Parallel communicating extended finite automata systems communicating by states
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Parallel communicating extended finite automata systems communicating by states Document Transcript

  • 1. International Journal of Computer Engineering (IJCET), ISSN 0976 – 6367(Print), International Journal of Computer Engineering and Technologyand Technology (IJCET), ISSN 0976 1, May - June (2010), © IAEME ISSN 0976 – 6375(Online) Volume 1, NumberISSN 0976 – 6375(Online) Volume 1 – 6367(print) IJCETNumber 1, May - June (2010), pp. 166-179 ©IAEME© IAEME, http://www.iaeme.com/ijcet.html PARALLEL COMMUNICATING EXTENDED FINITE AUTOMATA SYSTEMS COMMUNICATING BY STATES M.Ramakrishnan Department of Computer Science and Engineering Anna University of Technology, Coimbatore Email: ramkrishod@gmail.com S.Balasubramanian Director IPR Anna University of Technology, Coimbatore E-mail: s_balasubramanian@rediffmail.comABSTRACT In this paper, parallel communicating extended finite automata is introduced.Several extended finite automata are working in parallel and communicate each other byrequest. We investigate the computational power of these systems. We have proved thatrecursively enumerable languages and non context free languages are accepted byparallel communicating extended finite automata systems over K and this system is morepower than the existing systems.Keywords: Extended finite automata, multihead automata, parallel computation.1. INTRODUCTION A parallel computer is a collection of processing elements that communicate andcooperate to solve large problems fast parallel architectures will play an increasinglycentral role in information processing. In the commercial world, all of the major databasevendors support parallel machines for their high end products Several major databasevendors also offer shared nothing versions for large parallel machines and collections ofworkstations on a fast network often called clusters. Finite state machines (finiteautomata) are the formal systems for solving many tasks in computer science.Multiprocessor automata system consists of several finite automata, calledprocessors[1],which are coordinated by a central processing unit and it decides which 166
  • 2. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEprocessor is to become active or frozen at a given steps. The processors worksindependently from the other ones based on the internal transition function whichdepends on the internal state and current input symbols. The states achieved by theprocessors depend on their current input symbol and current state. Parallelcommunicating finite automata systems are finite collections of automata workingindependently but communicating their states to each other by request [12].Twoessentially different architectures, depending on the protocols of cooperating andcommunication among the components, have been studied[4] in the case of cooperatingdistributed grammar systems the cooperation is done by means of the sentential form;components may rewrite, in turn, the sentential form according to their own strategies.When a component is active, all the other are inactive. Quite different is the cooperationin parallel communicating (PC) grammar systems[3],[2] where the components work inparallel on their own sentential forms, and form time to time some components ask, bymeans of query symbols for the work of other ones. The contacted components have tosend their current work to those components which asked for it. The idea of consideringseveral automata which cooperate in the aim of recognizing a word, following differentstrategies, can be found in many papers though it is not explicitly asserted. We mentionhere some of them parallel communicating automata systems [5],[6],[10],or cooperatingmulti-stack pushdown automata[7]. Systems of finite automata work in parallel on thesame input tape and communicate with each other by states, in order to recognize theword placed on the common input tape [9]. These systems have components whichcommunicate with each other under similar protocols to those considered for parallelcommunicating grammar systems mentioned above[8]. Every component is entitled torequest the state of any other component; the contacted component communicates itscurrent state and either remains in the same state (in the case of the non-returningstrategy) or enters again the initial state (in the case of the returning strategy). Incentralized systems only one component (the master of the system) is allowed to ask astate form the other. We want to stress the each step in an automata system is either ausual accepting step or a communication step; moreover, the communication steps havepriority to the accepting ones. We also mention that whenever a component requests astate, the state must be communicated. The extended finite automaton is a generalization 167
  • 3. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEof the traditional finite automata model. The extended finite automata model can beviewed as a compact representation of a representation of a mechanism where the dataregisters are modeled in the state transitions. This model retains many advantages of thefinite automata model while overcoming the major limitation of the traditional model. In this paper, we introduced parallel communicating extended finite automatasystems and extend the concepts of parallelism and communication from the grammarsystems area to extended finite automata systems. The new model we propose in thispaper is based on a different view to computation, that is, it makes use of cooperation andcommunication. A parallel communicating extended finite automata system is atranslating device based on communication between extended finite automata working inparallel. It consists of several extended finite automata working independently butcommunicating with each other by request. The strategy of cooperation of finite automatasystems is modified for extended finite automata systems:. This proposed modelincreases the computational power of the components by cooperation and communicationto decrease the complexity of the different tasks by distribution and parallelism than theexisting moles. The transition function is differing from the existing models. That is thetransition function of each automaton depends on the input word and it changes thecurrent state to new state and read head red the word on the input tape and writes in theregister. In this paper we used the definition of extended finite automat system over thegroup K [11] .The working strategy is similar to that of parallel communicatinggrammar systems mentioned above.2. PRELIMINARIES An alphabet is a finite nonempty set of symbols. The set of all words over analphabet V is denoted by V∗. The empty word is written as ε and, V+ = V∗ - {ε}.For afinite set A, we denote by card (A) the cardinality of A. Let K = (M, ·, e) be a group under the operation denoted by ·with the neutralelement denoted by e. 168
  • 4. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME Motivation of this paper is the amount of memory required is not much longerthan the generalized finite automata systems. So we introduced parallel communicatingautomata systems and these automata read word instead of symbols.DEFINITION 2.1 A parallel communicating extended finite automata system of degree n is an (n+4) tuple, A = (V, A1, A2, , , , An, K,Z ) Where V is the input alphabets, and Ai = (Qi, V, fi, Fi), 1≤ i ≤n, are extendedfinite automata with the set of states Qi ,fi is the transition function form Q i × (V ∪{ є} )→ 2Qi× Mi This sort automaton i can be viewed as a finite automaton i having acounter in which any element of Mi can be stored. The relation (si,mi) ∈fi (si, ai ), qi, si ∈Qi , ai ∈ V ∪{ε },mi ∈Mi means that ithautomaton Ai changes the current state qi into si ,by reading the input symbol ai in theinput tape and writes in the register xi·mi, where xi is the old content of the register. The initial value of the ith register is ei. We shall write ( qi, aiw,mi) ├ ( si, aiw,mi ·ri) iff (si, ri) ∈ fi (si, ai ) Where Qi ,1≤ i ≤ n are not necessarily disjoint sets and K = {K1, K2,…,Kn } ⊆Uni =1Qi is the set of query states. A1, A2,…, An are called the components of the extendfinite automata system A. The system A is said to be centralized if K ⊆ Qi, the master of this system beingthe component i whenever a system is centralized, the first component of A is its master. The system A is said to be deterministic if the following conditions are satisfied (i) │ fi i(s,a,)│ ≤ 1 for all s ∈Qi , a ∈V ∪ {ε } (ii) If │fi (s, ε)│ ≠ 0 for some s ∈ Qi, then │fi (s,a)│ = 0 for all a ∈V, hold for all 1≤ i ≤ n,. 169
  • 5. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEDEFINITIONS 2.2 Configuration of a parallel communicating extended finite automata system isdefined as a 3n-tuple(s1,x1,e1, s2, x2,e2,… ,sn, xn , en ) Where si ∈Qi is the current state ofthe component i . xi ∈ V* is the remaining part of the input word which has not been read yet by thecomponent i, ei the register element of Mi.. We define the set of all configurations of A in the following way (s1,x1, m1,s2,x2,m2 , …,sn,xn , mn) ├ r (p1,y1, m1 ·r2 , p2, y2 , m2 ·r2......pn,yn , mn ·rn) if and only if one of the following two conditions must satisfied (i) K∩ { s1,s2,…….. sn}=0 and xi = aiyi, ai ∈ V ∪ {∈}, pi∈fi (si,ai), 1 ≤ i ≤ n, (ii) For all 1 ≤ i ≤ n, such that si = Kji and sji ∉K we put pi = sji, and pj = qji ,pr = sr, for all the other 1 ≤ i ≤ n, and yt = xt. 1 ≤ i ≤ n, and (s1,x1, m1, s2,x2,m2 , …,sn,xn , mn) ├ r (p1,y1, m1 ·r2 , p2, y2 , m2 ·r2......pn,yn , mn ·rn) if and only if one of the following two conditions must satisfied (i) K∩ { s1, s2,…….. Sn}=0 and xi = ai yi , ai ∈ V ∪ {∈}, pi∈fi (si,ai), 1 ≤ i ≤ n, (iii) For all 1 ≤ i ≤ n, such that si = Kji and sji ∉K we put pi = sji, pji = qji, pr=sr for all the other 1 ≤ i ≤ n, and yt = xt. 1 ≤ i ≤ n, From the above equations when the current states of some components are querystates these components enter into communication with those components which areidentified by the appearing quary states. The component identified by the query state isforced to send its current state to the requesting one, supposing that it is not a query state,and this state becomes the new current state of the receiver component. Note that PCEFSwith moves based only on the relation ├r is said to be returning, PCEFS with moves 170
  • 6. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEbased only on the relation├ is called non returning. We used the following notation ├and ├r by ├*and ├r* for reflexive and transitive closure in returning and non retuningsystems if A is a non returning communication, then ET(i)A(x) = { (s1,x1, m1, s2,x2,m2 , …,sn,xn , mn) ├* (p1,y1, m1 ·r2 , p2, y2 , m2·r2......pn,yn , mn ·rn), where 1 ≤ i ≤ n, sj ∈ Fj, if A is a returning communication, then ETR(i)A(x) = {yi ∈ U* ( s1,x1, m1, s2,x2,m2 , …,sn,xn , mn) ├r* (p1,y1, m1 ·r2 , p2, y2 , m2 ·r2......pn,yn , mn ·rn), where 1 ≤ i ≤ n, sj ∈ Fj, We define the following. RCPCEFS (n) the class of all retuning centralized parallel communicatingextended finite automata systems of size n; RPCEFS (n) the class of all returning parallel communicating extended finiteautomata systems of size n; CPCEFS (n) the class of all non-returning centralized parallel communicatingextended finite automata systems of size n; PCEFS (n) the class of all non-returning parallel communicating extended finiteautomata systems of size n; RCPCEFS (n) ⊆ RPCEFS (n) and CPCEFS (n) ⊆ PCEFS (n) where n ≥ 1.EXAMPLE Let A = ({a, b, c}, A1, A2, { K1, K2,}, Z), be a non-returning and non-centralizedPCEFS and its transition function of the system is f1(q1,ε ) = (K2 , e1), f2 (q2,a) =( q2 ,e2) f1(q1,a) =( K2 , e1), f2 (q2,b) = (s1 , e2) f1(q2, ε) =(K2,e1), f2 (s1,b) = (s1 ,e2) 171
  • 7. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME f1 (s2, b) = (K2 , e2) , f1 (sf,c) = (qf ,e) f1 (qf,c) =( qf , e) , f2 (s2, ε) = (sf , e) , f2 (sf , ε) = (sf ,e) Hence ET(1)A ({x)}= {anbncn / n ≥ 1}. Therefore a parallel communicating extended finite automata systems of size isable to compute a non-context-free language by reading an input consisting of a word.3. COMPUTATIONAL POWER Parallel communicating extended finite automata systems turn out to be powerfulcomputational devices. Among other things, it can be shown that these systems. Evenwith a very small number of components and with relatively simple input languages overgroup of a word, are able to determine any recursively enumerable language. In the sequel, we define two operations on words and languages useful in outconsiderations concerning the computational power of PCEFS. A homomorphism whicherases some symbols and leaves unchanged the others is said to be a projection. for twodisjoint alphabets V and V`, mapping h: (V ∪V`)* →V* is a projection, since it erasesthe symbols form V .Other reparation is a well-known operation in formal languagetheory and in parallel programming theory, called the shuffle operation. A shuffle of twostrings is an arbitrary interleaving of the substrings of the original strings, like shufflingtwo decks of cards.THEOREM 1 X(n) is included in the class of languages accepted by deterministic n-head finiteautomata for all X(n) is included in the class of languages accepted by n head finiteautomata for all X(n) ∈{RCPCEFS,RPCEFS, CPCEFS,CPCEFS,PCEFS }PROOF: Let X = RPCEFS the other classes of languages are related as similarly. 172
  • 8. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME Let the classes of returning parallel communicating extended finite automatasystem of size n is A ,A = (V, A1, A2….An, K, Z)Ai = (Qi, V, fi, qi, Fi), i∈[1, n] Now we construct the extended n head finite system is as followsA =( ( Q1∪ K ) × ( Q2 ∪K ) × ( Q3∪ K ) ×…… × ( Qn∪ K ) , V, f, (q1, q2,…..qn) , F1× F2×…..×Fn , n ,Z )Where f( ( s1,s2, ….sn), a1,a2,…,an) = { ( p1,p2,….pn ) | ( pi, mi) ∈ f(si,ai), ai ∈ V ∪{ε}if and only if { s1,s2, ….sn} ∩ K = 0f( ( s1,s2, ….sn), ε, ε,…, ε) = { ( p1,p2,….pn ) , (m1,m2,….,mn) }where pi = { sji is not in K , if si = Kj, = {qi, if there exist si =Ki , = {si, otherwise. Clearly that current state of of all multi head extended finite automata encodescurrent states of all extended finite automata systems. Obliviously that the multi head extended finite automata system is equal to thereturning multi head extended automata system.THEOREM 2 X(n) is included in the class of languages accepted by deterministic n-head finiteautomata for all X(n) is included in the class of languages accepted by n head finiteautomata for all X(n) ∈{DRCPCEFS, DRPCEFS, DCPCEFS,DCPCEFS,DPCEFS }PROOF: Obliviously that if A is deterministic retuning parallel communicating extendedautomata system then A is deterministic. 173
  • 9. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMETHEOREM 3 A language is accepted by an n head extended finite automaton iff it belongs toparallel communicating extended finite automata system of degree n.PROOF: Let A = (Q, V,f, q0, F,Z,n) be a n head extended finite automaton. A = (V, A1, A2….An, K, Z) parallel communicating extended finite automata systemof size n and is denoted by PCEFS (n) where for each i , Ai = (Qi, V, fi, qi, Fi) and thetransition function is different from the original automata system is defined earlier. i-1 I Qi = K ∪ Q ∪ (Q x (V ∪ {є}) ) ∪ (Q x (V∪ { є}i ) ∪ X ix YiWhere Xi = {o, i ≤ 2 = {pj: │p∪ Q, 1 < i < i-2},i>2 } Yi = {0, if i = n. = {{Si│i+1 ≤j < n}, if i<n} The transition mapping fi is defined asWhen i=1, f1(p, a)= (p, a, r1), a ∪ V ∪{ є }},r1∪ M1 p ∪ Q, (p, r1) ∪ f1(p, a) f1((p,a), є) ={(s2 , r1)},a∪ V ∪ { є }, r1∪M2 , f1(sj, є)={(sj+1 ,rj ) 2 ≤ j < n-1. f1(Sn, є )={(kn , rj) } From the above equations the first element belong to the state from the set of statesbelongs to Q, either it reads an input word and writes in the register. This state is sent to second element which has required it. The remaining elements arewaiting. When i = 2, f2(p, є)={(K1 , r2)f2( (p, b) ,a )= {(( p, b ,a ), r2 )}, a,b∪ V ∪ { є }, p∪ Q, 174
  • 10. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME (p, r2 ) ∪ f2(p, b) , r2 ∪M2 f2((p,a,b), є) ={(s3 , r2)},a,b ∪ V ∪ { є }, r2 ∪M2 , p, s3 ∪ Q, (s3 , r2) ∪f2((p,a,b), є) f2 (sj, є)={(sj+1 ,rj ) , 3 ≤ j < n-1, f2 (Sn, є) = {(kn , rn) } The second element to the same, all the symbols of a word read by reading headin the current stated words and written in the register and the other elements are waiting.When 2 < i < n, fi(p, є)={(pi , ri ) }fi(pj, є)={(pj+1 ,rj ) }, i+1 ≤ j < n-3,fi( (p, a1,a2, ……an-1) ,a )= {(( p, a1,a2, ……an-1 ,a ,ri )},a,aj∪ V ∪ { є }, ) 1 ≤ j < i-1. p∪ Q,(p, ri ) ∪ fi((p, a1,a2, ……an-1) , a) , ri ∪Mifi (sj, є)={(sj+1 ,rj ) , i+1 ≤ j <n-1,fi (Sn, є )={(kn , rn) }. Proceeding in this way, until the last element receives the states and it encodes thestate of the first element when the process is started and correspondingly the inputsymbols of a word read by read head and write in the register and the remaining elementsare waiting.When i= n, , fn (p, є) = {(pi , rj ) } fn(pj, є)={(pj+1 ,rj ) }, i+1 ≤ j < n-3, fn (pn-2, є)={(Kn-1,,rn-1 ) }, fn( (p, a1,a2, ……an-1) ,a )= {(( p, a1,a2, ……an-1 ,a ,rj )}, a,aj∪ V ∪ { є }, ) 1 ≤ j < i-1. p∪ Q, (p, ri ) ∪ fi((p, a1,a2, ……an-1) , a) , ri ∪Mi 175
  • 11. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME fn( (p, a1,a2, ……an) , є )= {(( p, a1,a2, ……an ,rn )}, rn ∪Mn From the above transition in the n head extended finite automata the last elementsenter a state from the set of all states from Q and it sent to all the other elements at thesame time. This system is similar to a n head extended finite automaton. This implies thatthe n head finite automaton is equal to the returning parallel communicating extendedfinite automata systems of degree n.THEOREM 4 A language is accepted by a deterministic n head extended finite automaton iff itbelongs to DPCEFS (n)PROOF: It is obliviously that is satisfied if A is deterministic.4. PARALLEL COMMUNICATING GRAMMAR SYSTEMSDEFINITION: A parallel communicating grammar system of size n ≥1 is n+3 tuple Γ(n) =(N,K,T,(S1,P1),(S2,P2),….,(Sn,Pn)),Where N,T are two disjoint alphabets, Si, 1≤ i ≤ n are the axioms of the components ofγ, Pi, 1≤ i ≤ n, are finite sets of production rules over N∩T=0, K ={ Q1Q2,….Qn) is theset of query symbol and (Pi,Si) are the components of the system where Moreover,N,T.K are pair wise disjoint. For two n–tuples (x1,x2,…..xn),(y1,y2,….yn),xi, yi ∈ (N∪T)*, 2 ≤ i ≤ n, the derivation in a parallel communicating grammar systemas above is defined as follows (x1, x2,…..xn), ⇒ (y1,y2,….yn) if the following conditions holds no query symbol appears in x1, and then we have a component-wise derivation,xi ⇒ pi yi, 1 ≤ i ≤ n, except in the case when xi ∈T* and then yi = xi In the case of query symbol appearing, a communication step is performed asthese symbols impose Each occurrence of Qj in xi is replaced by xj, supposing that xj does 176
  • 12. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEnot contain any query symbol, and, after that, the component resumes working from itsaxiom. Moreover, the communication has priority over the effective rewriting. A parallel communicating grammar system is said to be centralized if a requestsymbols are introduced by the first component and non centralized otherwise.DEFINITION: The language is generated by a system is defined asL(Γ(n)) = {x∈T* | (S1,S2,…Sn) ⇒* ( x, x1,x2 , ….. , xn} , xi ∈ (N∪T)*, 2 ≤ i ≤ n.DEFINITION 4.1 Let A = (V, A1, A2….An, K, Z) be a centralized parallel communicating extendedfinite automata system of degree n. We can associate with each configuration a numberbetween 1 and n which is 1 if no query symbol appears in the configuration, or 2 ≤ j ≤ nwhere Kj is the only query state in the configuration. That is the state of the mastercomponent is considered configuration. Now we define trace of the parallel communicating extended finite automata system Aof degree n. Trace (A) = { trace (q1, x,e, q2,x,e,….,qn.x,e) ├* (s1, x,e, s2,x,e,….,sn.x,e) and Trace (A) = { trace (q1, x,e, q2,x,e,….,qn.x,e) ├r* (s1, x,e, s2,x,e,….,sn.x,e)where si ∪Fi , i ≤ i≤ nGiven a cpcefs(n) | rcpcefs(n) A we say that trace(A) is the trace languageTHEOREM 5 The system rcpcefs (2) and cpcefs (2) accepting non context free languages buthaving regular trace languages.PROOF: Consider the deterministic cpcefs(3) f1(q,ε,e ) = (s1 , e), f2(q,a,e ) = (r1 , m6), 177
  • 13. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME f3(t1 ,a, m8) = (t1 ,m9), f1(s1,a,e ) = (s2 ,m1), f2(r1,a, m6 ) = (r1 ,e), f3(t1 ,b, e) = (t2 ,m10), f1(s2,ε,m1 ) = (s1 , e), f2(r1, є, e ) = (r2 ,e), f3(t1 ,b, m10) = (t2 ,e), f1(s1 , b, e) = (s3 , m2), f2(r3, є, e ) = (r2 ,e), f3(t2 , ε, e) = (t3 ,e), f1(s3 , b, m2), = (s3 , m3), f2(r1, c, e ) = (r3 ,m7), f3(t3 , c, e) = (t4 , m10), f1(s4, c, m3), = (s4 , m4), f2(r1, c, m7 ) = (r4 ,e), f3(t4 , ε, m10) = (t4 ,e), f1(s4, c, m4), = (s4 , m5), f3(q ,a, e) = (t1 ,m8),where the accepting states are s4, r4 and t4. The parallel communicating extended finiteautomata system recognized the languages {anbncn / n ≥ 1}.5. CONCLUSION AND FURTHER WORK Parallel communicating extended finite automata systems provide moreinteresting problems for further study. Finite automata without any external control havevery limited accepting power. These systems read the input words and write in theregisters and reducing the space and computation time .It is more powerful than thegeneralized automata system. We have proved that it accepts non regular languages andnon context free languages also.6. REFERENCES[1] Buda, A., (1987), Multiprocessor automata, Information processing Letters (4), 257-261.[2] Csuhaj –Varju, E. (1994), Grammar systems: a framework for natural language generation, in G. Paun (ed), Mathematical Aspects of Natural and Formal Language’s World Scientific Publishing, Singapore, pp 63-78. 178
  • 14. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME[3] Csuhaj – Varju, E. and Dassow, J. (1990), On cooperating / distributed grammar systems, Journal of Information processing and Cybernetics 26, 49-63.[4] Csuhaj – Varju, E., Dassow, J., Kelemen, J. and Paun, G. (1994), Grammar Systemss. Agrammatical approach to distribution and cooperation, Gordon and Breach, Amsterdam.[5] Csuhaj-Varju, E., Martin-Vide, C., Mitrana, V. and Vazil, G. (2000), Parallel communicating pushdown automata systems, International Journal of the Foundations of Computer Science II (4), 633-650.[6] Csuhaj-Varju, E. and Vaszil, g. (2002), Parallel communicating grammar systems with incomplete information communication, in A. Kuich and G. rozenberg (eds), Proc. Developments in Language Theory, LNCS 2295, Springer – Verlag, Berlin, pp. 359-368.[7] Dassow, J. and Mitrana, V. (1999), Stack operation un multi – stack pushdown automata, Journal of Computer System Science 58, 611-621.[8] Dassow, J. Paun, G. and Rozenberg, G. (1997), Grammar systems, in Rozenberg and Salomaa (1997).[9] Martin-Vide, C., Mateescu, A. and Mitrana, V. (2002), Parallel finite automata systems Communicating by states, International Journal of the Foundations of Computing Science 13 (5), 733-749.[10] Martin-Vide, C., Mitrana, V. (2001), some undecidable problems for parallel communicating finite automata systems, Information processing Letters 77, (5-6), 239- 245).[11] Victor Mitrrana , Ralf Stiebe , 2001, Extended finite automata over groups, Discrete Applied Mathematics 1008, 287-300.[12] Sathibalan, M., M.K.Krithivasan and M.Madhu2003, Some variants in communication of parallel communicating pushdown automata, J.Automata. Lang. Combinator 8(3): 401-416. 179