PETRI NETS GENERATING KOLAM PATTERNS

Transcription

1 . Lalitha et al / Idia Joural of omputer Sciece ad Egieerig (IJSE) PETRI NETS GENERATING KOLAM PATTERNS. Lalitha epartmet of Mathematics Sathyabama Uiversity, heai-119, Idia lalkrish_24@yahoo.co.i K. Ragaraja epartmet of MA harath Uiversity, heai-73, Idia kr2210@hotmail.com Abstract Array Toke Petri Nets are kow to geerate all the ie families geerated by Array Grammars [10]. Applicatio of two dimesioal array grammars i kolam geeratio has bee discussed [1]. Motivated by this we have used Array Toke Petri Nets to geerate kolam patters. Parallel geeratio ad sequetial/parallel geeratio are the two types of geeratio used. The Kolam patters which are geerated by Regular Grammars are geerated by Array Toke Petri Nets ad Kolam patters which are geerated by otext-free ad otextsesitive grammars are geerated by Array Toke Petri Nets with ihibitor arcs. Keywords: Kolam; array toke; parallel geeratio; sequetial geeratio; Petri et; ihibitor arc. 1. Itroductio Kolam is a traditioal art of decoratig courtyards i South Idia. Every kolam patter ca be treated as a two dimesioal laguage over the alphabet of tiles. Theoretical models geeratig two dimesioal arrays have bee used to geerate complicated kolam patters [1, 2, 3, 4, 5]. lassificatio of kolam patters has bee doe based o the differet array grammars which geerate them [1]. Several approaches have bee used to geerate these itricate desigs. O the other had Petri et is a abstract formal model of iformatio flow [7]. Petri ets have bee used for aalyzig systems that are cocurret, asychroous, distributed, parallel, o determiistic ad / or stochastic. Tokes are used i Petri ets to simulate dyamic ad cocurret activities of the system. A laguage ca be associated with the executio of a Petri et. y defiig a labelig fuctio for trasitios over a alphabet, the set of all firig sequeces, startig from a specific iitial markig leadig to a fiite set of termial markigs, geerates a laguage over the alphabet. Petri et model to geerate arrays of tiles has already [9] bee itroduced. I this model arrays over a give alphabet of square tiles of same size are used as tokes i the places of the et. Joiig of two square tiles ca be doe i four ways. (right), (left), (up) ad (dow). ateatio rule is used as label of trasitio. Firig a sequece of trasitios startig from a specific iitial markig leadig to a fiite set of termial markigs would joi tiles accordig to the rule ad move the array to the fial set of places. This model geerates the kolam patters that ca be geerated by regular grammars. To icrease the geerative capacity we have itroduced the cocept of ihibitor arcs. Ihibitor arcs are used to geerate the kolams geerated by cotextfree or cotext-sesitive grammars. This paper is orgaized as follows. Sectio 2 gives the prelimiary defiitios, defies Kolam Array Toke Petri Net Structure (KATPNS) ad gives a example. Sectio 3 defies KATPNS with ihibitor arcs ad gives a example. Sectio 4 compares the models with Regular ad otext-free array grammars geeratig kolams. 2. Kolam Array Toke Petri Nets I this sectio we defie Petri et, give the prelimiary otatios, defie Kolam Array Toke Petri Net Structure ad also give a example. ISSN : Vol. 3 No. 1 Feb-Mar

2 . Lalitha et al / Idia Joural of omputer Sciece ad Egieerig (IJSE) efiitio 1. A Petri Net structure is a four tuple = (P,T,I,O) where P = {p 1, p 2,..., p } is a fiite set of places, 0, T = {t 1, t 2,, t m } is a fiite set of trasitios m 0, P T= Ø, I:T P is the iput fuctio from trasitios to bags of places ad O:T P is the output fuctio from trasitios to bags of places. efiitio 2. A Petri Net markig is a assigmet of tokes to the places of a Petri Net. The tokes are used to defie the executio of a Petri Net.The umber ad positio of tokes may chage durig the executio of a Petri Net. efiitio 3. A ihibitor arc from a place p i to a trasitio t j has a small circle i the place of a arrow i regular arcs. This meas the trasitio t j is eabled oly if p i has o tokes. A trasitio is eabled oly if all its regular iputs have tokes ad all its ihibitor iputs have zero tokes. To geerate kolams we use alphabets cosistig of square tiles of same size with patters i it. A certai family of patters called Kambi Kolam over a square grid, is expressed with regular dotted square tiles. Nagata et al [8] have give a digital represetatio to such tiles. The set of primitives give below are primitives used i geeratig Kambi Kolam [8]. iamod Pupil p 1 p 2 rop d 1 d 2 d 3 d 4 f 1 f 4 f 2 f 3 Fa Saddle S 1 S 2 S 3 S 4 fig 1 We deote a square tile without ay patter by (blak). Arrays over the alphabet of tiles will form a patter. d 4 For example the array d 3 d1 with the alphabet take from fig 1, will form the kolam i fig 2. d 2 fig 2 I this model arrays over the alphabet of tiles is used as tokes i places. If all the iput places of a trasitio have the same array as tokes the firig the trasitio moves the array from the iput place to its output place. If all the iput places of a trasitio have differet arrays or if oe iput place has differet arrays the the trasitio without label caot fire. Two types of labels are used i this model. I. Name of a iput place: If the trasitio has may iput places with differet arrays as tokes the the label specifies a iput place which has sufficiet umber of tokes of the same array. The firig the trasitio will remove tokes from all its regular iput places ad the array which is i the (label specified) place will be moved ito the output place. A example of this is give i fig 3. ISSN : Vol. 3 No. 1 Feb-Mar

3 . Lalitha et al / Idia Joural of omputer Sciece ad Egieerig (IJSE) Positio of tokes before firig: P 1 A 3 Positio of tokes after firig: A 1, A 2 P P 3 2 P 1 A 3 P 2 A fig 3 II. ateatio rule: If T is a trasitio of a et the the label σ(t) is a set of cateatio rules. Whe ay cateatio rule is used as a label of a trasitio, all the iput places of the trasitio should have the same array as toke. A trasitio with iput places havig differet arrays ad a cateatio rule as label caot fire. ateatio of tiles or joiig of tiles ca be doe i four directios (right), (left), (up) ad (dow). For A, Σ ** the four cateatio rules are give below. A jois to the right of A so that right side of A ad left side of coicide. A jois to the left of A so that the left side of A ad right side of coicide. I the above two cases A ad should have the same umber of rows. A jois above A so that the top of A ad bottom of coicide. A jois below A so that the bottom of A ad top of coicide. I the above two cases A ad should have the same umber of colums. I applyig cateatio rules two types of geeratios are cosidered (i) Parallel ad (ii) Sequetial/ Parallel. Parallel geeratio: Rules are applied i a parallel maer i this geeratio. If A is the rule applied the all A i the last colum will be joied to i the right. If a right rule is used as a label all the tiles i the last colum of the array i the iput place will have to be cateated with some tile i the alphabet. If a rule is missig for ay of the tile, the a blak tile will be joied to the specific tile. For example let the iput place of a trasitio T have the array ad σ(t) = {, }. I parallel geeratio all o the last colum will be cateated to ad is cateated to. Hece firig the trasitio will joi to the right of all tiles of the array. The resultat array reachig the output place would be. If σ(t) = { }the firig the trasitio will joi to the right of all ad a blak tile to the right of. The resultat array reachig the output place would be. Similarly i all other three rules we apply the rules i a parallel maer ad missig rule is automatically joied to i the specified directio. Sequetial/ Parallel geeratio I kolam patters we fid two kids of pullis used. Oe is rectagular ad the other is i the form of diamod or triagular. Hece parallel geeratio may ot be able to geerate all kolams. Hece we use Sequetial/ Parallel geeratio. olum Rule: I case all i the last colum (first colum) of the array is ot required to be joied to usig right rule (left rule). If the topmost(tm) ad bottommost(bm) is to be joied to a blak tile the we use the otatio tm (tm ), bm (bm ). Except the first ad the last of the colum all the other are cateated to i a parallel maer. For example let the iput place of a trasitio T have the array. P 3 ISSN : Vol. 3 No. 1 Feb-Mar

4 . Lalitha et al / Idia Joural of omputer Sciece ad Egieerig (IJSE) If σ(t)={, tm, bm }. The resultat array reachig the output place would be. Row Rule: I case all i the first row (last row) of the array is ot required to be joied to usig up rule (dow rule). If the leftmost(lm) ad rightmost(rm) is to be joied to a blak tile the we use the otatio lm (lm ), rm (rm ). Except the first ad the last of the row all the other are cateated to i a parallel maer. For example let the iput place of a trasitio T have the array. If σ(t)={ lm, rm,, }. The resultat array reachig the output place would be For example cosider the et i the followig figure. P 1 t 1 P 2 A t 2 P 3 If A = Y reaches the place p 2 is Y Y, σ(t 1 ) = { Y} ad σ(t 2 ) = { lm, rm,, } the after firig t 1 the array that. All i the first colum is joied to y i the left. Oce this array reaches p 2 the trasitio t 2 is eabled ad so whe t 2 fires the array that reaches p 3 will be Y. The left most Y Y of ad the right most of are joied to a blak tile ad the middle is joied to. Sice o rule is give for Y it is beig joied with a blak tile. All these cateatios are doe i the upward directio. I geeral all rules are applied i a parallel maer uless the prefix rm, lm, tm ad bm is used alog with a rule. Hece we call it sequetial/parallel geeratio. efiitio 4. A Kolam Array Toke Petri Net Structure (KATPNS) is a four tuple (, Σ, σ, F) where = (P, T, I, O) is a Petri et structure, Σ a alphabet of square tiles of same size, σ partial fuctio o the set of trasitios which defie the labels, M 0 :P Σ ** is the iitial markig ad F P is a set of fial places. efiitio 5. The laguage geerated by the KATPNS is L() = {A Σ ** / A is i p for some p i F}. With arrays of Σ ** i some places as iitial markig all possible sequeces of trasitios are fired. The set of all arrays collected i the fial places F is called the laguage geerated by. Example 1. The KATPNS (, Σ, σ, F) where the graph of is give i fig 6, Σ ={, T, R,, L, 1, 2, 3, 4 }, σ the partial fuctio is give i fig 5, S = is the iitial markig i p 1 ad p 3 ad F = {p 2 }. The geeratio here is parallel. The tiles are give i fig 4. The arrays reachig p 2 are collected as the laguage geerated T R L fig 4 ISSN : Vol. 3 No. 1 Feb-Mar

5 . Lalitha et al / Idia Joural of omputer Sciece ad Egieerig (IJSE) σ(t 2 ) = { }, σ(t 3 ) = { }, σ(t 4 )= { T}, σ(t 5 ) ={ } σ(t 6 ) ={ T 1, R, 4 }, σ(t 7 ) = { T 2, L, 3 }, p 1 p 2 S t 2 S p 3 t 3 t 4 p 4 t 5 p 5 t 6 t 7 fig 5 fig 6 The firig sequece t 4 t 5 t 6 t 7 geerates the kolam referred to as Asaapalakai is give i fig 7. 2 L 3 T 1 R 4 fig 7 Trasitios t 2 ad t 3 ca be fired ay umber of times before we fire t 4. Firig t 2 ad t 3 geerates a array made up of the tile. The whe the sequece t 4 t 5 t 6 t 7 is fired we get all the boudary tiles. This kolam patter ca be geerated by a Regular Matrix Grammar [1] ad has used oly parallel geeratio. 3. KATPNS with ihibitor arcs I this sectio we defie KATPNS with ihibitor arcs ad give example. Whe parallel geeratio is used we ca geerate oly rectagular pictures. To geerate triagular or diamod shaped kolams we have to use i Sequetial/ Parallel geeratio. To geerate Kolams geerated by otext-free or otext-sesitive grammar we have to cotrol the sequece of firig. This cotrol is achieved by itroducig ihibitor arcs i the Petri et. efiitio 6. If the Petri et of a KATPNS has at least oe ihibitor arc the we defie the KATPNS as Kolam Array Toke Petri Net Structure with ihibitor arcs. With arrays of Σ ** i some places as iitial markig all possible sequeces of trasitios are fired. The set of all arrays collected i the fial places F is called the laguage geerated by. Example 2. The KATPNS (, Σ, σ, F) where the graph of is give i fig 8, Σ ={, S 1, S 2, S 3, S 4, d 2, d 4 }, σ the partial fuctio is give i fig 9, the iitial markig S i p 1 ad p 3 ad F = {p 2 }. The geeratio here is sequetial/parallel. The tiles are give i fig 1. ISSN : Vol. 3 No. 1 Feb-Mar

6 . Lalitha et al / Idia Joural of omputer Sciece ad Egieerig (IJSE) t 7 (p 8 ) p 4 p S 1 t 4 (p 6 ) t 2 S p 3 p 2 t p 6 3 t 6 t 5 p 8 p 5 p 7 σ(t 2 ) = { }, S = σ(t 5 ) = { tm S 1, bm S 2, }, σ(t 6 )= σ(t 5 ) σ(t 8 ) = { tm S 4, bm S 3, }, σ(t 9 ) = σ(t 9 ) σ(t 10 ) = { d 4 } σ(t 11 ) = { d 2 } p 9 t 9 t 8 t 11 p 11 t 10 p 10 fig 9 fig 8 The firig sequece t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10 t 11 geerates the Kolam patter i fig 10. To start with t 2 ad t 3 are both eabled. If t 2 is fired the the ihibitor iput p 4 of t 6 makes sure that the sequece t 4 t 5 is also fired the same umber of times. Similarly the ihibitor iput p 5 of t 9 makes sure that the sequece t 7 t 8 is also fired the same umber of times. The Kolam patters geerated are of size (2 + 4) x (2 +3)/ 0, where is the umber of times t 2 is fired. This kolam is cotext-sesitive [5]. Fig 10: Kambi kolam 4. omparative Results I this sectio we recall the defiitios of Array rewritig Grammar [6], compare KATPNS ad KATPNS with ihibitor arcs with some of the Kolam array grammars. efiitio 6: G = ( V, I,, S ) is a array rewritig grammar(ag), where V = V 1 V 2, V 1 a fiite set of otermials, V 2 a fiite set of itermediates, I a fiite set of termials, P = P 1 P 2 P 3, P 1 is the fiite set of otermial rules, P 2 is the fiite set of itermediate rules, P 3 is the fiite set of termial rules. S V 1 is the start symbol. P 1 is a fiite set of ordered pairs (u, v), u ad v i (V 1 V 2 ) + or u ad v i (V 1 V 2 ) +. P 1 is cotext-sesitive if there is a (u,v) i P 1 such that u = u 1 S 1 v 1 ad v = u 1 α v 1 where S 1 V 1, u 1, α, v 1 are all i (V 1 V 2 ) + or all i (V 1 V 2 ) +. P 1 is cotext-free if every (u,v) i P 1 is such that u V 1 ad v i (V 1 V 2 ) + or (V 1 V 2 ) +. P 1 is regular if u V 1 ad v of the form U V, U iv 1 ad V i V 2 or U i V 2 ad V i V 1. P 2 is a set of ordered pairs (u,v), u ad v i (V 2 {x 1,,x p }) + or u ad v i (V 2 {x 1,,x p }) + ; x 1,,x p i I ++ have same umber of rows i the first case ad same umber of colums i the secod case. P 2 is called S, F or R as the itermediate matrix laguages geerated are S, F or R. P 3 is a fiite set of termial rules are ordered pairs (u,v), u i (V 1 V 2 ) ad v i I ++. A Array Grammar is called (S:S)AG if the otermial rules are S ad at least oe itermediate laguage is S. A Array Grammar is called (S:F)AG if the otermial rules are S ad oe of the itermediate ISSN : Vol. 3 No. 1 Feb-Mar

7 . Lalitha et al / Idia Joural of omputer Sciece ad Egieerig (IJSE) laguage is S. A Array Grammar is called (S:R)AG if the otermial rules are S ad all the itermediate laguages are regular. Similarly all the other six families (F:S)AG, (F:F)AG, (F:R)AG, (R:S)AG, (R:F)AG ad (R:R)AG are defied. (:Y)AL refers to the laguage geerated by the (:Y)AG, where, Y { R, F,S}. Theorem 4.1. Ay Regular Matrix kolam ca be geerated by KATPNS. Proof. If the patter is geerated by a regular matrix grammar the it ca be geerated by (R:R)AG. Ay regular array grammar ca be geerated by a Array Toke Petri Net Structure (ATPNS) without ihibitor arcs [10]. The sequece of firig i a KATPNS without ihibitor arcs is i the form (t 1 t j m ) k. Thus we do ot require ihibitor arc to geerate this kolam patter. Theorem 4.2. Ay (F:R) array kolam ca be geerated by KATPNS with ihibitor arcs. Proof. The o termial rules of (F:R) array grammar are i the form ( A) ( A ) ( ) or. Have a subet to ( ) geerate (A) ad aother to geerate (). Use ihibitor arcs to make sure both the subets are executed the same umber of times. I example 2, the umber of times the sequeces t 2, t 4 t 5 ad t 7 t 8 are fired is beig cotrolled by the itroductio of the two ihibitor arcs. We ca restrict the firig of a trasitio by itroducig a ihibitor arc. Thus (F:R) array kolam ca be geerated by KATPNS with ihibitor arcs. 5. oclusio Iterestig kolam patters are described by Array grammars. Sice Array Toke Petri Net Structure is able to geerate all these families we have tried to geerate kolam patters with Petri et. KATPNS is able to geerate all regular kolams ad with the itroductio of ihibitor arcs the (F:R) kolam patters are also geerated. Refereces [1] G. Siromoey, R. Siromoey ad Kamala Krithivasa, Array Grammars ad Kolam, omputer Graphics ad Image Processig 3, 1974, pp [2] G. Siromoey, R. Siromoey ad Kamala Krithivasa, Abstract families of Matrices ad Picture Laguages, omputer Graphics ad Image Processig 1, 1972, pp [3] K.G.Subramaia, A Note o Regular Kolam Array Grammars Geeratig Right Triagles, Patter Recogitio Vol. 11, 1979, pp [4] K.G. Subramaiaa, Rosiha M. Ali, M. Geethalakshmi, Atulya K. Nagar, Pure 2 Picture Grammars Ad Laguages, iscrete Applied Mathematics,157(2009) 3401_3411. [5] G. Siromoey, R. Siromoey ad T. Robiso, Kambi kolam ad cycle grammars, A Perspective i theoretical computer sciece, 16, 1989, pp [6] G. Siromoey, R. Siromoey ad Kamala Krithivasa, Picture Laguages with Array Rewritig Rules, Iformatio ad otrol 22, pp , [7] J.L Peterso, Petri Net Theory a Modelig systems, Pretice Hall, Ic., Eglewood liffs, N J, [8] Shojiro Nagata, Robiso Thamburaj, igitalizatio Of Kolam Patters Ad Tactile Kolam Tools, Formal Models, Laguages Ad Applicatios, World Scietific, pp [9]. Lalitha ad K. Ragaraja, haracterisatio Of Pastig System Usig Array Toke Petri Nets. Iteratioal Joural of Pure ad Applied Mathematics, Vol 70, No. 3, [10]. Lalitha, K. Ragaraja ad.g. Thomas, Rectagular Arrays ad Petri Nets, Proceedigs of Natioal oferece o Theoretical omputer Sciece ad iscrete Mathematics, Kalasaligam Uiversity, Ja ISSN : Vol. 3 No. 1 Feb-Mar