. Let Lo be the set of strings initially present in the system, and R1 be the set of rules. Then the EOL System G = (V, T, w, Rl \ {a + x(out) 1 a E V, x E V*) U { A +A I A E V has no rule in Rl) U {a ---t x 1 a - x(out) isin R1) U {w - x 1 x E Lo)) generates the same set of strings as L(II). Note here that the set of rules is made complete by add- ing the rules {A --+A) for all those A which do not have any rule in II. Hence, L(G) = L(II).

THEOREM 2.2 EPPl(nPri, nCoo) n TOL # 4.

Proof Consider parallel rewriting P System II = ({a, all, {a, al), []]I, al, (R1, 4)) where R1 : {al + a) U {al + a(out)) U {a ---t aa) U {a --+ aaa) U

{a -- aa(out)} U {a - aaa(out)}. The language generated by II is {ai I i = 2"3", n, m 2 0), which is a TOL-

language.

THEOREM 2.3 EPP1(Pri, nCoo) n ETOL# 4.

Proof Consider the system.

II =({a, b, c, d, a', a", Al, A d ' , A", 4 , B2, C1, C2, C', ,C1', {a, b, c, 4, [dl, wl, (R1, pl)) where wl = {abc, adc, AIB1 C1, A2B2C2).

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The language generated is {akb'ck : l 2 k > 11 U {d"dck : k 2 1 2 11, which is ETOL. Initially, we have the strings abc, AlBl CI, adc, A2B2C2 in the system. The strings abc and adc can go out of the system as such. The rules r7 and rg have more priority than rg. Thls ensures the fact that the number of b's in the string is always greater than or equal to the number of a's and c's on either side. Now we show that Al and C1 generate an equal number of a's and c's respectively. Suppose we have a"AlbnBlcnC1 in the system at some point of time. Suppose we apply Al +A1 to terminate the a's on the leftside of bn. Then since the rule C1 + C has higher priority than Al --+A1 and since Al -d has higher priority than Al +A1, we apply Al + a' getting analbnBlcnC1. The rule B1 --. bB1 can be applied in all steps since, no rule has more priority than it. So, in the above step, applying the rule for b, the string becomes a n d b n + l ~ l c n ~ l . Here, only the rule C1 -+ C can be applied for C1 (and it has to be applied since rewrit- ing is carried out in parallel) as d + a > C1 + cC1. So the string in the above step on rewriting Al, Bl and C1 is a"dbnf 'B~C"C'. Now the rules d - a, C - c and B -+ X can be applied at the next step getting the string a n + l n + l n f l b c . Note that here the termination of b is optional. So, the

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language generated by AIBICl is {d"blck : 1 2 k 2 1). A similar argument with the string A2B2C2 (The priorities r2' > r16, r17 ensure that the number of d's in the string is less than or equal to the number of a's and c's. Also it can be shown that the number of a's and c's are equal) shows that the language generated is {akb'ck : 1 2 k 2 1) U {d"dck : k 2 1 2 1).

Proof Let G be a matrix grammar in binary normal form. Assume that the matrices are numbered from 1 to k. We construct the parallel rewriting P System

wl = XA such that ( S - - XA) is the initial matrix of G w2 = 4.

The rules are of the following forms:

R1 : rl : {X + i' Yi(in 2) 1 mi : (X + Y, A ---+ x) is of type 2 in G) r2 : {X+il'Y;(in 2) 1 mi : (X-Y, A++) is of type 3 in G}. 13 : {X + il(in 2) 1 mi: (X -4 Y, A --+ x) is of type 4 in G) r4 : {a -- a(out) I a E T) r5: {X- + I X E NI and there does not exist any mi containing a rule

for X) rg : { A + + I A E N2 and there does not exist any mi containing a rule for A) r~ : {A --' Ai(in 2) 1 A E N2, mi is a matrix of types 2 ,3 or 4 in G and contains

a rule for A) 4 : {Ai+Ai(in 2))

: {A-+A(in 2) IA E N2) r,:{B-+B(in 2)1BEN2,B#A) r+ : { + - +} pi : { J A > ~ A , ~ A > J ~ , ~ B ~ A # B )

R2 : r l : { i '+ i l l 5 i 5 ~ ) u { ~ " - - + X } U { Y , ' - YIYENI). r2: {Yi+ Yl YEN]) r3:{Y-+ Y(in 1)I Y E N1) ri :{A:+x(in l)lmi : (X+Y,A+x) or (X+X,A--+x) is a matrix of

type 2 or 4 in G ) u {A:+ + (in l)lmi : (X-+Yl A++) is a matrix of type 3 in G ) U { i - + X ( l 5 i 5 k)

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The system works as follows: In the initial configuration, we have the string XA in membrane 1. Then, rules are applied to X and A and, as both targets are the same, viz membrane 2, the string is placed in mem- brane 2. There are two steps of rewriting taking place in membrane two: In the first step, we rewrite Yi, i', Ai and the symbols from N2. As all the targets are the same, the string remains here itself. In the next step, we rewrite i, A:, B' and Y giving priority to i$ over ri. This checks whether any rule has been wrongly applied, that is whether the chosen symbol A E N2 does not correspond to X E N1. If the simulation has been done correctly, the string is placed in membrane 1 since, a majority of the rules (except 5) are targeted to membrane 1. Assume that in membrane 1, we have a string of the form Yw, YE N1, w E (N2 U T )*. The rule corresponding to Y is applied, and also to the symbol A E N2 which corresponds to Y. A11 other symbols of N2 are rewritten using rg, and if the symbol A occurs more than once, it is rewritten using 6 , since JA has higher priority than r~ and, since A can be rewritten in that step. After this step, the string is placed in membrane 2 and the computation proceeds in the manner described above. The rules a - - - a(out), a E T take the string out of the system, and it is listed in the language if it is purely over T*. The priorities ensure that all simu- lations are done correctly and hence, we collect outside the skin membrane exactly the terminal strings generated by the grammar G and hence, L(II) = L(G).

Note 2.1 We do not know whether this result is optimal. Whether a characterization of RE can be obtained using one membrane is left as an open problem.

THEOREM 2.5 RE C EPPl(nPri, Coo).

Proof Let G = (V, T, S, P) be a type-0 grammar in Kuroda normal form. We construct the system II = (VU {#},T,[l]l, #S#, (R1, 4)) where R1 : P U

{#aXb# + ab(out) ( X - X is in P,a, b f T*, X E N ) U {#aXb# + acb (out)IX+cisinP,a,b E T* ,XEN)U{X-- -+XIXE N}.

Initially, membrane 1 has #S#. Then, all rules in P can be applied, and, when the sentential form contains a single nonterminal X, the string can go

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out of the system if there is a derivation in P corresponding to X. Other- wise, the computation never halts, as the rule X-X, X E N can be applied. Clearly, the set of strings generated by ll are the same as those generated by G.

3. P SYSTEMS BASED ON SEQUENTIAL REWRITING, WITH EXTERNAL OUTPUT AND MEMBRANES OF VARIABLE THICKNESS

DEFINITION 3.1 A P System based on sequential rewriting, with external output and membranes of variable thickness is a construct II = (V, T, p, wl, w2,. . . ,wm,R1, RZ, . . . , R,) where:

(1) V is an alphabet (The total alphabet of the system); (2) T 5 V (The terminal alphabet); (3) p is a membrane structure; (4) wl, . . . , w, are finite languages over V, associated with the regions of p; (5) R1, RZ, . . . ,R, are sets of evolution rules for the regions of p. The rules

are of the following forms:

(a) X + v f , X ~ V,vt=v or v6 or V T , v E V . (b) X -+ v(tar), X E V, v E V * and tar E {here, out, in,). (c) U--+vf , IuI > l ,vf=v or v6 or V T , U ,V E V*. (d) u ---t v(tar), 1 u I > 1, u, v E V*, tar E {here, out, in,).

The language generated by the system consists of all strings coming out of the skin membrane, after a complete computation. Everytime, we apply only one rule to a string, if more than one rule can be applied to a string at a time, we nondeterministically choose one rule and apply it. The parallel- ism refers to the fact that we are processing all available strings in the system simultaneously. The way the membrane changes its thickness on introduction of T in the membrane are the same as in the paper.

We denote by ERP(a,P,y), the family of languages generated by P Systems based on sequential rewriting and having membranes of variable thickness, a E {Coo, nCoo), P E {S, nS}, 7 E {r, nr). If rules of types c and d are not used, the system is non cooperating.

Proof Let G be a matrix grammar in binary normal form. Correspond- ing to each rule of the form (X- A, A + x), we consider the rule (X + D, A -+ x), D a new symbol. Let there be k matrices numbered 1 to

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k, of types 2 and 4 n - k matrices of type 3, numbered k + l to n. We construct the system of degree k + 3 l l = (V, T, p, wl , w2, ~ 1 1 , . . . , w k ~ ,

w3, R1, RZ, Rl1,. . . , Rp, R3) with the following components:

V = N 1 U N 2 U { + , D ) U { X i : X ~ N 1 , 1 5 i 5 n)U{i: l 5 i 5 k) P = [1[2[1~11~ . . . ~ ~ ~ ~ ~ 2 [ 3 ~ 3 ~ i wl = XA such that (S -+ XA) is the initial matrix of G. w i ~ = 4, 1 5 i <k,w2=w3=+.

The rules are as follows:

R1 : {X + Xi(in 2) ( mi = (X ---, Y, A + x ) is a matrix of type 2 or 4 con- taining a rule for X) U {X + Xi(in 3) 1 mi = (X --+ Y, A - - + +.) is a matrix of type 3 containing a rule for X) U {a + a(out) : a E T).

Ri, : {Xi+Ylmi is a matrix of type 2 or 4 for which there is a rule X + Y, X E N1) U {A -+ B(out) I mi is a matrix of type 2 or 4 for whlch there exists a ruleofthe form A--.B,AEN~},~ 5 i s k.

R3 : {Xi+ Y 1 mi is a matrix of type 3 having the rule X + Y for X E Nl) U { A + + I mi is a matrix of type 3 having the rule A + + for A€N2)U{+ + +).

Membrane 1 contains XA in the initial configuration. In membrane 2, we simulate rules corresponding to matrices of types 2 and 4. If matrix mi is simulated correctly in membrane i, the string comes out of membrane 2. Likewise, a matrix of type 3 is simulated in membrane 3 and then, the com- putation never halts. If there are symbols from N2 present in membrane 1, and if the corresponding symbol X is not present, the computation can- not proceed. Likewise, if in membrane i, the symbol X in N1 is present, but the corresponding symbol A in N2 is not present, the computation cannot proceed. Even if both X and A are present in membrane i and if X is not rewritten first, the system halts without any fruitful output. This ensures that the simulation of the matrices are done correctly and hence, L(I1) = L(G ).

4. CONCLUSION

We have introduced a new computability model, the P System based on parallel rewriting. The objects are strings, and the evolution is defined

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330 S. N. KRISHNA AND R. RAMA

in terms of string transformations. We have considered rewriting as the underlying operation with strings. Characterizations of RE languages are obtained by simple P Systems with one membrane. Several questions naturally arise in this framework. Of definite interest is to consider what happens if targets are nondeterministically chosen when a string is trans- formed; does this have any significant effect in increasing the power of the system. We can also think of other possible ways to choose the target membrane when a rewriting rule is applied to a string. How this affects the power of the system is a problem worthwhile considering.

Acknowledgement

The authors are much indebted to Gh. P h n , for many useful remarks about previous versions of this paper.

References

[l] Dassow, J. and PBun, Gh. (1999). On the power of membrane computing, J. of Universal Computer Sci., 5(2), 33 - 49 (www.iicm.edu/jucs).

[2] Freund, R., Generalized P systems, Fundamentals of Computation Theory, FCT'99, Iag i, (Ciobanu, G. and PBun, Gh. Eds.), LNCS 1684, Springer, 1999, pp. 281 -292.

[3] Freund, R., Generalized P systems with splicing and cutting/recombination, Workshop on Formal Languages, FCT99, Iag i, 1999.

[4] Krishna, S. N. and Rama, R. (1999). A variant of P systems with active membranes: Solving NP-complete problems, Romanian J. of Information Science and Technology, 2, 4.

[5] Malita, M., Membrane Computing in Prolog, Manuscript, 1999. [6] Mitrana, V., P systems with membrane division, Manuscript, 1999. [q PBun, Gh., Computing with membranes, Journal of Computer and System Sciences, to ap-

pear, and Turku Center for Computer Science, TUCS Report No. 208, 1998 (www.tucs.fi). [8] PBun, Gh., Computing with membranes - A variant: P Systems with Polarized

Membranes, Intern. J. of Foundations of Computer Science, in press, and Auckland University, CDMTCS Report No. 098, 1999 (www.cs.auckland.ac.flz/CDMTCS).

[9] PBun, Gh., P systems with active membranes: Attacking NP complete problems, Sub- mitted, 1999, and Auckland University, CDMTCS Report No. 102,1999 (www.cs.auckland. ac.nz/ CDMTCS).

[lo] PBun, Gh., Rozenberg, G. and Salomaa, A,, Membrane computing with external output, Fundamenta Informaticae, to appear, and Turku Center for Computer Science, TUCS Report No. 218, 1988 (www.tucs.fi).

[l 11 PBun, Gh., Sakakibara, Y. and Yokomori, T., P systems on graphs of restricted forms, ZFZP Con$ on TCS: Exploring New Frontiers of Theoretical Informatics, Sendai, Japan, 2000.

[12] PBun, Gh. and Yokomori, T., Membrane computing based on splicing, Preliminary Proc. of Fifth Intern. Meeting on D N A Based Computers (Winfree, E. and Gifford, D. Eds.), MIT, June, 1999, pp. 213-227.

[13] PBun, Gh. and Yu, S., On synchronizat~on In P systems, Fundamenta Informaticae, 38(4), (1999), 397-410, and University of Western Onrario Report T R 539, 1999 (www.csd. uwo.ca/faculty/syu/TR539.html). D

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