A Note on Parallel Communicating T0l Array Grammar Systems

Parallel communicating L systems are systems having the communicating components as L-systems. In parallel communicating L-array systems we have 0L-(T0L) Array systems as communicating components. These models generates interesting pictures, some results on generative capacity are given.

communicate in some way with each other. Thus the Parallel communicating grammar systems originate (Paun, 1992). In Parallel Communicating (PC) grammar system, the components of the grammar system work in parallel (each having its own sentential form) and communicate in some way (i.e when the query occurs, sending the currently generated string to other component). This Parallel Communicating grammar system whose components are L-systems (of various type), is mainly motivated by theoretical reasons, but seems to have biological applications in symbiosis, parasitism, simultaneous interrelated growing of plants and animals etc . Motivated by this we have proposed Parallel Communicating L-Array systems and Parallel Communicating Tabled L-Array system. We consider a specific type of L-Array systems defined in (Nirmal and Krithivasan, 1981) as the components of the system. This model is capable of generating interesting pictures. We give some hierarchy results in communicating array grammar and also examine some generative capacity.

Definition, examples and hirerarchy
In this section we define PC0LAS (Parallel Communicating 0L Array System), PCT0LAS (Parallel Communicating Tabled 0L Array System). We assume the reader to be familiar with the theory of 2D languages. We illustrate the model with examples. We present some more results on hierarchy.

Definition
Parallel Communicating 0L Array System (PC0LAS) is a 3-tuple G = ( , , P) where 1.  =  K ,  is basic alphabet, K = {Q 1 ,Q 2 ,Q 3 ,…, Q n } are query symbols. 2.  = { 1 , 2 ,…,  n } is the axiom of n-tuple and w i **, 1 i n. 3. P = (P 1 ,P 2 ,P 3 , …, P n ) where P i = P i1 P i2 . Each P i1 is a nonempty finite subset of  X ** , is a finite set of pairs (a,x) with a in å and x in å** of dimensions r x s such that for each a in å at least one such pair is in P, 1 i n. The pairs (a,x) are called the rules or productions and are written as a x. The communicating rules in P i2 are of the form Query symbol, and C. The sets P i2 maybe empty, for 1 i n. The derivation step is of two types as i.
x i  y i if x i has no query symbols 1 in. P i ii.
(x 1 ,x 2 ,…,x n )  (y 1 ,y 2 ,…,y n ) where some x i has query symbols Q j and communicates with jth component for 1 i,j n.
In other words an n-tuple (x 1 ,x 2 ,x 3 ,…,x n ) directly yields (y 1 ,y 2 ,…,y n ) if either no query symbol appears in x i and then we have a component wise derivation x i  y i in G i , 1 in, or x i contains query symbols and then a communication step is performed ,as these symbols impose. Specifically each symbol Q i is replaced by the corresponding component x i , whereas the component G i , resumes working from its axiom. The communication has priority over rewriting. The Language generated by g is In other terms, a derivation consists of repeated rewriting and communication steps, starting from the n-tuple of axioms. We retain in L() the array generated in this way on the first component, G 1 (Which is considered the master of the system) without containing query symbols. We shall denote by PC n X be the family of languages generated by PC grammar having at most n components, n1.
i.e. To derive the configuration (y 1 ,y 2 ,y 3 ,…,y n ) either (i) No query symbol appears in x 1 ,x 2 ,x 3 ,…,x n and then we have a component wise derivation, x i = y i , in each component P ij , 1 in, 1 jn (one rule is used in each component P i ), except for the case when x i is terminal , x i  T* then x i = y i . or (ii)Query symbols occur in some x i . Then a communication step is performed. Each occurrence of Q j in x i is replaced by x j, provided x j does not contain query symbols. In a communication step, the communicated string x j replaces the query symbol Q j. Then the grammar G j resumes rewriting beginning again from its axiom. The communication has priority over the effective rewriting: no rewriting is possible as long as at least one query symbol is present. If some query symbols are not satisfied at a given communication step, then they may be satisfied at the next step.

Definition
Parallel Communicating Tabled 0L Array System (PC0LAS) is a 3-tuple G = ( , , P) where ,  are as in definition of PC0LAS. P= (P 1 ,P 2 ,P 3 ,…,P n ) where P i = P i1 U P i2 . Each P i1 consists of a finite set {P i1 1 ,P i1 2 ,Pi1 3 ,...,P i1 f }, f ³ 1. Each P i1 t is a finite subset of  x ** called a table with the following two conditions. i.
(P) pi (a )  ( a) ** (<a,> P); ii (a )  ( <a , >) P , a's are of the same dimension. P i2 is defined as in definition of PC0LAS. The derivation step is of two types as i x i  y i if x i has no query symbols 1 i n. p i ii (x 1 ,x 2 ,x 3 ,…,x n )(y 1 ,y 2 ,y 3 ,…,y n ) where some x i has query symbols, Q j and communicates with j th component for 1 i,j n.
The derivations are defined as follows P = P r = {P 1 ,P 2 ,…,P i } where P i =P i1 U P i2 , then we apply to ù the rules from the tables of Pr (Pc) row by row (column by column) i.e., we choose a rule in P r and apply the rules to the first row (column). Then we choose another set of rules in P r and apply the rules to the second row (column). Proceeding in this manner we apply the rules from a table to row (column) of . If the query symbol appears, the communication takes place so as to get a rectangular array and if no query symbol appears in the master system then the communication does not work and the derivation stops. If the resultant array is rectangular then the rules from the table Pr (Pc) can be applied again, otherwise the derivation comes to an end. In PCT0L array system, the query symbols can occur anywhere by rule provided. The resultant after the communication step, is rectangular. The row or column rules are applied in such a way that the rectangular format is maintained. The following figures are the models to form the rectangle pattern after the communication. iii) h is a partial coding from V into . Starting from , arrays are derived by part PCTMLS and then coding h is applied to these arrays is undefined if there is at least one a ij for which h(a ij ) is not defined.

Clearly
for all X  { 0LAL ,T0LAL }. We shall show that X  PC 2 X (two components are strictly powerful than one) for X  { 0LAL ,T0LAL }.
PC 2 0LAL -0LAL  . Proof: Follows from the definition of PC0LAL. In particular, model-1 cannot be generated by any E0LAL.
Clearly can be obtained from G 1 , then L(  )L . Hence at least a communication step is necessary.
x a x can be used in a  x and a  x. a x But the arrays in the communication must be 'a'. If an array (a) t , t 1 is communicated then either  or a 2t , a 3t are also generated. Hence from  1 we get only  1 and no other array. i.e., L  PC n T0LAL.
but not in PC n T0LAL. We present below some more hierarchy and the properties of the above family. 0LAS  PCT0LAS Proof Inclusion follows from definition. For proper inclusion, consider a PCT0LAS  generated by PCT0LAS array system (of degree two)  = (G 1 ,G 2 ) let G = ({ a,b,c,Q 2 }, {c,a}, P) P = P 1 P 2 .
is not an 0LAS.

Corollary
We can prove the following corollary trivially from the definition i.
T0LAS  PCT0LAS The following hierarchy is true obviously, when the number of tuple is more any model can be done in fewer steps.
There exist some finite languages, which are not PC0L array languages. Proof: As L, G should be a propagating system. Hence w = a. To get a b, we This will generate the language So arrays, which are not in L, will be generated by G. Hence L is not a PC0LAL.

Theorem
There exists an algorithm which for an arbitrary PCTAS G produces G° and a partial coding h such that L (G)  h ( L (G°) ) in the normal form Proof: Let G = (V,P,,) be an PCTAS, then Let G' =(V{S,F},P,S', ',h) be an part CPCTAS, where {S,F}V and P' = P 0 {P i ' P i  P} Let h be a partial coding h: V', such that h is defined if h(a ij ) defined for all i,j h is not defined if atleast one a ij is not defined.
The table in P' was constructed in such a way as to provide a possibility for the following simulation (in G°) of derivations in G. If D is a derivation in G, then we construct a corresponding derivation D° in G° in such a way that h gives the partial coding. We keep track of all productive occurrences such that the partial coding h is defined for all.

Conclusion
A new type of communicating L-array model is formulated in this paper. This model generates interesting pictures. Some results on hierarchy are given. Seven other variation of this model can be made using the concept of E0LAL and ET0LAL.