Parallel communicating extended finite automata systems communicating by states

International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),

ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME

166

PARALLEL COMMUNICATING EXTENDED FINITE

AUTOMATA SYSTEMS COMMUNICATING BY STATES

M.Ramakrishnan

Department of Computer Science and Engineering

Anna University of Technology, Coimbatore

Email: [email protected]

S.Balasubramanian

Director IPR

Anna University of Technology, Coimbatore

E-mail: [email protected]

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

International Journal of Computer Engineering

and Technology (IJCET), ISSN 0976 – 6367(print)

ISSN 0976 – 6375(Online) Volume 1

Number 1, May - June (2010), pp. 166-179

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167

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



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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.



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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} )

→ 2Q

i× 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 } ⊆

Un

i =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,.



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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



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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)



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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)}= {anb

nc

n / 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.



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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.



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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.

Qi = K ∪ Q ∪ (Q x (V ∪ {є})i-1

) ∪ (Q x (V∪ { є}iI ) ∪ 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,



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(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



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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



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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),



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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 {anb

nc

n / 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.

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