Interntionl Journl of Computer Applictions (0975 8887) Volume 57 No.5 Novemer 0 Domino Recognizility of ringulr icture Lnguges V. Devi Rjselvi Reserch Scholr Sthym University Chenni 600 9. Klyni Hed of Deprtment of Mthemtics St. Joseph Institute of echnology Chenni 600 9 D.G. homs Deprtment of Mthemtics Mdrs Christin College Chenni 600 059 ABSRAC he notion of locl iso-tringulr picture lnguges nd recognizle iso-tringulr picture lnguges re introduced. Domino recognizility of iso-tringulr picture lnguges nd HRL-domino systems re defined. Also the concept tht recognizle iso-tringulr picture lnguges re chrcterized s projections of hrl-locl tringulr picture lnguges is derived. heorems re proved. Keywords Iso tringulr domino system overlpping of iso tringulr pictures.hrl domino systems. INRODUCION A generliztion of forml lnguges to two dimensions is possile to different wys nd severl forml models to recognize or generte two dimensionl ojects hve een proposed in the literture. hese pproches were initilly motivted y prolems rising in the frmework of pttern recognition nd imge processing [3] ut two dimensionl ptterns re lso pper in studied concerning cellulr utomt nd other models of prllel computing [5]. Alredy notion of recognizility of set of pictures in terms of tiling systems is introduced [4]. he underlying ide is to define recognizility y projection of locl properties. Informlly recognition in tiling system is defined in terms of finite set of squre pictures of side two which correspond somehow to utomton trnsitions nd re clled tiles. In picture to e recognized (over the lphet ) ech qudruple of positions form squre to e covered y tile (with symols sy in the lphet ) such tht coherent ssignment of picture positions to lels in is uilt up nd such tht projection from to reestlishes the considered picture. hen the tiles cn e viewed s locl utomton trnsitions nd tiling given picture mens to construct run of the utomton on it []. he locl lnguges re lnguges given y finite set of uthorized tiles of size ( ). he use of locks of size ( ) implies tht in computtionl procedure to recognize given picture the horizontl nd verticl controls re done t the sme time. hen it is nturl to sk wht hppens when the two scnning re done seprtely nd in prticulr wht this cn imply when pply the projections fterwrds. In [] the so clled hv-locl picture lnguges re defined where the squre tiles of side re replced y dominoes tht correspond to two kinds of tiles: horizontl dominoes of size ( ) nd verticl dominoes of size ( ). In this pper the notion of domino systems to recognize isotringulr picture lnguges.. RELIMINARIES In this section some definitions of tiling systems re recollected [6]. Let e finite lphet of symols. A picture over is rectngulr rry of symols over. he set of ll pictures over is denoted y. Given p l (p) nd l (p) denote the numer of rows nd columns respectively of p. he pir (l (p) l (p)) is the size of p; p(i j) denotes the symol t row i nd column j i l (p) nd j l (p). A picture lnguge L is suset of. Let p e picture of size (m n). Let pˆ e the picture of size (m+ n+) otined y ordering p with specil symol. B hk (p) denotes the set of ll supictures of p of size (h k). A tile is picture of size ( ). mn denotes the set of ll pictures of size (m n) over the lphet. Here some sic concepts of iso-tringulr picture lnguges re given. Definition. An iso-tringulr picture p over the lphet is n isosceles tringulr rrngement of symols over. he set of ll isotringulr pictures over the lphet is denoted y Σ. An iso-tringulr picture lnguge over is suset of Σ. Given n iso-tringulr picture p the numer of rows (counting from the ottom to top) denoted y r(p) is the size of n iso-tringulr picture. he empty picture is denoted y. Iso-tringulr pictures cn e clssified into four ctegories.. Upper iso-tringulr picture. Lower iso-tringulr picture 3. Right iso-tringulr picture 4. Left iso-tringulr picture he upper tringulr iso-picture cn e represented in the co-ordinte system s follows: A lower tringulr iso-picture cn e represented in the coordinte system s follows: 6
Interntionl Journl of Computer Applictions (0975 8887) Volume 57 No.5 Novemer 0 tiles) over n lphet nd projection : such tht (L) = L. he fmily of recognizle iso-tringulr picture lnguges will e denoted y IREC. Definition.8 An iso-tringulr tiling system is 4-tuple ( ) where nd re finite set of symols : is projection nd is set of iso-tringulr tiles over the lphet {}. Definition. If p Σ then pˆ is the iso-tringulr picture otined y surrounding p with specil oundry symol. Definition.3 Let p Σ is n iso-tringulr picture. Let nd e two finite lphets nd : e mpping which is clled s rejection. he projection y mpping of iso-tringulr picture is the picture p such tht (p(i j k)) = p(i j k). Definition.4 Given n iso-tringulr picture p of size i for k i. Denote B k (p) the set of ll iso-tringulr supictures of p of size k. B (p) is in fct n iso-tringulr tile. Definition.5 Let L Σ e n iso-tringulr picture lnguge. projection of mpping of L is the lnguge (L) = {p / p = (p) p L}. he Definition.6 Let e finite lphet. An iso-tringulr picture lnguge L Σ is clled locl if there exists finite set of isotringulr tiles over {} such tht L = {p Σ / B ( pˆ ) }. he fmily of locl iso-tringulr picture lnguges will e denoted y ILOC. Exmple. Let = { } e finite lphet. he iso-tringulr picture lnguge L Σ is tiling recognizle if there exists tiling system = ( ) such tht L = (L()). It is denoted y L(). he fmily of iso-tringulr picture lnguges recognizle y iso-tringulr tiling system is denoted y L(IS). 3. DOMINO RECOGNIZABILIY OF ISO-RIANGULAR ICURES 3. Overlpping of iso-tringulr pictures Definition 3. Horizontl Overlpping he horizontl overlpping is etween U iso-tringulr picture nd D iso-tringulr picture of equl size nd denoted y the symol over. Exmple 3. Definition 3. Verticl Overlpping he verticl overlpping is defined etween L nd R iso-tringulr picture of sme size nd it is denoted y the symol over. Exmple 3. U over D = 3 3 Δ L over R = 3 hen L = L() =... he lnguge L() is the set of tringles with size k with lterntive nd in the rows. Clerly L() is locl. Definition.7 Let e finite lphet. An iso-tringulr picture lnguge L is clled recognizle if there exists iso-tringulr locl picture lnguge L (given y set of iso-tringulr Definition 3.3 Right Overlpping he right overlpping is defined etween ny two glule iso-tringulr pictures of sme size nd is denoted y the symol over. his overlpping includes the following. () D over U () R over U (c) D over L (d) R over L. Exmple 3.3 D over U = 3 3 D U 3 7
Interntionl Journl of Computer Applictions (0975 8887) Volume 57 No.5 Novemer 0 Definition 3.4 Left Overlpping he left overlpping is defined etween ny two glule isotringulr pictures of sme size nd it is denoted y the symol over. his overlpping includes the following. () U over R () U over L (c) L over R (d) R over D. Exmple 3.4 R over D = Definition 3.5 he set of ll pictures otined y overlpping n isotringulr pictures of sme size is denoted y O(p). Here dominoes of the following types re considered. (i) Horizontl dominoes (ii) Right nd left dominoes. Horizontl Left dominoes Right Definition 3.6 Let L e n iso-tringulr picture lnguge. he lnguge L is hrl-locl if there exists set of dominoes over the lphet {} such tht L = {p In this cse we write L = L(). Σ / O(B ( pˆ )) }. Definition 3.7 An iso-tringulr domino system (IDS) is 4-tuple ( ) where nd re two finite lphets is finite set of dominoes over the lphet {} such tht : is projection. he iso-tringulr domino system recognized y n isotringulr picture lnguge L over the lphet nd is defined s L = (L) where L = L() is the hrl-locl isotringulr picture lnguge over. he fmily of isotringulr picture lnguges recognized y iso-tringulr domino system is denoted y L(IDS). roposition 3. If L Γ is hrl-locl iso tringulr picture lnguge then 3 3 3 3 3 3 3 3 L is locl iso-tringulr picture lnguge. ht is L(IDS) L(IS). roof Let L Γ e hrl-locl iso tringulr picture lnguge. hen L = L() where is finite set of dominoes. Here construct finite set of iso-tringulr tiles of size nd show tht L = L(). Define s follows θ Γ {} / θ Where the symol O denotes overlpping. Let L = L(). Now show tht L = L. Let p L then y definition Hence p L. B (pˆ) O(B (pˆ)) O(θ) Δ. Conversely let p L nd q B (pˆ ). hen O(θ) Δ. his implies tht O(q) O (B(pˆ)) Δ. herefore q nd p L. Hence L = L. Remrk 3. he converse of the roposition. is not true. ht is there re lnguges tht re in ILOC ut not in hrl-locl. Lemm 3. Let L e locl iso-tringulr picture lnguge over n lphet. hen there exists n HRL-locl lnguge L over the lphet nd mpping : such tht L = (L). roof Let L = L() where is finite set of iso-tringulr tiles of size over {}. By definition contins ll llowed supictures of size of pictures in L. he ide of the proof is to show tht the property of eing n llowed supicture of size of picture in L y mens of domines over n lrger lphet cn e expressed. his is ccomplished y choosing s the set itself nd defining the set of dominoes. Let = ( {}) nd let = 3 3 c c c 3 f 3 3 c 3 d f f 3 d d d 3 c c 3 c c c 3 8
Interntionl Journl of Computer Applictions (0975 8887) Volume 57 No.5 Novemer 0 d d d 3 c c c 3 c 3 f c 3 f f f 3 Let L = L(). hen define mpping etween the two lphets nd such tht 3 3 It is esy to verify tht p L nd (p) =. Conversely let p L nd let q B ( pˆ ) e supicture of pˆ of size. o prove tht (p) L. It suffices to show tht (q). Suppose the iso-tringulr picture q is the following o complete the proof first (L) = L to e proved. Before proving it formlly give n exmple to clrify how picture pl nd picture p L such tht (p) = p re relted. Suppose pˆ hen the corresponding picture pˆ will e the following pˆ 3 In the definition of pˆ severl different order symols re used. More precisely the order symols for re ll isotringulr tiles of size contining. Now L = (L) will e proved. Let p L e of size m. Consider picture pˆ over s follows. 3 3 3 3 3 q 5 4 4 6 where ll the tiles 0 5 nd the four iso-tringulr tiles of size. 7 3 3 4 5 4 6 hen (q) =. Similrly q cn lso e ny lock (D-iso tringulr tile) nd in this cse (q). heorem 3. L(IS) = L(IDS). 4 5 4 6 7 5 3 3 9 5 8 8 6 0 9 6 0 9
Interntionl Journl of Computer Applictions (0975 8887) Volume 57 No.5 Novemer 0 roof he inclusion L(IDS) L(IS) is n immedite consequence of roposition 3.. he inverse inclusion follows from Lemm 3.. Before concluding n exmple s n ppliction of the theorem s given. Exmple 3. Consider the lnguge L of iso-tringulr picture over = { }. In the exmple given elow we oserve tht L L(IS). In order to show tht L L(IDS) it suffices to verify tht it cn e otined s projection of the lnguge L over = { } of tringles in which the medin of the tringle crry the symols nd nd the other symol crry the symol nd. It is cler tht L is hrl-locl. In fct it is represented y the following set of dominoes. Now L = (L) where : is such tht ( ) = ( ) = nd ( ) = ( ) =. Hence L is recognizle y isotringulr domino system. ht is L L(IDS). 4. CONCLUSION In this pper the overlpping of iso-tringulr pictures hve een introduced nd the notion of recognizility of isotringulr pictures y new formlism clled domino system hve een investigted. he theorem L(IDS) = L(IS) is proved. ringulr picture lnguges cn generte ll pictures in picture lnguges. he lerning of iso-tringulr pictures nd unry iso-tringulr picture lnguges nd their complexity deserve to e studied further. 5. REFERENCES [] M. Ltteux nd D. Simplot Recognizle picture lnguges nd domino tiling Internl Report I-94-64 Lortoire d Informtique Fond. [] D. Gimmrrsi wo dimensionl nd recognizle functions In roc. Developments in Lnguge heory Finlnd 993. [3] G. Rozenerg nd A. Slom (Eds) World Scientific ulishing Co. Singpore 994 pp. 9030. [4]. Klyni V.R. Dre nd D.G. homs Locl nd Recognizle iso picture lnguges Lecture notes in Computer science 004 volume 336/004 pp. 738743. [5] D. Gimmrresi nd A. Restivo Recognizle picture lnguges Interntionl Journl pttern Recognition nd Artificil Intelligence Specil Issue on prllel imge processing M. Nivt nd A. Soudi nd.s.. Wng (Eds) pp. 346 99. Also in roc. First Interntionl Colloquium on rllel imge processing 99 Sme journl vol. 6 No. & 3 pp. 456 99. [6] D. Gimmerresi nd A. Restivo wo dimensionl finite stte recognizility Fundment Informtice Specil Issue: Forml lnguge heory volume 5 no. 3 4 (996) pp. 3994. 0