petri nets generating kolam patterns - Indian Journal of Computer ...

D. Lalitha et al / Indian Journal of Computer Science and Engineering (IJCSE)t 1 (p 1 )p 5p 4p S 1t 4 (p 6 )t 2Sp 3p 2t 7 (p 8 )t p 6 3p 9p 8t 6t 5p 7t 9t 8t 11 p 11t 10 p 10D σ(t 2 ) = { D↑ D },S = DD σ(t 5 ) = { tm D→ S 1 , bm D→S 2 , D→D},σ(t 6 )= σ(t 5 )σ(t 8 ) = { tm D← S 4 , bm D←S 3 , D←D},σ(t 9 ) = σ(t 9 )σ(t 10 ) = { D↑d 4 }σ(t 11 ) = { D↓d 2 }fig 9fig 8The firing sequence t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10 t 11 generates the Kolam pattern in fig 10. To start with t 2 and t 3 are bothenabled. If t 2 is fired then the inhibitor input p 4 of t 6 makes sure that the sequence t 4 t 5 is also fired the samenumber of times. Similarly the inhibitor input p 5 of t 9 makes sure that the sequence t 7 t 8 is also fired the samenumber of times. The Kolam patterns generated are of size (2n + 4) x (2n +3)/ n ≥ 0, where n is the number oftimes t 2 is fired. This kolam is context-sensitive [5].Fig 10: Kambi kolam4. Comparative ResultsIn this section we recall the definitions of Array rewriting Grammar [6], compare KATPNS and KATPNSwith inhibitor arcs with some of the Kolam array grammars.Definition 6: G = ( V, I, , S ) is an array rewriting grammar(AG), where V = V 1 V 2 , V 1 a finite set ofnonterminals, V 2 a finite set of intermediates, I a finite set of terminals, P = P 1 P 2 P 3 , P 1 is the finite set ofnonterminal rules, P 2 is the finite set of intermediate rules, P 3 is the finite set of terminal rules. S V 1 is the startsymbol. P 1 is a finite set of ordered pairs (u, v), u and v in (V 1 V 2 ) + or u and v in (V 1 V 2 ) +.P 1 is context-sensitive if there is a (u,v) in P 1 such that u = u 1 S 1 v 1 and v = u 1 α v 1 where S 1 V 1 , u 1 , α, v 1 areall in (V 1 V 2 ) + or all in (V 1 V 2 ) +. P 1 is context-free if every (u,v) in P 1 is such that uV 1 and v in (V 1 V 2 ) + or (V 1 V 2 ) +. P 1 is regular if u V 1 and v of the form U V, U inV 1 and V in V 2 or U in V 2 and V inV 1 .P 2 is a set of ordered pairs (u,v), u and v in (V 2 {x 1 ,…,x p }) + or u and v in (V 2 {x 1 ,…,x p }) + ;x 1 ,…,x p in I ++ have same number of rows in the first case and same number of columns in the second case. P 2 iscalled CS, CF or R as the intermediate matrix languages generated are CS, CF or R.P 3 is a finite set of terminal rules are ordered pairs (u,v), u in (V 1 V 2 ) and v in I ++ .An Array Grammar is called (CS:CS)AG if the nonterminal rules are CS and at least one intermediate languageis CS. An Array Grammar is called (CS:CF)AG if the nonterminal rules are CS and none of the intermediateISSN : 0976-5166 Vol. 3 No. 1 Feb-Mar 2012 73