Picture Array Generation Based on Membrane Systems and 2D Context-Free Grammars

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1 Jounl of Mhemics nd Infomics Vol. 7, 07, 7-44 ISSN: (P), (online) Pulished Mch 07 DOI: hp://dx.doi.og/0.457/jmi.v75 Jounl of Picue y Geneion Bsed on Memne Sysems nd D Conex-Fee Gmms P.S. zeezun Nish, S. Hemlh, N.Gnnml Dvid nd K.G. Sumnin 4 Depmen of Mhemics, J.B..S. College fo Women, Chenni, Indi Depmen of Mhemics, S.D.N.B Vishnv College fo Women, Chenni, Indi Depmen of Mhemics, Mds Chisin College, Chenni, Indi Emil: kgsmni948@gmil.com Received 4 Mch 07; cceped 0 Mch 07 sc. The heoy of uom nd foml lnguges is n impon nch of heoeicl compue science. In wo-dimensionl (D) lnguge heoy, exended D conex fee picue gmm (EDCFPG) consiues heoeicl model of picue y geneion mking use of noneminl symols nd goups of conex-fee sing gmm ules in ewiing column o ow of noneminl symols in picue y wih no pioiy in he ewiing. Hee io-inspied model of P sysem in he fmewok of memne compuing is poposed y involving he EDCFPG kind of ules in he egions of he P sysem in geneing picue ys. The geneive powe of he esuling model of y P sysem is exmined y comping wih some well-known D gmm models. Keywods: Two-dimensionl y; conex-fee gmm; memne compuing. Inoducion The heoy of foml lnguges nd uom is consideed o e he ckone of heoeicl compue science. Moived, y vious pplicion polems in he es of imge nlysis, picue pocessing, pen ecogniion nd sevel ohes (see, fo exmple, [,4,9,]), wo-dimensionl lnguge heoy ws developed s n exension of foml sing lnguge heoy wih diffeen models of picue y geneion hving een poposed nd invesiged. In [0], simple ye genel enough wo-dimensionl picue y geneing model, known s Pue D conex-fee gmm (PDCFG) ws inoduced. In geneing picue y lnguge L, his D gmm involves only eminl symols nd ewies deivion sep, ll he symols in column o ow of picue y mking use of finie se of conex-fee sing gmm ules, heey yielding he picue ys of L. Thee hs een ohe invesigion [] on he mhemicl popeies of he picue lnguge fmily of PDCFG. n exension o his D gmm model, clled exended D conex-fee picue gmm (EDCFPG), wih moe picue geneive powe hn he PDCFPG ws poposed in [] y llowing 7

2 P.S.zeezun Nish, S.Hemlh, N.Gnnml Dvid nd K.G.Sumnin viles in he ules of his picue gmm model nd collecing he picue ys geneed ove se of eminl symols. On he ohe hnd, io-inspied compuing model, iniilly clled memne sysem u le nmed s P sysem (in honou of is oigino), ws inoduced y Păun [] inspied y he sucue nd funcioning of living cells. mong diffeen oles of P sysems, ewiing P sysems consiue specific kind wheein finie sings ove n lphe e he ojecs nd conex-fee ewiing ules e he evoluion ules. Exending he ewiing P sysems o ys sevel P sysem models fo picue y geneion, hve een consideed in he lieue (see fo exmple [9]). Hee we conside n y P sysem wih y ojecs nd EDCFPG kind of les of ules in is egions. We exmine he geneive powe of his y P sysem model y comping i wih some welleslished D picue geneing models.. Bsic definiions Fo noions of foml lnguge heoy we efe o [5] nd o [,4,0] fo y gmms. Le V e n lphe, which is finie se of symols. wod o sing u ove V is of he fom u u... un, whee u i V, i n, fo some n. The lengh of wod u is denoed y u. The empy wod wih no symols is denoed y λ. The se of ll * wods (lso clled hoizonl wods) ove V including he empy wod, is denoed y V. u Fo ny wod ( u) u = nd λ = λ. is of he fom u = u u... u, he veicl wod u is given y u =. We lso define m n m n picue y (lso clled n y) α ove n lphe V n M = whee V, i m, j n. The se of ll ** picue ys ove V is denoed y V, ** ++ λ. We denoe V λ y V. mn ij which includes he empy y lso denoed y We infomlly menion some of he picue geneing models h e needed in he susequen secion. In he D mix gmm model inoduced in [6], clled conex-sensiive mix gmm (CSMG), hee e wo phses of deivion. In he fis phse hoizonl wods ove eminl symols of his phse nd efeed o s inemedie symols, is geneed y Chomsky conex-sensiive gmm. Then in he second phse, fom ech of he inemedie symols in such hoizonl wods, veicl wods of he sme lengh ove eminl symols (of he second phse) e deived o consiue he columns of picue y ove eminl symols. The deivion in he second phse kes plce in pllel wih ll he ules used in sep eing he igh-line ules. We denoe he geneed picue lnguge clss y CSML. u n 8

3 Picue y Geneion Bsed on Memne Sysems nd D Conex-Fee Gmms In he fis phse of his D gmm model, if Chomsky conex-fee gmm o egul gmm is used insed of conex-sensiive gmm, hen i is especively clled conex-fee mix gmm (CFMG) o egul mix gmm (RMG) nd he coesponding picue lnguge clsses e especively denoed y CFML nd RML. Exending he D mix gmm, Tled mix gmm ws inoduced in [7] y specifying finie se of les of ules in he second phse of geneion wih ech le hving eihe igh-line noneminl ules o igh-line eminl ules. The ules in single le e used sep of deivion in he second phse fo geneing he columns in pllel. The esuling fmilies of picue y lnguges e denoed y TCSML, TCFML nd TRML. I hs een shown h XML TXML fo X {R, CF, CS }. We now ecll exended D conex-fee picue gmm consideed in []. In n EDCFPG, unlike D mix gmm, hee is single phse of deivion. Sing wih n xiom picue y, eihe column of noneminls o ow of noneminls pesen in he y is ewien especively y column le o ow le of conexfee ules sep of deivion. The igh sides of ll he ules in le e equied o e of equl lengh o ensue h he ecngul fom of he picue y ewien is minined. When he deivion emines yielding picue y ove eminl symols, he geneed y is colleced in he lnguge. Definiion.. [] n exended D conex-fee picue gmm (EDCFPG) is 5- uple G = ( V, T, P c, P, Μ) whee V is finie se of symols ; The elemens of V T e clled viles; T V is he se of eminl symols; P = i m, P = j n Ech, i m, clled column le, is c { } { }; c i j * se of conex-fee ules of he fom α, V T, α V such h fo ny wo ules α, B β in, we hve α = β ; Ech, j n, clled ow le, is se of conex-fee ules of he fom wo ules c i c i j C γ, C V T, γ V * such h fo ny ** C γ, D δ in, we hve γ = δ ; Μ V λ is finie se of j xiom ys. Deivions e defined s follows: Fo ny wo ys M, M, we wie M M, if M is oined fom M y eihe ewiing evey symol of column of M y ules of some column le in P c o of ow of M y ules of some ow le j in. c i P The eflexive nsiive closue of is denoed y *. The picue y lnguge L ( G) geneed y G is he se of picue ys * ** { M M 0 M T, fo some M 0 Μ}. The fmily of picue y lnguges geneed y EDCFPGs is denoed y EDCFPL. 9

4 P.S.zeezun Nish, S.Hemlh, N.Gnnml Dvid nd K.G.Sumnin Exmple.. Conside he EDCFPG G ( V, T, P, P,{ M }) = c 0, whee V = { S, X,, B, C,, }, T = {, }, {, S P c = c }, {, }, c P = M = X S c = { S B, X X }, { S C, X }, c = =,, B X 40 0 ; = {, C }. n illusion of how he les of ules e pplied is given S elow. Sing fom he xiom y M 0 = if he sequence of les of ules X pplied is,,, hen he successive ys oined in he deivion e given y, c c B X, S X X, wih he finl y ove eminl symols { }, eing colleced in he picue lnguge geneed y he EDCFPG G. On inepeing he symol s lnk, he y genee epesens he digiized lee T ove he symol, of he fom. y P sysem wih EDCFPG We now inoduce n y P sysem involving exended D conex-fee picue gmm ules nd ecngul y ojecs in is egions. Definiion.. n y P sysem (of degee m ) wih exended D conex-fee picue gmm ules is consuc Π = ( V, T, µ, F,, Fm, R,, Rm, i0 ) whee V is he lphe, T V is finie se of eminl symols; µ is memne sucue wih m memnes lelled in one-o-one mnne wih,,, m ; F,, F e m finie ses of picue ys ove V ssocied wih he m egions of µ ; R,, R e m finie ses of column les o ow les of exended D conex-fee picue gmm ules ove V (s in EDCFPG) ssocied wih he m egions of µ; he les hve one of he ched ges: hee, ou, in (in genel, he ge indicion hee is no menioned explicily nd is undesood); finlly, i 0 is he lel of n elemeny memne of µ (he oupu memne). compuion in Π is defined in he sme wy s in n y-ewiing P sysem wih he successful compuions eing he hling ones: ech y, fom ech egion of C,

5 Picue y Geneion Bsed on Memne Sysems nd D Conex-Fee Gmms he sysem, which cn e ewien y column/ow le of ules (s done in EDCFPG) ssocied wih h egion (memne), should e ewien; his mens h one le of ules is pplied; he y oined y ewiing is plced in he egion indiced y he ge ssocied wih he le used (if he ched ge is hee, i mens h he y emins in he sme egion, if he ge is ou, i mens h he y exis he cuen memne; nd if he ge is in, i mens h he y is immediely sen o one of he diecly lowe memnes, nondeeminisiclly chosen if sevel exis hee; if no inenl memne exiss, hen le wih he ge indicion in cnno e used. ) compuion is successful only if i sops, configuion is eched whee no le of ules cn e pplied o he exising picue ys. The esul of hling compuion consiss of picue ys ove T plced in he memne wih lel i 0 in he hling configuion. The se of ll such ys compued o geneed y sysem Π is denoed y L ( Π). The fmily of ll y lnguges L ( Π) geneed y sysems Π s ove, wih mos m memnes, is denoed y P m ( EDCFPG). Exmple.. Conside he y P sysem wih wo memnes nd exended D conex-fee picue gmm ules given y ({ B, X, Y, Z,,, c},{,, c},[ [ ] ], F, φ, R,, ) Π c =, R whee F = B ; R consiss of column les, c c ech wih ge in nd R consiss of he column le wih ge ou, he column le nd he ow les. The les of ules c c 4 4, e s follows: c = { XY }, { XY }, c = { B ZB}, c = c c c = { B Z}, = X, Y, Z, 4,,. = X Y Z X Y Z c compuion ss wih xiom y B in egion. The egion iniilly hs no y in i. If ules of le c e pplied, hen he y X Y B poduced is sen ino he inne egion due o he ge indicion in of he le c. Hee if he ules of le c e pplied, hen he y X Y B poduced is sen ck o egion, due o he ge indicion ou of he le c. The pocess cn epe. On he ohe hnd pplicion of he le c in egion (sending he y o egion ) followed y he pplicion of he le c 4 in egion will yield n y in egion of he fom X XY YZ Z. The pplicion of he ow le cein nume of imes followed y he pplicion of ow le,, c wih will yield n y ove { } m ( m ) ows nd n ( n ) columns wih he symols in he fis n columns ove c. { }, he nex n columns ove { } nd he ls n columns ove { }

6 P.S.zeezun Nish, S.Hemlh, N.Gnnml Dvid nd K.G.Sumnin 4. Compison esuls We now oin compison esuls on he geneive powe of y P sysem wih exended D CF picue gmm ules nd ecngul y ojecs in is egions. Theoem 4.. EDCFPL = P ( EDCFPG) P ( EDCFPG) Poof: The equliy E DCFPL P ( EDCFPG) = in he semen is sighfowd. In fc he y P sysem wih one memne nd EDCFPG ules will genee excly he picue y lnguges in he clss EDCFPL since hee is only one memne nd he les of ules h ewie ys e only EDCFPG kind of les of ules. The inclusion P ( E DCFPG) P ( EDCFPG) follows fom he definiion of P m ( EDCFPG). In ode o pove he pope inclusion, conside he picue lnguge L consising of m n,( m, n ) picue ys wih he popey p given y he following: In m n picue y, he fis n columns e ove { }, he nex n columns e ove { }, nd he ls n columns e ove { c }. This lnguge L is in P ( EDCFPG) s shown in Exmple. u L cnno e in EDCFPL. In fc L cnno e geneed y ny EDCFPG, s he kind of dependence in he columns given y popey p cnno e hndled y ny EDCFPG. I is known [6] h RML CFML CSML. We now show h he y P sysem P ( EDCFPG) conins picue lnguge h cnno e geneed y ny CSMG. Theoem 4.. P ( EDCFPG) ( CSML CFML). φ Poof: We conside he picue y lnguge L consising of picue ys, ech of which hs hee ys of he fom M, M, M wih equl nume of columns such h M is elow M nd M is elow M. Hee M is n ( m + ) n, ( m, n ) y wih he fis n columns ove { }, he nex n columns ove { } nd he ls n columns ove { c }. The ( p + ) n, ( p, n ) y M is simil o M hving he sme nume of columns s M u he nume of ows of M need no e equl o he nume of ows of M. The y M wih one ow nd n columns is n given y d (wih he sme n defining M o M ). n y P sysem wih wo memnes nd EDCFPG ules h genees L cn e consuced y slighly modifying he les of ules in Exmple.. The memne sucue [ [ ] ], he iniil y B in egion e he sme s in Exmple.. The les of ules in egion e column les ech wih ge in nd in egion, he column le, c c wih ge ou, he column le c c 4 nd he ow le s in Exmple. while he 4

7 Picue y Geneion Bsed on Memne Sysems nd D Conex-Fee Gmms d d d ow le in egion is modified s follows: = X, Y, Z. In X Y Z egion, he following ow les e included whee, 4 c c = X, Y, Z, =,,. X Y Z X Y Z The ow le c genees ow of d's while he ddiionl ow le of ules genee he ows, 4 elow he ow of d's. Noe h he ps of he y ove nd elow he ow of d's e like lhough he nume of ows ove nd elow he ow of d's need no e he sme. This picue lnguge L cnno e geneed y ny CSMG s in ny geneed y, he feue, nmely, ow of d's wih he ys ove nd elow i eing idenicl hving he sme nume of ows wih ech ow hving n 's, followed y n 's nd n c's, cnno e hndled y ny CSMG. I is known [7] h TRML TCFML TCSML. We now exhii non-ivil P EDCFPG nd TCSML u no in TCFML. picue lnguge which is oh in ( ) Theoem 4.. P ( EDCFPG) ( TCSML TCFML) φ. Poof: We conside he sme picue y lnguge L consideed in Exmple., which shows h L P ( EDCFPG). TCSMG G cn e consuced o genee L. In fc he fis phse of he TCSMG G will hve o genee he well-known [5] n n n conex-sensiive lnguge { B C n } whee, B, C e he inemedie symols. In he second phse he following les of igh-line ules = {, B B, C cc}, = { d, B db, C dc }, = {, B B, C cc }, { B C 4 =,, c} cn e included in ode o genee he ows of he picue ys in he lnguge L. 5. Conclusion The y P sysem wih les of conex-fee ules nd EDCFPG ype of ewiing is seen o genee picue lnguges h cnno e geneed y he wo-dimensionl conex-sensiive mix gmms [, 6]. Compisons wih ohe kinds of D picue geneing models such s he clss of ecognizle picue lnguges [] emin o e exploed. Possile pplicion o geneion of picue pens cn lso e exmined s done in [8]. cknowledgemen. The uhos hnk he eviewes fo he useful commens. The uho K.G. Sumnin is geful o UGC, Indi, fo he wd of Emeius Fellowship (No.F.6-6/06-7/EMERITUS-05-7-GEN-59 /(S-II)) o him o execue his wok in he Depmen of Mhemics, Mds Chisin College. 4

8 P.S.zeezun Nish, S.Hemlh, N.Gnnml Dvid nd K.G.Sumnin REFERENCES. M.M.Besni,.Figei nd.cheuini, Expessiveness nd complexiy of egul pue wo-dimensionl conex-fee lnguges, In. J. Compu. Mh., 90 (0) D.Gimmesi nd.resivo, Two-dimensionl lnguges, In Hndook of Foml Lnguges" Vol., Eds. G. Rozeneg nd. Slom, Spinge Velg, 997, Gh.Păun, Compuing wih Memnes: n Inoducion, Spinge-Velg, Belin, Rosenfeld, Picue Lnguges-Foml Models fo Picue Recogniion, c-demic Pess, New Yok, G.Rozeneg,.Slom, (Eds.): Hndook of Foml Lnguges Volumes -, Spinge-Velg, Belin, G.Siomoney, R.Siomoney nd K.Kihivsn, sc fmilies of mices nd picue lnguges, Compu. Gphics Imge Pocess., (97) R.Siomoney, K.G.Sumnin nd K.Rngjn, Pllel/Sequenil ecngul ys wih les, In. J. Compu. Mh., 6 (977) G.Smdnielhompson, N.Gnnml Dvid nd K.G.Sumnin, lpheic fl splicing pue conex-fee gmm sysems, J. Mhemics nd Infomics, 7 (07) K.G.Sumnin, P sysems nd picue lnguges, Lecue Noes in Compue Science, Spinge Velg, Vol (007) K.G.Sumnin, R.M.li, M.Geehlkshmi nd.k.ng, Pue D picue gmms nd Lnguges, Discee ppl. Mh., 57(6) (009) K.G.Sumnin, M.Geehlkshmi,.K.Ng nd S.K.Lee, Two-dimensionl picue gmm models, Poceedings of he nd Euopen Modelling Symposium (EMS008), Livepool Hope univesiy, Englnd, 008, pp P.S.P.Wng, y gmms, Pens nd ecognizes, Wold Scienific,