Flat Splicing Array Grammar Systems Generating Picture Arrays

Transcription

1 Intrntionl Journl of Computr Informtion Systms n Inustril Mngmnt Applictions. ISSN Volum 8 (2016) pp c MIR Ls, Flt Splicing Arry Grmmr Systms Gnrting Pictur Arrys G. Smnilthompson 1, N. Gnnmlr Dvi 1, Atuly K. Ngr 2 n K.G. Surmnin 2 1 Dprtmnt of Mthmtics, Mrs Christin Collg, Tmrm, Chnni Ini {smnilthompson,ngmcc}@gmil.com 2 Dprtmnt of Mthmtics n Computr Scinc, Fculty of Scinc, Livrpool Hop Univrsity, Livrpool, L16 9JD, UK ngr@hop.c.uk, kgsmni1948@yhoo.com Astrct: Whil stuying th rcominnt hviour of DNA molculs, H (1987) introuc nw oprtion, cll s- plicing on wors or strings, which r finit squncs of symols. Thr hs n intnsiv rsrch using th concpt of splicing on strings in th contxt of DNA computing, stlishing importnt thorticl rsults on computtionl univrslity. A prticulr clss of splicing, known s flt splicing on strings ws rcntly consir n this oprtion ws xtn to provi pictur rry gnrting two-imnsionl mols. Mking us of th oprtion of flt splicing on rrys, w propos hr grmmr systm, cll flt splicing rgulr rry grmmr systm (FSRAGS), s nw mol of pictur gnrtion. Th componnts of FSRAGS gnrt pictur rrys working in prlll using th ruls of two-phs grmmr cll 2RLG n with two iffrnt componnts of th FSRAGS communicting using th rry flt splicing oprtions on columns n rows of th rrys. W stlish som comprison rsults ringing out th gnrtiv powr of FSRAGS n lso xhiit th powr of FSRAGS in gnrting crtin floor signs. Kywors: Flt splicing, Pictur rry, Pictur lngug, Forml lngugs, Grmmrs, Grmmr systms I. Introuction In th qust for crting ltrntivs for th clssicl silicons computing, th fil of DNA computing [13, 14], sis othrs, ws orn with Almn solving smll instnc of Hmilton pth prolm [1] s on th us of DNA molculs n oprtions on DNA squncs. In th stuy of molling th DNA rcomintion procss unr th ction of rstriction nzyms n ligs, H [8] strct this phnomnon in his sminl ppr on forml mol of th rcominnt hviour of DNA molculs n fin th oprtion of splicing of wors, which r strings of symols. This oprtion of splicing of two wors u = u 1 u 2 n v = v 1 v 2 involvs cutting u, twn u 1 n u 2 n v, twn v 1 n v 2, s ictt y crtin ruls known s splicing ruls n psting th prfix u 1 of u with th suffix v 2 of v yiling nw wor w = u 1 v 2. Susquntly, this opn up svrl thorticl invstigtions [6, 10, 11] giving ris to importnt n p lngug-thortic rsults in th r of forml lngug thory, which is wily ccpt to th ckon [5, 12] of thorticl computr scinc. Bs on th fct tht DNA occur in circulr form, H [9, 10] introuc th oprtion of splicing on circulr wors. Motivt y spcific form of splicing on circulr wors, rcntly, iffrnt form of splicing, cll flt splicing on wors hs n introuc in [2]. Th oprtion of flt splicing on pir of wors (α, β) involvs cutting α n insrting β into it, s irct y rul, cll flt splicing rul. Inspir y prolms in img procssing, svrl gnrtiv mols of two-imnsionl pictur rrys s on lngug thory, hv n propos n stui (s, for xmpl, [7, 15, 16, 21, 24]). Rcntly, th oprtion of flt splicing on wors hs n xtn to pictur rrys [23] n n rry flt splicing systm (AF S) hs n fin to scri pictur rry lngugs. On th othr hn, th notion of grmmr systm consisting of svrl grmmrs or othr lngug intifying mchnisms cooprting ccoring to som wll-fin protocol, ws introuc s forml frmwork for molling istriut complx systms [3]. Bs on iffrnt motivtions, vrity of grmmr systm mols hv n propos n intnsivly invstigt. A iffrnt typ of grmmr systm, cll Splicing Grmmr Systm ws introuc in [4] with th componnt grmmrs working in prlll n communiction twn componnts ing on y th io-inspir oprtion of splicing on strings of symols introuc y H [8]. This systm ws xtn to rrys in [22] y introucing splicing rry grmmr systm. Hr w mk us of th rry flt splicing oprtion n introuc grmmr systm cll flt splicing rgulr rry grmmr systm (F SRAGS) n xmin th powr MIR Ls, USA

2 337 Smnilthompson t l. of this systm in gnrting pictur rry lngugs. Th componnts of th F SRAGS involv th ruls of th twophs grmmr 2RLG [7, 19] working in prlll n th rry flt splicing oprtion [23] is us for communiction mong two iffrnt componnts in trms of insrtion of n rry gnrt in componnt into n rry gnrt in iffrnt componnt. W stlish som comprison rsults ringing out th powr of (F SRAGS). W lso xhiit th us of (F SRAGS) in gnrting crtin floor signs. A prliminry vrsion of this ppr pprs in [18]. II. Prliminris For notions n rsults rlt to forml string lngugs n rry lngugs, w rfr to [7, 17]. A linr wor or simply, wor w = 1 2 n is finit squnc of symols i, 1 i n, tkn from finit lpht Σ. Th mpty wor with no symols, is not y λ. Th st of ll wors ovr Σ is not y Σ. Th lngth w of wor w is th numr of symols in th wor, incluing rptitions of symols. For xmpl,th lngth of th wor is 5. For ny wor w, w not y t w th wor w writtn vrticlly. For xmpl, if w = ovr {, }, thn t w = A pictur rry or simply, n rry M of siz m n ovr n lpht Σ is rctngulr rry with m rows n n columns n is of th form M = 11 1n..... m1 mn whr ch symol ij Σ, 1 i m, 1 j n. W not y M r n M c, th numr of rows n th numr of columns of M rspctivly. Th st of ll rctngulr rrys ovr Σ is not y Σ, which inclus lso th mpty rry λ with no symols. Σ ++ = Σ {λ}. A pictur lngug is sust of Σ. Th column ctntion X Y of n rry X of siz p q n n rry Y of r s is fin only whn p = r n likwis th row ctntion X Y is fin only whn q = s. In othr wors, in column ctntion X Y, Y is ttch to th right of X whil in row ctntion X Y, Y is ttch low X. Also X λ = λ X = X λ = λ X = X, for vry rry X. For xmpl, if X =, Y =., Z = thn X Y =, Y Z = An rry grmmr mol of th non-isomtric vrity gnrting rctngulr rrys of symols is th two-imnsionl right-linr grmmr (originlly cll 2D mtrix grmmr [19]), which w cll hr s two-imnsionl right-linr grmmr, consistnt with th trminology us in [7]. Dfinition 1 A two-imnsionl right-linr grmmr (2RLG) [7, 19] is G = (V h, V v, V i, T, S, R h, R v ) whr V h, V v, V i r finit sts of symols, rspctivly cll s horizontl nontrminls, vrticl nontrminls n intrmit symols; V i V v ; T is finit st of trminls; S V h is th strt symol; R h is finit st of horizontl ruls of th form X AY, X A X, Y V h, A V i ; R v is finit st of vrticl ruls of th form X Y or X, X, Y V i, T λ. Thr r two phss of rivtion in 2RLG. In th first horizontl phs, strting with S th horizontl ruls r ppli (s in rgulr grmmr) gnrting strings ovr intrmits, which ct s trminls of this phs. In th scon vrticl phs ch intrmit in such string srvs s th strt symol for th scon phs. Th vrticl ruls r ppli in prlll in this phs for gnrting th columns of th rctngulr rrys ovr trminls. In this phs t rivtion stp, ithr ll th ruls ppli r of th form X Y, T or ll th ruls r of th form X Y or ll th ruls r th trminting vrticl ruls of th form X, T or ll of th form X λ. Whn th vrticl gnrtion hlts, th rry otin ovr T, is collct in th pictur lngug gnrt y th 2RLG. Not tht th pictur lngug gnrt y 2RLG consists of rctngulr rrys of symols. W not y L(2RLG) th fmily of rry lngugs gnrt y two-imnsionl right-linr grmmrs. W illustrt th working of 2RLG with n xmpl. Exmpl 1 Consir th 2RLG G L with rgulr horizontl ruls S AX, X BX, X B whr A, B r th intrmit symols n th vrticl ruls A A, A, B B, B whr, r th trminl symols. In th first phs, s in string grmmr, th horizontl ruls gnrt wors of th form AB n, n 1. For xmpl, th gnrtion of AB 3 is s givn low: S AX ABX ABBX ABBB = AB 3 on pplying th thr horizontl ruls on ftr nothr, with th rul X BX ppli twic. In th scon phs vry symol in such wor AB n is rwrittn in prlll ithr y using th ruls A A, B B in which cs th rivtion cn likwis continu ing rows of th form or y using th ruls A, B,

3 Flt Splicing Arry Grmmr Systms Gnrting Pictur Arrys 338 in which cs th rivtion trmints ing row of th form, thus gnrting pictur rrys, on mmr of which is shown in Fig. 1. In fct, for this pictur rry, th rivtion in th scon phs is s follows: A B B B A B B B A B B B A B B B on pplying th nontrminl ruls A A, B B. Whn th trminl ruls A, B B, B r ppli togthr, rivtion trmints yiling th rry in Fig. 1. Figur. 1: A pictur rry gnrt in Exmpl 1 An xtnsion of th 2RLG, which w cll hr s two-imnsionl tl right-linr grmmr (2T RLG) (originlly rfrr to s 2D tl mtrix grmmr in [20]) ws introuc in [20] to gnrt pictur lngugs tht cnnot gnrt y ny 2RLG. A 2T RLG is fin similr to 2RLG with th following iffrnc: In th scon phs, th right-linr ruls r group into iffrnt sts, cll tls such tht i) ll th ruls in tl r nontrminl ruls of th form ithr A B only or A B only or ii) ll r trminl ruls of th form ithr A only or A λ only n t tim ruls from only on tl r us in th scon phs. W not y L(2T RLG) th fmily of rry lngugs gnrt y two-imnsionl tl right-linr grmmrs. As n illustrtion, w giv n xmpl of 2T RLG. Exmpl 2 Consir th 2T RLG G H with rgulr horizontl ruls S AX, X BX, X BY, Y A whr A, B r th intrmit symols n th tls of vrticl ruls r t 1 = {A A, B B}, t 2 = {A C, B cd}, t 3 = {C C, D D}, t 4 = {C, D } whr,, c r th trminl symols. In th first phs, th horizontl ruls gnrt wors of th form AB n A, n 1. In th scon phs vry symol in such wor AB n A is rwrittn in prlll y using th tls of ruls. Appliction of th tl t 1 crtin numr of tims, follow y t 2 onc which is thn follow y t 3 gin crtin numr of tims n finlly trminting th rivtion in this scon phs, gnrts pictur rrys, on mmr of which is shown in Fig. 2. An oprtion, cll flt splicing on linr wors, is consir y Brstl t l. [2]. A flt splicing rul r is of th form (α γ δ β), whr α, β, γ, δ r wors ovr givn lpht Σ. For two wors u = xαβy, v = γzδ, n ppliction of th flt splicing rul r = (α γ δ β) to th pir (u, v) yils th wor w = xαγzδβy. In othr wors, th scon wor v c c c c Figur. 2: A pictur rry gnrt in Exmpl 2 is insrt twn α n β in th first wor u s rsult of pplying th rul r. Th notion of flt splicing on wors [2] hs n xtn to rrys in [23], y introucing two kins of flt splicing ruls, cll column flt splicing rul n row flt splicing rul n thus nw mol of pictur gnrtion, cll rry flt splicing systm is introuc in [23]. Dfinition 2 [23] Lt V n lpht. (i) A column flt splicing rul is of th form ( t ( 1 2 ) t (x 1 x 2 ) t (y 1 y 2 ) t ( 1 2 )) whr 1, 2, 1, 2 Σ {λ} with 1 = 2 n 1 = 2, x 1, x 2, y 1, y 2 Σ {λ} with x 1 = x 2 n y 1 = y 2. (ii) A row flt splicing rul is of th form (c 1 c 2 u 1 u 2 v 1 v ) whr c 1, c 2, 1, 2 Σ {λ} with c 1 = c 2 n 1 = 2, u 1, u 2, v 1, v 2 Σ {λ} with u 1 = u 2 n v 1 = v 2. (iii) Lt r 1, r 2,, r m 1 squnc of (m 1) column flt splicing ruls givn y r i = ( t (α i α i+1 ) t (γ i γ i+1 ) t (δ i δ i+1 ) t (β i β i+1 )), for 1 i (m 1). Lt X, Y two pictur rrys, ch with m rows, for som m 1, n givn y X = X 1 t (α 1 α 2 α m ) t (β 1 β 2 β m ) X 2, Y = t (γ 1 γ 2 γ m ) Y t (δ 1 δ 2 δ m ), whr X 1, X 2, Y r rrys ovr Σ with m rows, α i, β i, Σ {λ} (1 i m), with α 1 = α 2 = = α m, β 1 = β 2 = = β m, γ i, δ i, (1 i m), Σ {λ} with γ 1 = γ 2 = = γ m, δ 1 = δ 2 = = δ m. An ppliction of th column flt splicing ruls r 1, r 2,, r m 1 to th pir of rrys (X, Y ) yils th rry Z = X 1 t (α 1 α 2 α m ) t (γ 1 γ 2 γ m ) Y t (δ 1 δ 2 δ m ) t (β 1 β 2 β m ) X 2. Th pir (X, Y ) yiling Z (X, Y ) c Z. is thn not y (iv) Lt s 1, s 2,, s n 1 squnc of (n 1) row flt splicing ruls givn y s j = (η j η j+1 (µ j µ j+1 ) (ν j ν j+1 ) θ j θ j+1 ),

4 339 Smnilthompson t l. for 1 j (n 1). Lt U, V two pictur rrys, ch with n columns, for som n 1, n givn y U = U 1 (η 1 η 2 η n ) (θ 1 θ 2 θ n ) U 2, V = (µ 1 µ 2 µ n ) V (δ 1 δ 2 δ n ) whr U 1, U 2, V r rrys ovr Σ with n columns, η j, θ j, (1 j n), Σ {λ} with η 1 = η 2 = = η n, θ 1 = θ 2 = = θ n, µ j, ν j, (1 j n), Σ {λ} with µ 1 = µ 2 = = µ n, ν 1 = ν 2 = = ν n. An ppliction of th row flt splicing ruls s 1, s 2,, s n 1 to th pir of rrys (U, V ) yils th rry W = U 1 (η 1 η 2 η n ) (µ 1 µ 2 µ n ) V (δ 1 δ 2 δ n ) (θ 1 θ 2 θ n ) U 2. Th pir (U, V ) yiling W (U, V ) r W. is thn not y (v) An rry flt splicing rul is ithr column flt splicing rul or row flt splicing rul. Th nottion nots ithr c or r. (vi) For pictur lngug L Σ n st R of rry flt splicing ruls, w fin f(l) = {M Σ (X, Y ) M, forx, Y L, n som rul in R}. (vii) An rry flt splicing systm (AF S) is A = (Σ, M, R c, R r ) whr Σ is n lpht, M is finit st of rrys ovr Σ, cll initil st, R c is finit st of column flt splicing ruls n R r is finit st of row flt splicing ruls. Th pictur lngug L(A) gnrt y A is itrtivly fin s follows: f 0 (M) = M; For i 0, f i+1 (M) = f i (M) f(f i (M)); L(A) = f (M) = i 0 f i (M). Th fmily of pictur lngugs gnrt y rry flt splicing systms is not y L(AF S). An illustrtion of th ppliction of column n row flt splicing ruls s wll s th working of n rry flt splicing systm is givn in th following xmpl. Exmpl 3 Consir th rry flt splicing systm A R with lpht {, } n th initil st consisting of th rry M = { }. Th column flt splicing rul is c whr c = ( ). Th row flt splicing ruls r r 1, r 2, r 3 whr r 1 = ( ), r 2 = ( ), r 3 = ( ). Th column flt splicing rul c is pplicl to th pir (M, M). Not tht oth th componnts in th pir ing th sm initil rry M, th rquirmnt of qul numr of rows for th ppliction of column flt splicing rul c is stisfi. Also th scon rry in th pir gins with th column n ns with th column, s rquir in th rul c. Th first rry is cut twn th columns n whil th scon rry is insrt twn thm, yiling th rry. On th othr hn, th row flt splicing ruls r 1, r 2 cn us to xpn th rry rowwis. For xmpl, th squnc of ruls r 1, r 2, r 3 coul ppli to th pir of rrys with oth componnts hving th sm rry. Agin not tht oth th componnts in th pir ing th sm rry, th rquirmnt of qul numr of columns for th ppliction of squnc of row flt splicing ruls, is stisfi. In fct, th first rry is cut twn th first row n th scon row which is lso. Th scon rry stisfis th rquirmnts of th squnc of ruls r 1, r 2, r 3. Th scon rry thrfor cn insrt into th first rry to yil th rry. Procing lik this, w comput th succssiv trms f 0 (M), f 1 (M),. In fct f 0 (M) = M = { }, f 1 (M) = M f(m) = {,,., }, Thus th pictur lngug L(A R ) = f (M) consists of rctngulr rrys of vn si lngth ovr th symols,. Also if th numr of columns in such n rry is 2n, thn th first n columns r ovr whil th nxt n columns r ovr. Figur. 3: A mmr of th lngug of A R On such pictur rry is shown in Fig. 3. III. Flt Splicing Rgulr Arry Grmmr Systm W now introuc nw mol of pictur gnrtion, cll flt splicing rry grmnmr systm. Dfinition 3 A flt splicing rgulr rry grmmr systm (F SAGS) is construct G s = (V h, V v, V i, T, (S 1, R h 1, R v 1),..., (S n, R h n, R v n), F ) whr V h, V v, V i r rspctivly, th finit sts of horizontl, vrticl n intrmit nontrminls; V i V v ; T is th trminl lpht; S i, 1 i n is th strt symol of

5 Flt Splicing Arry Grmmr Systms Gnrting Pictur Arrys 340 th corrsponing componnt; Ri h, 1 i n is finit st of horizontl ruls, which r rgulr of th forms X AY, X A, X, Y V h, A V i ; Ri v, 1 i n is finit st of right-linr vrticl ruls of th forms A B, A B, A, A, B V v, T ; F is finit st of rry flt splicing ruls Th rivtions tk plc in two phss s follows : Ech componnt grmmr gnrts wor cll intrmit wor, ovr intrmits strting from its own strt symol n using its horizontl ruls ; th rivtions in this phs r on with th componnt grmmrs working in prlll. In th scon phs ny of th following stps cn tk plc: (i) ch componnt grmmr cn rwrit s in two imnsionl mtrix grmmr using th vrticl ruls, s- trting from its own intrmit wor gnrt in th first phs. (Th componnt grmmrs rwrit in prlll n th ruls r ppli togthr). Not tht th componnt grmmrs togthr trmint with ll ruls us in th form A or togthr continu rwriting in th vrticl irction with ll ruls us ithr in th form A B or in th form A B. (ii) At ny instnt th pictur rry X gnrt in th i th componnt for som i, 1 i n n th pictur rry Y gnrt in th j th componnt for som j, 1 j n cn flt splic using rry flt splicing ruls, ithr column or row flt splicing ruls s in finition 2, thus yiling pictur rry Z in i th componnt. In ny othr componnt (othr thn i th componnt), th rrys gnrt t this instnt will rmin unchng uring this flt splicing procss. Thr is no priority twn stps (i) n (ii). Th lngug L i (G s ) gnrt y th ith componnt of G s consists of ll pictur rrys, gnrt ovr T, y th rivtions scri ov. This lngug will cll th iniviul pictur rry lngug of th systm n w my choos this to th lngug of th first componnt. Th fmily of iniviul pictur rry lngugs gnrt y F SRAGS with t most n componnts is not y L n (F SRAGS). W illustrt th working of F SRAGS with n xmpl. Exmpl 4 Consir th F SAGS G 1 with two componnts givn y whr ({S 1, S 2, X, Y }, {A, C, D}, {A, C, D}, {, c, }, (S 1, R h 1, R v 1), (S 2, R h 2, R v 2), F ) R h 1 = {S 1 AX, X CX, X A, }, R h 2 = {S 2 Z, Z DZ, Z D}, R v 1 = {A A, C cc, A, C c}, R v 2 = {A A, D D, A, D } c c c c c c c c c c c c () () Figur. 4: () Pictur rry gnrt in th first componnt of G 1 () Pictur rry gnrt in th scon componnt of G 1 c c c c c c c c c c c c Figur. 5: A pictur rry gnrt y G 1 n F consists of th column flt splicing rul c 1 = ( c c ). Strting from S 1, th horizontl rgulr ruls in th first componnt gnrt wor of th form AC n 1 A on pplying th rul S 1 AX onc follow y th ppliction of th rul X CX (n 1) tims n finlly th rul X A onc, trminting th rivtion in th first componnt in th horizontl phs. At th sm tim in th scon componnt rivtions in th horizontl phs tk plc in prlll strting from S 2 n pplying th rul S 2 Z onc follow y th ppliction of th rul Z DZ (n 1) tims n finlly th rul Z D onc, yiling th wor D n for th sm n. In th scon phs vrticl rivtions in oth th componnts tk plc in prlll. In th first componnt n m (n + 1) pictur rry s in Fig. 4() is gnrt whil in th scon componnt n m n pictur rry s in Fig. 4() is gnrt. At this point, using th column flt splicing rul c 1 s mny tims s n, th rry in Fig. 4() is insrt in th rry in Fig. 4() twn th first column of s n th scon column of c s to yil pictur rry of th form s in Fig. 5. Thorm 1 L 2 (F SRAGS) \ L(AF S). Proof 1 Th pictur rry lngug L gnrt y th F SRAGS G 1 in Exmpl 4 consists of pictur rrys M ch of which hs th following proprty P : M hs xctly two columns of s with on, th lftmost column n th othr, th rightmost column, sis othr columns m of ithr c s or s. But L cnnot gnrt y ny AF S sinc th flt splicing ruls will insrt on rry of L into nothr, thry yiling pictur rrys which will violt th proprty mntion ov. Thorm 2 L(2RLG) = L 1 (F SRAGS) L 2 (F SRAGS).

6 341 Smnilthompson t l. Proof 2 Th qulity L(2RLG) = L 1 (F SRAGS) follows y noting tht th F SRAGS with on componnt will hv rwriting ruls s in 2RLG n th st of rry flt s- plicing ruls cn tkn to n mpty st s ths ruls cn ppli only whn thr r t lst two componnts. Th pictur rry lngug L of Exmpl 4, is gnrt y F SRAGS with two componnts. An rry of L hs th proprty tht thr r (n + 1) conscutiv columns of s (strting from th scon column) follow y n columns of c s. But this proprty which involvs proportion in th numr of columns of c s n th numr of columns of s (nlogous to th contxt-fr string lngug { (n+1) c n n 1}) cnnot mintin y rgulr horizontl ruls of 2RLG n hnc L cnnot gnrt y ny 2RLG. Sinc L(2RLG) = L 1 (F SRAGS) it follows tht ny F SRAGS with only on componnt cnnot gnrt th pictur rry lngug L. This provs th propr inclusion L 1 (F SRAGS) L 2 (F SRAGS). Thorm 3 L 2 (F SRAGS) \ L(2T RLG) Proof 3 As not in th proof of Thorm 2, th pictur rry lngug L of Exmpl 4, which is gnrt y F SRAGS with two componnts, hs th ftur tht n rry of L involvs proportion in th numr of columns of c s n th numr of columns of s. This proprty cnnot hnl y ny 2T RLG s th ruls in th first phs of 2T RLG r only rgulr ruls. Thus L cnnot gnrt y ny 2T RLG. In [20], nlogous to 2T RLG, two-imnsionl grmmr, hr cll two-imnsionl tl contxt-fr grmmr 2T CF G (originlly cll 2D tl CF mtrix grmmr) ws consir in [20] n this 2D grmmr hs contxt-fr grmmr kin of ruls in th first phs whil th scon phs hs tls of right-linr ruls s in 2T RLG. Drivtions r on s in 2T RLG. W not y L(2T CF G), th fmily of pictur rry lngugs gnrt y 2T CF G. Also, whn thr r xctly two tls in th scon phs of 2T CF G, with on tl consisting of ll right-linr ruls of th form A B n nothr tl consisting of ll trminl ruls of th form A, whr A, B r nontrminls n is trminl symol, th 2T RLG is in 2RLG. W now show tht th fmily L 3 (F SRAGS) of pictur lngugs gnrt y flt splicing rgulr rry grmmr systms with thr componnts,contins pictur lngugs tht cnnot gnrt y ny 2T CF G. Thorm 4 L 3 (F SRAGS) \ L(2T CF G) Proof 4 Consir th pictur rry lngug L 2 whos lmnts r pictur rrys with two rows of th forms or M 1 = M 2 = c c c c with th columns of s ing qul in numr to th columns of s in th formr rry whil this numr lso quls th numr of columns of c s in th lttr rry. This kin of pictur lngug cnnot gnrt y ny 2T CF G s th ftur prsnt in th rrys of th form M 2, nmly, qul numr of columns of s, s n c s is nlogous to th contxt-snsitiv string lngug { n n c n n 1} cnnot hnl y th contxt-fr kin of ruls in th first phs of 2T CF G. But th lngug L 2 is gnrt y th following F SRAGS G 2 with thr componnts: Dfin whr G 2 = (V h, V v, V i, T, (S 1, R h 1, R v 1), (S 2, R h 2, R v 2), (S 3, R h 3, R v 3), F ) T = {,, c,,, f}, V h = {S 1, S 2, S 3, X, Y, Z}, V i = {A, D, B, E, C, F }, V v = {A, D, B, E, C, F, A, D, B, E, C, F, A, D, B, E, C, F }. R h 1 = {S 1 AS 1, S 1 AX, X D}, R h 2 = {S 2 BS 2, S 2 BY, Y E}, R h 3 = {S 3 CS 3, S 3 CZ, Z F }, R v 1 = {A A, D D, A A, B B, D D, E E, A, B, D, E, C c, F f} R v 2 = {B B, E E, B B, E E, B, E } R v 3 = {C c C, F f F, C C, F F, C c, F f} F consists of th following column flt splicing ruls: ( A B E D ), ( B c C f F E ) Drivtions in th thr componnts tk plc in prlll with th ppliction of th horizontl ruls S 1 AS 1, S 1 AX, X D in th first componnt yiling th wor A n D for som n 1, whil, likwis, in th scon n thir componnts, th horizontl ruls yil rspctivly th wors B n E n C n F. At tim ll thr wors riv involv th sm n. If th vrticl ruls A A, D D A D in th first componnt r ppli, n rry of th form A is gnrt in th first componnt whil, likwis, ppliction of similr ruls in th scon n thir componnts gnrt th rrys rspctivly c c f of th forms B B E n C C F. If t this stg, th column flt splicing rul ( A E D ) is ppli, thn th rry otin is of th form A B A B B E D If now th vrticl

7 Flt Splicing Arry Grmmr Systms Gnrting Pictur Arrys 342 u u u u u u u u l h r u u u u u u u u l h r Figur. 8: A pictur rry gnrt y G F using rry flt splicing Figur. 6: A floor sign u u u u l h u u u u l h Figur. 9: A pictur rry gnrt y G F givn in Fig. 8, long with pictur rrys s th on givn in Fig. 9, cn gnrt y F SRAGS G F with two componnts. W o not giv th forml tils of G F ut informlly scri th i. Figur. 7: Primitiv Pttrns ruls such s A A tht chng th prim symols into oul prim symols r ppli follow y th ppliction of trminl ruls, thn pictur rrys of th form r gnrt which r collct in th pictur lngug of G 2. On th othrhn, prior to th ppliction of th trminl ruls, if th column flt splicing rul ( c B C f F E ) is ppli n th oul prim symols r chng into trminls y th ppliction of trminl ruls, thn rrys of th form c c c c r gnrt n r lso collct in th pictur lngug of G 2. IV. Appliction to Gnrtion of Floor Dsigns Gnrtion of crtin pictur pttrns, such s floor signs (s, for xmpl, Fig. 6), using rry gnrting grmmr mols hs n on [19]. Th i is to consir th pictur pttrn s n rry ovr crtin trminl symols n gnrt th rry with n rry grmmr. Thn sustitut for ch symol som rlvnt primitiv pttrn of th pictur to gnrt, yiling th pttrn. Hr w xplin how to construct F SRAGS with two componnts to gnrt th collction of floor signs, on mmr of which is shown in Fig. 6. Th primitiv pttrns involv in this floor sign r shown in Fig. 7. Th pictur rry ncoing th floor sign is givn in Fig. 8. Th pictur rry lngug consisting of pictur rrys such s th on f f Th i is to llow th first componnt gnrt th lft hlf of th pictur rry (with th rightmost column m of only h ) whil th scon componnt gnrts th scon hlf n rry flt splicing ruls r crt to conctnt th right hlf to th right of th lft hlf, yiling th pictur rry of th form in Fig. 8. Whn th rlvnt primitiv pttrns r sustitut for th corrsponing symols, this yils th floor sign pttrns s in Fig. 6. V. Conclusion n Discussion W hv consir hr grmmr systm tht involvs rgulr ruls s fin in th two-phs 2RLG n rry flt splicing ruls introuc in [23]. On coul lso consir in th flt splicing rry grmmr systm, CF horizontl ruls s in th two-phs two-imnsionl grmmr cll contxt-fr mtrix grmmr consir in [19], n xmin th gnrtiv powr of th rsulting systm, which might hlp scri mor complx floor signs. In fct it will of intrst to xplor th thorticl mol propos hr for othr possil pplictions in trms of xprimntl stuis. Acknowlgmnts Th uthors r grtful to th rviwrs for thir vry usful commnts. Rfrncs [1] Almn, L.M.: Molculr Computtion of Solutions to Comintoril Prolms, Scinc Nw Sris, 266(5187) (1994) [2] Brstl, J., Bosson, L., Fgnot, I. : Splicing systms n th Chomsky hirrchy. Thort. Comput. Sci. 436, 2-22 (2012).

8 343 Smnilthompson t l. [3] Csuhj-Vrjú, E., Dssow, J., Klmn, J., Păun, Gh.: Grmmr systms: A grmmticl pproch to istriution n cooprtion. Goron n Brch Scinc Pulishrs, [4] Dssow, J., Mitrn, V.: Splicing grmmr systms. Computrs n Artificil Intll. 15, (1996). [5] Esik, Z., Mrtin-Vi, C., Mitrn, V. (Es.): Rcnt Avncs in Forml Lngugs n Applictions, S- tuis in Computtioinl Intllignc, Springr-Vrlg, [6] Frun, R., Kri, L., Păun, Gh.,: DNA Computing Bs on Splicing: Th Existnc of Univrsl Computrs. Thory Comput. Syst. 32(1): (1999) [7] Gimmrrsi, D., Rstivo, A.: Two-imnsionl lngugs. In: [7], Vol. 3, pp (1997). [8] H, T.: Forml lngug thory n DNA: n nlysis of th gnrtiv cpcity of spcific rcominnt hviours. Bull. Mth. Biol. 49, (1987). [9] H, T., Circulr suggstions for DNA Computing, In: Pttrn Formtion in Biology, Vision n Dynmics(E. Cron, A. t l.), Worl Scintific, (2000) [10] H, T., Păun, Gh., Pixton, D. : Lngug thory n molculr gntics: gnrtiv mchnisms suggst y DNA rcomintion, In: [7], Vol. 2, pp , [11] H, T., Pixton, D.: Splicing n Rgulrity. Rcnt Avncs in Forml Lngugs n Applictions 2006: [12] Mrtin-Vi, C., Mitrn, V., Păun, Gh. (Es.): Forml Lngugs n Applictions. Springr-Vrlg, 2004 [13] Păun, Gh.: DNA computing s on splicing: univrslity rsults. Thor. Comput. Sci. 231(2): (2000) [14] Păun, Gh., Roznrg, G., Slom, A. : DNA Computing - Nw Computing Prigms. Txts in Thorticl Computr Scinc. An EATCS Sris, Springr [15] A. Rosnfl, Pictur Lngugs. Acmic Prss, Ring, MA, [16] A. Rosnfl, R. Siromony, Pictur lngugs survy, Lngugs of Dsign, 1 (1993) [17] Roznrg, G., Slom, A. (Es.): Hnook of Forml Lngugs Vol. 1 3, Springr, Brlin, (1997). [18] Surmnin, K.G., Smnilthompson, G., Dvi, N.G., Ngr, A.K. : Gnrting Pictur Arrys Bs on Grmmr Systms with Flt Splicing Oprtion,In: V. Snl t l. (s.), Innovtions in Bio-Inspir Computing n Applictions, Avncs in Intllignt Systms n Computing 424,Springr Intrntionl Pulishing Switzrln 2016, [19] Siromony, G., Siromony, R., Krithivsn, K. : Astrct fmilis of mtrics n pictur lngugs. Comput. Grph. Img. Procss. 1, (1972). [20] Siromony, R., Surmnin, K.G., Rngrjn, K. : Prlll/Squntil rctngulr rrys with tls. Int. J. Comput. Mth., 6A, (1977) [21] Surmnin, K.G., Ali, R.M., Gthlkshmi, M., Ngr, A.K. : Pur 2D pictur grmmrs n lngugs. Discrt Appl. Mth. 157 (2009) [22] Surmnin, K.G., Roslin Sgy Mry, A., Drsnmik, K.S. : Splicing Arry Grmmr Systms, Proc. Intrntionl Colloquium Thorticl Aspcts of Computing 2005, Lctur Nots in Computr Scinc 3722, Springr (2005). [23] Surmnin, K.G., Ngr, A.K., Pn, L. : A Pictur Arry Gnrting Mol Bs on Flt Splicing Oprtion, Proc. of th 10th Intrntionl Confrnc on Bioinspir Computing:Thoris n Applictions (BIC- TA), (2015). [24] Surmnin, K.G., Rngrjn, K., Mukun. M. (Es.): Forml Mols, Lngugs n Applictions. Sris in Mchin Prcption n Artificil Intll. Vol. 66, Worl Scintific Pulishing, Author Biogrphis G. Smilthompson ws orn on Mrch 13, H is pursuing his PhD progrmm from Jnury 2015 in th Dprtmnt of Mthmtics, Mrs Christin Collg, Ini with Prof. N. Gnnmlr Dvi s his suprvisor. H rciv his M.Phil gr in th yr His rsrch intrsts inclu DNA computing, pictur lngugs n grph lngugs. N. Gnnmlr Dvi ws orn on Dcmr 22, H hs n on th fculty of th Dprtmnt of Mthmtics, Mrs Christin Collg, Ini sinc 1980 n is currntly th H of this prtmnt. H ws th Dn of Scincs, Mrs Christin Collg uring th yrs H rciv his Ph.D. gr from th Univrsity of Mrs in H hs pulish numr of rsrch pprs in rput journls. His rsrch rs inclu grph grmmrs, pictur lngugs, DNA computing n stronomy. His - cmic visits inclu Univrsity of Brmn, Grmny n othr institutions in Frnc, Nthrlns n Grc. Atuly Ngr ws orn on Jnury 25, H is th Dn, Fculty of scinc, Livrpool Hop Univrsity, UK n lso hols th Fountion Chir s Profssor of Mthmticl Scincs in this univrsity. H rciv his Doctort (DPhil) in 1996 from th Univrsity of York. Prior to joining Livrpool Hop, h ws with th Dprtmnt of Mthmticl Scincs, n ltr t th Dprtmnt of Systms Enginring, t Brunl Univrsity, Lonon. Erlir h hs lctur t th Tt Institut of Funmntl Rsrch (TIFR), Bnglor, BITS Pilni n th Inin Institut of Tchnology (IITs) in Ini.

9 Flt Splicing Arry Grmmr Systms Gnrting Pictur Arrys 344 His rsrch is multi-isciplinry with xprtis in Nonlinr Mthmtics, Nturl Computing, Bio-Mthmtics n Computtionl Biology, Oprtions Rsrch, n Control Systms Enginring. H hs it volums on Intllignt Systms, n Appli Mthmtics. H is th Eitor-in- Chif of th Intrntionl Journl of Artificil Intllignc n Soft Computing (IJAISC) n srvs on itoril ors for numr of prstigious journls such s th Journl of Univrsl Computr scinc. H hs mor thn 200 pulictions in prstigious pulishing outlts n journls such s th Journl of Appli Mthmtics n Stochstic Anlysis, th Intrntionl Journl of Avncs in Enginring Scincs n Appli Mthmtics, th Intrntionl Journl of Fountions of Computr Scinc, th IEEE Trnsctions on Systms, Mn, n Cyrntics, Discrt Appli Mthmtics, Funmnt Informtic, IET Control Thory n Applictions n othrs. K.G. Surmnin ws orn on July 6, H rciv his PhD in 1980 from th Univrsity of Mrs, Ini. H ws on th fculty of th Dprtmnt of Mthmtics, Mrs Christin Collg, Chnni, Ini from 1970 to H hs srv s suprvisor for numr of PhD scholrs of Univrsity of Mrs. His min r of rsrch flls unr th ro topic of Mthmticl spcts of Computr Scinc n his rsrch intrsts inclu Comintorics on wors, pplictions of th thory of forml lngugs to pictur gnrtion, lrning n cryptogrphy n iologiclly motivt computing mols. H hs visit mny institutions in Frnc, Spin, Grmny, Vitnm, Chin, Mlysi, Jpn, UK on iffrnt prios for collortiv rsrch. H hs mny pulictions in rput journls. H ws visiting Profssor t th School of Mthmticl Scincs ( ) n th School of Computr Scincs ( ) t th Univrsiti Sins Mlysi, Mlysi. Currntly, h hs visiting profssor position t Livrpool Hop Univrsity, UK.