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Parallel communicating extended finite automata systems
1.
International Journal of
Computer Engineering (IJCET), ISSN 0976 – 6367(Print), International Journal of Computer Engineering and Technology and Technology (IJCET), ISSN 0976 1, May - June (2010), © IAEME ISSN 0976 – 6375(Online) Volume 1, Number ISSN 0976 – 6375(Online) Volume 1 – 6367(print) IJCET Number 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.com ABSTRACT In this paper, parallel communicating extended finite automata is introduced. Several extended finite automata are working in parallel and communicate each other by request. We investigate the computational power of these systems. We have proved that recursively enumerable languages and non context free languages are accepted by parallel communicating extended finite automata systems over K and this system is more power 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 and cooperate to solve large problems fast parallel architectures will play an increasingly central role in information processing. In the commercial world, all of the major database vendors support parallel machines for their high end products Several major database vendors also offer shared nothing versions for large parallel machines and collections of workstations on a fast network often called clusters. Finite state machines (finite automata) are the formal systems for solving many tasks in computer science. Multiprocessor automata system consists of several finite automata, called processors[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), © IAEME processor is to become active or frozen at a given steps. The processors works independently from the other ones based on the internal transition function which depends on the internal state and current input symbols. The states achieved by the processors depend on their current input symbol and current state. Parallel communicating finite automata systems are finite collections of automata working independently but communicating their states to each other by request [12].Two essentially different architectures, depending on the protocols of cooperating and communication among the components, have been studied[4] in the case of cooperating distributed 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 cooperation in parallel communicating (PC) grammar systems[3],[2] where the components work in parallel on their own sentential forms, and form time to time some components ask, by means of query symbols for the work of other ones. The contacted components have to send their current work to those components which asked for it. The idea of considering several automata which cooperate in the aim of recognizing a word, following different strategies, can be found in many papers though it is not explicitly asserted. We mention here some of them parallel communicating automata systems [5],[6],[10],or cooperating multi-stack pushdown automata[7]. Systems of finite automata work in parallel on the same input tape and communicate with each other by states, in order to recognize the word placed on the common input tape [9]. These systems have components which communicate with each other under similar protocols to those considered for parallel communicating grammar systems mentioned above[8]. Every component is entitled to request the state of any other component; the contacted component communicates its current state and either remains in the same state (in the case of the non-returning strategy) or enters again the initial state (in the case of the returning strategy). In centralized systems only one component (the master of the system) is allowed to ask a state form the other. We want to stress the each step in an automata system is either a usual accepting step or a communication step; moreover, the communication steps have priority to the accepting ones. We also mention that whenever a component requests a state, 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), © IAEME of the traditional finite automata model. The extended finite automata model can be viewed as a compact representation of a representation of a mechanism where the data registers are modeled in the state transitions. This model retains many advantages of the finite automata model while overcoming the major limitation of the traditional model. In this paper, we introduced parallel communicating extended finite automata systems and extend the concepts of parallelism and communication from the grammar systems area to extended finite automata systems. The new model we propose in this paper is based on a different view to computation, that is, it makes use of cooperation and communication. A parallel communicating extended finite automata system is a translating device based on communication between extended finite automata working in parallel. It consists of several extended finite automata working independently but communicating with each other by request. The strategy of cooperation of finite automata systems is modified for extended finite automata systems:. This proposed model increases the computational power of the components by cooperation and communication to decrease the complexity of the different tasks by distribution and parallelism than the existing moles. The transition function is differing from the existing models. That is the transition function of each automaton depends on the input word and it changes the current state to new state and read head red the word on the input tape and writes in the register. In this paper we used the definition of extended finite automat system over the group K [11] .The working strategy is similar to that of parallel communicating grammar systems mentioned above. 2. PRELIMINARIES An alphabet is a finite nonempty set of symbols. The set of all words over an alphabet V is denoted by V∗. The empty word is written as ε and, V+ = V∗ - {ε}.For a finite 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 neutral element 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 longer than the generalized finite automata systems. So we introduced parallel communicating automata 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 extended finite 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 a counter 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 ith automaton Ai changes the current state qi into si ,by reading the input symbol ai in the input 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 extend finite automata system A. The system A is said to be centralized if K ⊆ Qi, the master of this system being the 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), © IAEME DEFINITIONS 2.2 Configuration of a parallel communicating extended finite automata system is defined as a 3n-tuple(s1,x1,e1, s2, x2,e2,… ,sn, xn , en ) Where si ∈Qi is the current state of the component i . xi ∈ V* is the remaining part of the input word which has not been read yet by the component 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 query states these components enter into communication with those components which are identified by the appearing quary states. The component identified by the query state is forced 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 PCEFS with moves based only on the relation ├r is said to be returning, PCEFS with moves 170
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International Journal of
Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print), ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME based 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 retuning systems 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 communicating extended finite automata systems of size n; RPCEFS (n) the class of all returning parallel communicating extended finite automata systems of size n; CPCEFS (n) the class of all non-returning centralized parallel communicating extended finite automata systems of size n; PCEFS (n) the class of all non-returning parallel communicating extended finite automata 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-centralized PCEFS 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
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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 is able 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 powerful computational devices. Among other things, it can be shown that these systems. Even with a very small number of components and with relatively simple input languages over group of a word, are able to determine any recursively enumerable language. In the sequel, we define two operations on words and languages useful in out considerations concerning the computational power of PCEFS. A homomorphism which erases some symbols and leaves unchanged the others is said to be a projection. for two disjoint alphabets V and V`, mapping h: (V ∪V`)* →V* is a projection, since it erases the symbols form V .Other reparation is a well-known operation in formal language theory and in parallel programming theory, called the shuffle operation. A shuffle of two strings is an arbitrary interleaving of the substrings of the original strings, like shuffling two decks of cards. THEOREM 1 X(n) is included in the class of languages accepted by deterministic n-head finite automata for all X(n) is included in the class of languages accepted by n head finite automata for all X(n) ∈{RCPCEFS,RPCEFS, CPCEFS,CPCEFS,PCEFS } PROOF: Let X = RPCEFS the other classes of languages are related as similarly. 172
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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 automata system 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 follows A =( ( 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 = 0 f( ( 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 encodes current states of all extended finite automata systems. Obliviously that the multi head extended finite automata system is equal to the returning multi head extended automata system. THEOREM 2 X(n) is included in the class of languages accepted by deterministic n-head finite automata for all X(n) is included in the class of languages accepted by n head finite automata for all X(n) ∈{DRCPCEFS, DRPCEFS, DCPCEFS,DCPCEFS,DPCEFS } PROOF: Obliviously that if A is deterministic retuning parallel communicating extended automata system then A is deterministic. 173
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International Journal of
Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print), ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME THEOREM 3 A language is accepted by an n head extended finite automaton iff it belongs to parallel 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 system of size n and is denoted by PCEFS (n) where for each i , Ai = (Qi, V, fi, qi, Fi) and the transition 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 Yi Where 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 as When 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 states belongs 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 are waiting. When i = 2, f2(p, є)={(K1 , r2) f2( (p, b) ,a )= {(( p, b ,a ), r2 )}, a,b∪ V ∪ { є }, p∪ Q, 174
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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 head in 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 ∪Mi fi (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 the state of the first element when the process is started and correspondingly the input symbols of a word read by read head and write in the register and the remaining elements are 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
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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 elements enter a state from the set of all states from Q and it sent to all the other elements at the same time. This system is similar to a n head extended finite automaton. This implies that the n head finite automaton is equal to the returning parallel communicating extended finite automata systems of degree n. THEOREM 4 A language is accepted by a deterministic n head extended finite automaton iff it belongs to DPCEFS (n) PROOF: It is obliviously that is satisfied if A is deterministic. 4. PARALLEL COMMUNICATING GRAMMAR SYSTEMS DEFINITION: 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 of γ, Pi, 1≤ i ≤ n, are finite sets of production rules over N∩T=0, K ={ Q1Q2,….Qn) is the set 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 system as 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 as these symbols impose Each occurrence of Qj in xi is replaced by xj, supposing that xj does 176
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International Journal of
Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print), ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME not contain any query symbol, and, after that, the component resumes working from its axiom. Moreover, the communication has priority over the effective rewriting. A parallel communicating grammar system is said to be centralized if a request symbols are introduced by the first component and non centralized otherwise. DEFINITION: The language is generated by a system is defined as L(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 extended finite automata system of degree n. We can associate with each configuration a number between 1 and n which is 1 if no query symbol appears in the configuration, or 2 ≤ j ≤ n where Kj is the only query state in the configuration. That is the state of the master component is considered configuration. Now we define trace of the parallel communicating extended finite automata system A of 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≤ n Given a cpcefs(n) | rcpcefs(n) A we say that trace(A) is the trace language THEOREM 5 The system rcpcefs (2) and cpcefs (2) accepting non context free languages but having regular trace languages. PROOF: Consider the deterministic cpcefs(3) f1(q,ε,e ) = (s1 , e), f2(q,a,e ) = (r1 , m6), 177
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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 finite automata system recognized the languages {anbncn / n ≥ 1}. 5. CONCLUSION AND FURTHER WORK Parallel communicating extended finite automata systems provide more interesting problems for further study. Finite automata without any external control have very limited accepting power. These systems read the input words and write in the registers and reducing the space and computation time .It is more powerful than the generalized automata system. We have proved that it accepts non regular languages and non 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
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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
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