Triangular Tile Rewriting Grammars and Triangular Picture Languages

Transcription

1 Globl Journl of Pure nd pplied Mthemtics. IN Volume 12, Number 3 (216), pp Reserch Indi Publictions Tringulr Tile Rewriting Grmmrs nd Tringulr Picture Lnguges. Kuberl Deprtment of pplied Mthemtics, Pilli Institute of Informtion Technology, New Pnvel, Nvi Mumbi-41 26, Mhrstr. Dr. T. Klyni Deprtment of Mthemtics, t. Joseph s Institute of Technology, Chenni Dr. D.G. Thoms Deprtment of Mthemtics, Mdrs Christin College, Chenni bstrct Tile Rewriting Grmmrs (TRG) re new model for defining picture lnguges introduced in [1]. rewriting rule chnges homogeneous rectngulr sub-picture into n isometric one tiled with specified tiles. Derivtion nd lnguge genertion with TRG rules re similr to context-free grmmrs. F. weety et l. [4] hve introduced Hexgonl Tile Rewriting Grmmrs (HTRG). In this chpter we propose Tringulr Tile Rewriting Grmmrs (TTRG) for generting Tringulr Picture Lnguges. TTRG is the extension of HTRG. We study the derivtion of TTRG nd lso lnguge genertion with TTRG rules. ome closure properties re studied. TTRG is lso compred with TLOC nd TT. M subject clssifiction: Keywords: Tringulr picture lnguges, Tile Rewriting Grmmrs, tiling systems.

2 1966. Kuberl, Dr. T. Klyni, nd Dr. D.G. Thoms 1. Introduction picture is rectngulr rry of terminl symbols (the pixels). survey of forml models for picture lnguges is [3] where different pproches re compred nd relted: tiling systems, cellulr utomt nd grmmrs. Clssicl 2D grmmrs cn be grouped into two ctegories clled mtrix nd rry grmmrs. The mtrix grmmrs introduced by. Rosenfeld [7], impose the constrint tht the left nd right prts of rewriting rule must be isometric rrys; this condition overcomes the inherent problem of shering which pops up while substituting sub-rry in host rry. iromoney s rry grmmrs [6] re prllel-sequentil in nture, in the sense tht first horizontl string of non-terminls is derived sequentilly, using the horizontl production; nd then the verticl derivtions proceed in prllel, pplying set of verticl productions. TRG rule is schem hving to the left non-terminl symbol nd to the right locl 2D lnguge over terminls nd non-terminls; tht is the right prt is specified by set of fixed size tiles. s in mtrix grmmrs, the shering problem is voided by n isometric constrint but the size of TRG rule needs not to be fixed. The left prt denotes ny rectngle filled with the sme non-terminl. Whtever size the left prt tkes, the sme size is ssigned to the right prt. To mke this ide effective, we impose tree prtil order on the res which re rewritten. progressively refined equivlence reltion implements the prtil ordering. Derivtions cn then be visulized in 3D s well nested prisms, the nlogue of syntx trees on string grmmrs. In this pper we propose tringulr tile rewriting grmmrs using tringulr tiles. We lso study some closure properties of it. It is compred with other models. Tringulr Pictures In this section, we recll some bsic definition of tringulr pictures nd the recognizbility of the tringulr pictures. Definition 1.1. tringulr picture p over the lphbet is tringulr rry of symbols of. The set of ll tringulr rrys over the lphbet is denoted by T. tringulr picture lnguge over is subset of T. Given tringulr picture p, the number of rows (counting from the bottom to top), denoted by r(p), is the size of tringulr picture. The empty picture is denoted by. Definition 1.2. If p T, then ˆp is tringulr rry obtined by surrounding p with specil boundry symbol #. Here #. tringulr picture over the lphbet {} of size 4 surrounded by # is shown in Figure 1. Definition 1.3. Let p T. Let nd Ɣ be two finite lphbets nd π : Ɣ be mpping, which we cll projection. The projection by mpping π of tringulr picture p is the picture p ƔT, such tht π(p(i,j,k)) = p (i,j,k).

3 Tringulr Tile Rewriting Grmmrs 1967 Definition 1.4. tringulr picture of the bove form is clled tringulr tile over n lphbet {}. Definition 1.5. Given tringulr picture p of size k, for i k, we denoted by i (p), the set of ll tringulr subpictures of p of size i. Definition 1.6. Let L T be tringulr picture lnguge. The projection by mpping π of L is the lnguge π(l) ={p /p = π(p), p L} Ɣ Definition 1.7. Let be finite lphbet. tringulr picture lnguge L T is clled locl if there exist finite set of tringulr tiles over {#} such tht L ={p T / 2( ˆp) }. The fmily of Tringulr Locl Picture Lnguge is denoted by TLOC. Exmple 1.8. Let ={, 1} be finite lphbet. { = # # #, # # 1, # #, # 1 #, # #, # #, 1 # #, 1, 1 1 T. } Then L 1 = L( ) = 1 1, 1 1,..., 1 1, 1 1 1,... The Lnguge L(G) is the set of ll tringles of size k 2 with lterntive nd 1 in the rows. Clerly L( ) is locl. Definition 1.9. tringulr picture lnguge L ƔT is clled recognizble if there exists tringulr locl picture lnguge L (given by set of tringulr tiles) over n lphbet Ɣ nd projection π : Ɣ such tht π(l ) = L. The fmily of recognizble tringulr picture lnguges will be denoted by TREC. Definition 1.1. tringulr tiling system T is 4-tuple (,Ɣ,π,θ)where nd Ɣ re two finite set of symbols, π : Ɣ is projection nd θ is set of tringulr tiles over the lphbets Ɣ {#}. The tringulr picture lnguge L ƔT is tiling recognizble if there exists tiling system T = (,Ɣ,π,θ), such tht L = π(l(θ)). It is denoted by L(T ). The fmily of tringulr picture lnguge recognizble by tringulr tiling system is denoted by L(TT). 2. Tringulr Tile Rewriting Grmmrs (TTRG) In this section we introduce TTRG nd study its properties.

4 1968. Kuberl, Dr. T. Klyni, nd Dr. D.G. Thoms Definition 2.1. For finite lphbet, the set of tringulr picture is T.Fork 1, T k denotes the set of tringulr pictures of size k. (We will use the nottion p =k). # is used when needed s boundry symbol. ˆp refers to the bordered version of picture p. pixel is n element p(i, j, k). If ll pixels re identicl to C the picture is clled homogeneous nd denoted s C-picture. Let, Y, Z be the tringulr xes. If the co-ordintes of point is ( 1, 2, 3 ) then the co-ordintes of its neighbour in the direction x be ( 1 + 1, 2, 3 1) nd in the opposite direction be ( 1 1, 2, 3 + 1). In the direction of y be ( 1 1, 2 + 1, 3 ) nd opposite direction of y be ( 1 + 1, 2 1, 3 ). imilrly for z, the co-ordintes re ( 1, 2 1, 3 + 1) nd to the opposite direction ( 1, 2 + 1, 3 1). # # # Y # # # (,,) # # (, 1,1) (1, 1,) # # (, 2,2) (1, 2,1) (2, 2,) # (, 3,3) (1, 3,2)(2, 3,1) (3, 3,) # # # # # # # Z Figure 1: Definition 2.2. Let p be tringulr picture of size t. subpicture of p t position (i,j,k)is picture q such tht if t is the size of q then t t nd there exist integers i, j, k such tht, p(i, j, k) = q[i l,{(j m) + (t t ), k n],l,m,n here, l position of subpicture from the left m position of subpicture from the bottom n position of subpicture from the right We will lso write q (i,j,k) p, or the shortcut q p i,j,k (q (i,j,k) p). Moreover, if q (i,j,k) p, we define coor (i,j,k) (q, p) s the set of coordintes of p where q is locted. Conventionlly, coor(i,j,k)(q,p) = φ, if q is not subpicture of p. If q coincides with p we write coor(p) insted of coor (,,) (p, p). Exmple 2.3. Let p be tringulr picture of size 5 nd let q be subpicture of p t position (1,-2,1) of size 2 then the coor (1, 2,1) (q, p) ={(1, 2, 1), (1, 3, 2), (2, 3, 1)}. Definition 2.4. Let γ be n equivlence reltion on coor(p), written (x,y,z) γ (x,y,z ). Two subpictures q (i,j,k) p, q (x,y,z ) p re γ -equivlent, written q γ q,iff for ll pirs (x,y,z) coor (i,j,k) (q, p) nd (x,y,z ) coor (x,y,z )(q,p) it holds

5 Tringulr Tile Rewriting Grmmrs 1969 p(,,) p(, 1,1) p(1, 1,) p(, 2,2) p(1, 2,1) p(2, 2,) p(, 3,3) p(1, 3,2) p(2, 3,1) p(3, 3,) p(, 4,4) p(1, 4,3) p(2, 4,2) p(3, 4,1) p(4, 4,) Figure 2: (x,y,z) γ (x,y,z ). homogenous C-subpicture q p is clled mximl with respect to reltion γ iff for every γ -equivlent C-subpicture q it is, coor(q, p) coor(q,p)= φ coor(q,p) coor(q, p). In other words, q is mximl if ny C-subpicture which is equivlent to q is either subpicture of q or it is not overlpping. Definition 2.5. For tringulr picture p T size k is: k (p) ={q T k /q p}. the set of subpictures (or tiles) with Definition 2.6. Consider set of tiles ω T k. The loclly testble lnguge in the strict sense defined by ω (written Loc u (ω) u stnds for unbordered picture) is the set of pictures p T such tht k(p) ω. The loclly testble lnguge defined by finite set of tiles Loc u,eq ({ω 1,ω 2,...,ω n }) (eq stnds for the equivlity test) is the set of pictures p T such tht for some i, k( ˆp) = ω i. The bordered loclly testble lnguge defined by finite set of tiles Loc eq ({ω 1,ω 2,...,ω n }) is the set of pictures p T such tht for some i, k( ˆp) = ω i. Definition 2.7. [ubstitution] If p, q, q re pictures, q (i,j,k) p nd q,q hve the sme size, then p[q /q] (i,j,k) denotes the picture obtined by replcing the occurrence of q t position (i,j,k)in p with q. The min definition follows: Definition 2.8. There re four types of tringulr rrys nd they re clssified s U- rry, D-rry, L-rry nd R-rry. U D L R

6 197. Kuberl, Dr. T. Klyni, nd Dr. D.G. Thoms Horizontl overlpping It is defined between U-rry nd D-rry of equl size nd it is denoted by the symbol over.l(g) = L(G 1 ) over.l(g 2 ). Exmple: U over D imilrly defined for verticl overlpping over, left overlpping \ over nd right overlpping / over. Definition 2.9. Tringulr Tile Rewriting Grmmr is tuple (,N,,R)where is the terminl lphbet, N is set of non-terminl symbols, N is the strting symbol, R is set of rules. R my contin two kinds of rules: Fixed size: {t 1,t 2 }, where N, t 1,t 2 ( N) k with k>. Vrible size: ω, where N, ω ( N) k with 1 k 2. Intuitively fixed size rule is intended to mtch subpicture of (smll) bounded size, identicl to the right prt {t 1,t 2 }. vrible size rule mtches nd subpicture of ny size which cn be tiled using ll the elements {t 1,t 2 } of the tile set ω. However, fixed size rules re not specil cse of vrible size rules. Definition 2.1. Consider grmmr G = (,N,,R), let p, p ( N) k be pictures of identicl size, nd let γ,γ be equivlence reltions over coor(p). We sy tht (p,γ ) derives in one step from (p, γ ) written (p, γ ) (p,γ ) iff for some G N nd for some rule ρ : R there exist in p -subpicture r (i,j,k) p, mximl with respect to γ, such tht: p is obtined substituting γ with picture s, tht is p = p[s/r] (i,j,k) where s is defined s follows: Fixed size: if ρ = {t 1,t 2 }, then s ={t 1,t 2 }; Vrible size: if ρ = ω, then s Loc u,eq (ω). Let z be coor (i,j,k) (r, p). Let Ɣ be the γ -equivlence clss contining z. Then, γ is equl to γ, for ll the equivlence clss not equl to Ɣ; Ɣ in γ is divided in two equivlence clsses, z nd its complement with respect to Ɣ (= φ if z = Ɣ). More formlly: γ = γ \{(x 1,y 1,z 1 ), (x 2,y 2,z 2 )/(x 1,y 1,z 1 ) z or (x 2,y 2,z 2 ) z} The subpicture r is nmed the ppliction re of the rule ρ in the derivtion step. We sy tht (q, γ ) is derivble from (p, γ ) in n steps, written (p, γ ) n G (q, γ ) iff p = q nd γ = γ when n =, or there re picture r nd n equivlence reltion γ such tht (p, γ ) n 1 G (q, γ ) nd (γ, γ ) G (q, γ ). We use the bbrevition (p, γ ) G (q, γ ) for derivtion with n steps.

7 Tringulr Tile Rewriting Grmmrs 1971 Definition The picture lnguge defined by grmmr G (written L(G)) isthe set of p T such tht, if p =k, then ( k, coor(p) coor(p) coor(p) G (p, γ ) (1) where the reltion γ is rbitrry. For short we write G p. Notice tht the derivtion strts with -picture isometric with the terminl picture to be generted nd with the universl equivlence reltion over the co-ordintes. The equivlence reltion computed by ech step of (1) is clled germinl reltion. When writing exmples by hnd, it is convenient to visulize the equivlence clsses of germinl reltion, by ppending the sme numericl subscript to the pixels of the ppliction re rewritten by derivtion step. The fmily of lnguges generted by TTRG is denoted by L(TTRG). To illustrte we present two exmples. Exmple [Chinese tringulr rrys] G = (,N,,R), where ={,<,>, }, N ={} nd R consists of one fixed size, one vrible size rule: { < >, > }, {,,, <,,, picture in L(G) is: < > < > < > ize 8 } Obtined by repeted ppliction of fixed size nd vrible size rules. imilrly, we cn get the picture lnguge L(G), for the size of 5, 8, 11, 14,... In generl, the size of the picture lnguges form the generl form is in rithmetic progression (.P). Exmple [Dyck nlogue] The next lnguge, superset of Chinese tringulr rrys cn be defined by sort of blnking rule. ut since the terminls cnnot be deleted without shering the picture, we replce them with chrcter. To obtin the

8 1972. Kuberl, Dr. T. Klyni, nd Dr. D.G. Thoms grmmr, we dd the following rules to the Chinese tringulr rrys grmmr: / ( ) To illustrte, in Figure 2 we list the derivtion steps of picture. Non-terminls in the sme equivlence clss re mrked with the sme subscript. We need to use indices in the production of the grmmr: ize ^ < >

9 Tringulr Tile Rewriting Grmmrs 1973 ^ < > picture in L(G) is: < > < > < > < > < > < > nd is obtined by repeted ppliction of the vrible size rule nd fixed size rule. sic property The fmily L(TTRG) is closed under union, overlpping, closures, rottion nd projection. Proof. Consider two grmmrs G 1 = (, N 1,,R 1 ) nd G 2 = (, N 2,,R 2 ). uppose for simplicity we ssume tht N 1 N 2 = φ, N 1 N 2, nd tht G 1, G 2 generte pictures hving size t lest 2. Then it is esy to show tht the grmmr G = (, N 1 N 2 {},,R 1 R 2 R), where Union. { [[ ]] [[ ]]} R =, is such tht L[G] =L[G 1 ] L[G 2 ]. Overlpping. Tringulr rrys of sme size cn be overlpped using the following overlpping opertions.

10 1974. Kuberl, Dr. T. Klyni, nd Dr. D.G. Thoms Horizontl overlpping. It is defined between U-rry nd D-rry of equl size nd it is denoted by the symbol over. R = { [[ ]] [[, Verticl overlpping. It is defined between L nd R rrys of sme size nd it is denoted by the symbol over. R =, Right overlpping. It is defined between ny two glule iso-tringulr rrys of sme size nd it is denoted by the symbol / over. R = { [[ ]] [[, Left overlpping. It is defined between ny two glule iso-tringulr rrys of sme size nd it is denoted by the symbol \ over. R = { [[ ]] [[, ]]} ]]} ]]} Closures: over / over / / over / \ over G = (, N 1 {},,R 1 R), where, { [[ ]] [[ ]]} R =, is such tht L(G) = L(G) over. imilrly over, / over \ over cses re nlogous. Rottion bout 9. Construct the grmmr G = (,N,,R ), where R, is such tht, if t R 1 is fixed size rule, then t R is in R ; ω R 1 is vrible size rule, then ω is in R, with t ω imply t R ω. It is esy to verify tht L(G) = L(G 1 ) R. Projection. Consider grmmr G 1 = ( 1,N 1,,R 1 ) nd projection π : 1 2. It is possible to build grmmr G 2 = ( 2,N 1,,R 2 ), such tht L(G 2 ) = π(l(g 1 )). imply pply π to unitry rules. Tht is, if x R 1, then π(x) R 2, while the other rules of G 1 remin in R 2 unchnged.

11 Tringulr Tile Rewriting Grmmrs Comprison Results In this section we compre TTRG with TLOC nd TT. Theorem 3.1. L(T LOC) L(T T RG). Proof. Consider locl two-dimensionl lnguge over defined by the set of llowed tringulr tiles. / Let = then n equivlent TTRG is, G ={,{},,R} where R is the set { θ/θ }. Lemm 3.2. L(T LOC u,eq ) L(TTRG). Proof. Consider Locl Tringulr picture lnguge over defined (without boundries) by the sets of llowed tiles {ω 1,ω 2,...,ω n }, ω i T 2. n equivlent grmmr is ω 1 /ω 2 /.../ω n. Exmple 3.3. Consider the lnguge TLOC u,eq.l(ω) = 1 1, 1 1,... generted by { ω = 1 1 It cn be generted by the following TTRG. Let G ={,N,,R}, when ={, 1}, N ={}, nd R consists of two fixed size rule nd one vrible size rule: { { Clerly, L(T LOC u,eq ) L(TTRG). }. 1 1, 1, 1 We now consider the nottions TT eq nd TT u,eq. Definition 3.4. The Tringulr tiling system TT eq nd TT u,eq re the sme s TT, with the following respective chnges; }, }.

12 1976. Kuberl, Dr. T. Klyni, nd Dr. D.G. Thoms Replce the locl lnguge defined by Definition 1.7 with TLOC eq ({ω 1,ω 2,...,ω n }) where ω i (1 i n) is finite set of tringulr tiles over Ɣ. Replce the locl lnguge defined by Definition 1.7 with TLOC u,eq ({ω 1,ω 2,...,ω n }) where ω i (1 i n) is finite set of tringulr tiles over Ɣ. InTT u,eq there is no boundry symbol #. Let L(TT eq ), L(TT u,eq ) be the fmilies of lnguges generted by TT eq nd TT u,eq respectively. Lemm 3.5. L(TT eq ) L(TT). Proof. We first prove L(TT) L(TT eq ). This is trivil, becuse if we consider the tringulr tile set ω of TT by tking {omeg 1,ω 2,...,ω n }=P(ω)(the power set), then we obtin n equivlent TT eq.nextwehvetoprovel(tt eq ) L(TT).In [1], the fmily of lnguges L(T LOC eq ( )) where is the set of tiles, is proved to be proper subset of L(TT), is closed with respect to projection, nd L(TT eq ) is the closer with respect to projection of L(T LOC eq ( )). Therefore L(TT eq ) L(TT). Hence proved. Lemm 3.6. L(TT u,eq ) L(TT eq ). Proof. First we prove L(TT eq ) L(TT u,eq ). Let T = (, Ɣ, {ω 1,ω 2,...,ω n },π) be TT eq. For every tringulr tile set ω i seprte its tiles contining the boundry symbol # (cll this subset ω i ) from the other tiles (ω i ). This is ω i = ω i ω i. Introduce new lphbet Ɣ nd bijective mpping br : Ɣ Ɣ, we use symbols in Ɣ to encode boundry nd new tile set δ i to contin them for every tile t in ω i, if there is tile in ω i which overlps with t, then encode this boundry in new tile t nd put it in the set δ i. For exmple, suppose b c ω i overlps with then ll # #, # b #, # # c ω i, br(b) br(c), br() b br(c), br() br(b) re in δ i. Consider TT u,eq, T = (,Ɣ Ɣ,,π ) where π extends π to π s follows: π (br()) = π () = π(), Ɣ nd ubr : Ɣ Ɣ Ɣ is defined s ubr() = br 1 () if Ɣ otherwise =, nd it is nturlly extended to tiles nd tile sets. is the set {ω/ω ω i δ i ubr(ω) = ω i ω δ i = φ 1 i n}. The proof of L(T ) = L(T ) is obvious. To prove L(TT u,eq ) L(TT eq ) c

13 Tringulr Tile Rewriting Grmmrs 1977 Let T = (, Ɣ, {ω 1,ω 2,...,ω n },π) be TT u,eq. To construct n equivlent TT eq, we introduce the boundry tile sets δ i, defined s follows. For every tile b c ω i, the following tiles re in δ i. # # #, # #, # #, # # b, b c, # c #, b # #, c # #. Consider TT eq, T = (,Ɣ,,π), where is the set {ω ω i /ω δ i ω = φ 1 i n}. It is esy to show tht L(T ) = L(T ). Theorem 3.7. L(TT) L(TTRG). Proof. It follows the bove theorem nd lemms nd the fct tht L(TT u,eq ) is the closure of L(T LOC u,eq ) with respect to projection. Remrk 3.8. L(TTRG) = L(TT). To show this we give the following exmple. Let G = (,N,,R)be TTRG where ={,x}, N ={}, nd R consists of the rules x {, x,,, x, x, x, x, x picture in L(G) is x x x x x x x x x x x x size 8 It is known tht the lnguge L(G) is generted by the grmmr (R : CF )T G, G = (N, I, {,x},p 1 P 2, 1, 2,T} N ={ 1, 2 } I ={ 1, 2, 2 } T = x P 1 ={ 1 T 1 }[( or )P 1 ={ 1 T. 1 } }

14 1978. Kuberl, Dr. T. Klyni, nd Dr. D.G. Thoms (or) P 1 ={ 1 T. 1 }] P 2 ={ , } L 1 ={<x> n <x> n < x >, n 2} L 2 ={ n <x> n <x> n <x> n,n 2} L 3 ={ n 1 <x> n <x> n <x> n <x> n <x> n 1,n 2}. ut L(G) is note generted by ny (R : R)T G. Clerly it is not generted by ny TT. i.e., L(G) is not tiling recognizble. Here TTRG is more powerful. 4. Conclusion In this pper we introduced Tringulr Tile Rewriting Grmmrs (TTRG). The closure properties of TTRG re proved for some bsic opertions. The expressive power of TTRG is greter thn the previous model TT. We investigte further interesting properties. References [1] Crespi Reghizyi,., Prdell, M., 25, Tile Rewriting Grmmrs nd Picture Lnguges, Theoreticl Comp. cience, 34, pp [2] Dor Gimmrresi nd ntonio Restivo, 1992, Recognible Picture Lnguges, Interntionl Journl Pttern Recognition nd rtificil Intelligence 6(2-3), pp pecil issue on Prllel Imge Processing. [3] Dor Gimmrresi nd ntonio Restivo, 1997, Two-dimensionl Lnguges, In rto lom nd Grzegorz Rozenberg, editors Hndbook of Forml Lnguges, 3, eyond Words, pp , pringer-verlg, erlin. [4] weety, F., Klyni, T., Thoms, D.G., 27, Hexgonl Tile Rewriting Grmmrs, 7th Ntionl Conference on Emerging Trends in utomt (ET 7), pp [5] tefno Crespi Reghizzi nd Mtteo Prdell, Tile Rewriting Grmmrs, DEI- Politecnico di Milno nd CNR IEIIT-MI, Pizz Leonrdo d Vinci, 32, I-2133 Milno, Itly. [6] Rni iromoney, 1987, dvnces in rry Lnguges, In Hrtmut Ehrig, Mnfred Ngl, Grzegorz Rozenberg, nd zriel Rosenfeld, editors, Proc. 3rd Int. Workshop on Grph-Grmmrs nd Their ppliction to Computer cience, 291, Lecture Notes in Computer cience, pp pringer-verlg. [7] Henning Fernu nd Rudolf Freund, 1996, ounded prllelism in rry grmmrs used for chrcter recognition. In Petr Perner, Ptrick Wng, nd zriel Rosenfeld, editors, dvnces in tructurl nd yntcticl Pttern Recognition (Proceedings of the PR 96), 1121, pp. 4 49, pringer-verlg.