Pure 2D Context-free Puzzle P system with Conditional Communication

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1 IOSR Journl of Mthemtis (IOSR-JM) e-issn: p-issn: X. Volume 0 Issue 6 Ver. III (Nov - De. 04) PP Pure D Context-free Puzzle P system with Conditionl Communition S.Hemlth P.S.AzeezunNish (Deprtment of Mthemtis S.D.N.B Vishnv College for Women Chenni Indi) (Deprtment of Mthemtis J.B.A.S. College for Women Chenni Indi) Astrt: The feture of onditionl ommunition in memrne omputing hs een introdued in symol ojets nd rrys in [] [4] nd [6]. Pure ontext-free grmmrs hve een introdued in [7]. In [8] this ide hs een pplied on rry ojets with n dditionl ontrol on regulr grmmrs. In this pper we use the onditionl ommunition of memrne omputing to pure D ontext free grmmrs nd study the power of the system y ompring with the existing system. Keywords: Conditionl ommunition ontext free grmmr P systems Pure grmmr Puzzle grmmr I. Introdution The re of memrne omputing ws initited y Păun [3]. Now this omputility model is lled s P system whih is distriuted highly prllel theoretil omputing model sed on the memrne struture nd the ehvior of the living ells. P systems with string-ojets nd string-proessing rules s known in forml lnguge theory hve een onsidered [8] nd investigted extensively. Arry lnguges hve lso een studied in [9 0]. The prolem of hndling rry lnguges using P systems hs een onsidered y Ceterhi et l [6]. Arry grmmrs hve een introdued nd investigted with the tools of P system [7 3 5]. Among vrints of P system onditionl ommunition is feture introdued in []. On the other hnd rewriting P systems with string-ojets hve een extended to rry-rewriting P systems with rry-ojets nd the power of suh systems hs een studied in [ 6]. Conepts of pure grmmrs hs een introdued in [7]. Pure D piture grmmrs hve een studied in [8]. In this pper we hve inorported onditionl ommunition in Puzzle P system with rry ojets [4] nd further properties hve een studied. In setion we rell some preliminry definitions with exmples. Setion 3 dels with the min results of this pper whih omines the Pure D ontext free grmmrs with P systems with onditionl ommunition. II. Preliminries We rell some sis of forml lnguge theory s in []. In this setion we rell some si definitions s in [] Let e finite lphet. A word or string w over is sequene of symols from. The set of ll words over inluding the empty word with no symols is denoted y *. An rry over onsists of finitely mny symols from pled in the points of Z (the two-dimensionl plne) nd the points of the plne not mrked with symols of re ssumed to hve the lnk symol. We will pitorilly represent the rrys inditing their non lnk pixels whenever possile. The set of ll rrys over will e denoted y *. An rry over {} desriing the piture pttern T is shown in Figure. Fig An rry desriing piture token T. An rry in prtiulr n e retngulr. A retngulr m n rry M over is of the form M m n mn 6 Pge

2 Pure D Context-free Puzzle P system with Conditionl Communition where eh ij i m j n. The set of ll retngulr rrys over is denoted y ** whih inludes the empty rry. Now we rell the definitions of pure grmmrs nd pure ontext-free grmmr s in [7 8]. A pure grmmr is triple G = ( P S) where is finite lphet S is finite set of words over nd P is finite set of ordered pirs (x y) of words over. The elements of P re referred to s prodution nd denoted y x y. If in eh prodution x y of P the left side x is symol then we sy tht G is pure ontext-free grmmr. Lnguges generted y Pure Context Free (PCF) grmmrs re referred s PCF Lnguges. A pure ontext free grmmr is G = ( P set of xioms words) where is finite lphet set of xiom words nd P is finite set of ontext free rules of the form *. Derivtions re done s in ontext free grmmr exept tht unlike ontext free grmmr there is only one kind of symol nmely the terminl symol. The lnguge generted onsists of ll words generted from eh xiom word. Also we rell the notions of string rewriting P systems with onditionl ommunition [] nd rry rewriting P systems with onditionl ommunition [4]. An extended (string) rewriting P system (of degree m ) with onditionl ommunition [] is onstrut = (V T M M M m R P F R P F... R m P m F m ) where: V is the lphet of the system T V is the terminl lphet is memrne struture with m memrnes (injetively lelled y... m) M i re finite lnguges over V representing the strings initilly present in the regions i i =... m of the system R i re finite sets of ontext-free rules over V present in regions i i =... m of the system P i re permitting onditions nd F i re foridding onditions ssoited with regions i i =... m. The onditions n e of the following forms:. empty : No restrition is imposed on strings they either freely exit the urrent memrne or enter ny of the diretly inner memrnes ; An empty permitting ondition is denoted y (true ) {in out} nd n empty foridding ondition y (flse not ) {in out}.. symols heking : eh P i is set of pirs ( ) {in out} for V nd eh F i is set of pirs ( not ) {in out} for V: string w n go to lower memrne only if there is pir ( in) P i with lph(w) nd for eh ( not in) F i we hve lph(w); similrly for the string to go out of memrne i it is neessry to hve lph(w) for t lest one pir ( out) P i nd lph(w) for ll ( notout) F i 3. sustrings heking : eh P i is set of pirs (u ) {in out} for u V + nd eh F i is set of pirs (v not ) {in out} for v V + : string w n go to lower memrne only if there is pir (u in) P i with u Su(w) nd for eh (v notin) F i we hve v Su(w); similrly for the string to go out of memrne i it is neessry to hve u Su(w) for t lest one pir (u out) P i nd v Su(w) for ll (v not out) F i. Thus we hve onditions of the types empty sym su k respetively where k is the length of the longest string in ll P i F i ; when no upper ound is imposed we reple the susript y *. A system is sid to e non-extended if V = T. The trnsitions in system s ove re defined in the following wy. In eh region eh string whih n e rewritten y rule from tht region is rewritten. The rule to e pplied nd the symol rewritten y it re non-deterministilly hosen. Eh string otined in this wy is heked ginst the onditions P i F i from tht region. If it fulfils the requested onditions then it will e immeditely sent out of the memrne or to n inner memrne if ny exists; if it fulfils oth in nd out onditions then it is sent either out of the memrne or to lower memrne non-deterministilly hoosing the diretion- nd non-deterministilly hoosing the inner memrne in the se when severl diretly inner memrne exist. If string does not fulfil ny ondition or it fulfills only in onditions nd there is no inner memrne then the string remins in the sme region. If string nnot e rewritten (this n e the se with strings from other memrnes) then it is diretly heked ginst the ommunition onditions nd s ove it leves the memrne (or remins inside forever) depending on the result of this heking. The lnguge generted y the ove system is denoted y L(). The fmily of ll lnguges L() generted y the system of degree t most m with permitting onditions of type nd foridding onditions of type is denoted y [E]LSP m (rw ) {empty sym} {su k k }. If the degree of the systems is not ounded then the susript m is repled y *. Exmple. [] Consider the system = ({ }{ } [ [ ] ] {} R P F R P F ) 7 Pge

3 Pure D Context-free Puzzle P system with Conditionl Communition R = { } P = {(true in) (true out)} F = {( notin) ( notout)} R = { } P = {(true out)} F = {( notout)} n genertes the lnguge L(Π( { n 0} An rry is mpping A : Z V with finite support supp(a) where supp(a)= {v Z A(v) }. In order to speify n rry it is suffiient to speify the set (v A(v)) for v supp(a). For exmple the L-shped ngle with equl rms is given y the sets {((0 0) ) (( 0) ) (( 0) ) ((3 0) ) ((0 ) ) ((0 ) ))) 0 3) )} ll the other elements hve symol. The empty rry is denoted y nd the set of ll rrys over V is given y V * = V + {}. Any suset of V * is n rry lnguge. An rry prodution over V is triple = (W A B) where W is finite suset of Z nd A B re rrys with the supports inluded in W. For two rrys C D over V nd prodution s ove we write C D if D n e otined y repling su rry of C identil to A with B; ll pixels of W whih re lnk in A should e lnk lso in C. An Arry P System [] is onstrut = (V T F... F m R... R m i 0 )where: V is the lphet of the system T V is the terminl lphet is memrne struture with m memrnes (lelled injetively y... m) F i re finite set of rrys over V representing the rrys initilly present in the regions i i =... m of the system R R... R m re finite sets of rry-rewriting rules over V of the form A B (tr); nd i 0 is the output elementry memrne of. The system is sid to e non-extended if V = T. A omputtion in n rry P system is defined in the sme wy s in string rewriting P system with the suessful omputtions eing the hlting ones. A omputtion is suessful only if it stops onfigurtion is rehed where no rule n e pplied to the existing rrys The result of hlting omputtion onsists of the rrys omposed only of symols from T pled in the memrne with lel i 0 in the hlting onfigurtion. Exmple.Consider the non-extended ontext-free system [] = ({} {} [ [ [ 3 ] 3 ] ] R R 3) R (in) R (out) (in) R 3 = The ove system genertes ll L-shped ngles with equl rms eh rm eing of length t lest three. An extended rry-rewriting P system (of degree m ) with onditionl ommunition is onstrut = (V T M... M m R P F... R m P m F m ) where V is the lphet of the system T V is the terminl lphet is memrne struture with m memrnes (injetively lelled y... m) M... M m re finite set of rrys over V representing the rrys initilly present in the regions... m of the system R R... R m re finite sets of rry-rewriting rules over V present in regions... m of the system P i re permitting onditions nd F i re foridding onditions ssoited with region i i m. The onditions n e in the forms empty symol heking surry heking whih re defined nlogous to the orresponding forms in the string se. The differene is tht the ojets re rrys. Thus we hve the onditions of the types empty sym surr in ll P i F i. The trnsitions in n rry-rewriting P system with onditionl ommunition re nlogous to the string se. But the result of hlting omputtion is s defined for rry-rewriting P systems. The set of ll rrys omputed y n rry-rewriting P system with onditionl ommunition is denoted y [E]AL() with E eing omitted when the system is non-extended. The fmily of rry lnguges generted y systems s ove is denoted y [E]ALP m (rw ) {empty sym surr}. We illustrte with n exmple. Exmple 3.Consider the non-extended system = ({} {} [ ] R P F ) R P = {( in)} F = { notin} 8 Pge

4 genertes ll L-shped pitures over. Pure D Context-free Puzzle P system with Conditionl Communition III. Pure d Context Free P System With Conditionl Communition A Pure D Context free P system with onditionl ommunition is onstrut = ( M M... M m (R P F ) (R P F )... (R m P m F m ) O) where: is the lphet of the system with only terminls is the speil symol for representing lnks is memrne struture with m memrnes (injetively lelled y... m) M i re finite lnguges over V representing the strings initilly present in the regions i i =... m of the system R i re finite sets of pure ontext-free rules over ** present in regions i i = m of the system P i re permitting onditions nd F i re foridding onditions ssoited with regions i i =... m s disussed in the preliminries nd O the output memrne. We denote the new system s PDCFP m ( ) where {empty symol surry} the onditionl ommuniting onditions. The lnguges generted y this system is given y L(). Exmple 4.Consider the onstrut = ( [ [ ] ] M M (R P F ) (R P F ) ) with = { d} M d M = d P = {true in} F notin notout P = {true in} F notin notout d d d Applying the ove rules the output genertes piture pttern of token H. Theorem.The PDCFP 3 (surr surr) genertes token T ptterns Proof. Consider the onstrut = ( [ [ ] ] M M (R P F ) (R P F ) ) with = { d} M d e M = P in d d F d d notin notout P = {true in} F notin notout d d d Applying the ove rules the output genertes piture pttern of token T. IV. Comprisons And Closure Properties Theorem.The lss PDCFP m { } intersets the lss PDCFL with regulr ontrol. Proof. The result follows from exmple 4. Theorem 3.The lss PDCFPP m { } is inomprle with the fmilies of RML nd CFML. Proof: L = () m n n e generted y PDPP (surr urr) with 9 Pge

5 Pure D Context-free Puzzle P system with Conditionl Communition The orresponding permitting nd foridding rules n e frmed s erlier to void irregulr ptterns. This lnguge n lso e generted y regulr Siromoney Mtrix grmmr. But H tokens nd T tokens nnot e generted CFML [0]. Consider L = { n n / n } is CFML ut nnot e generted y the system PDCFPP m ( ). Also L 3 = n () * is RML ut not pure CFL. Theorem 4.The lss PDCFPP m { } intersets with LOC nd REC. Proof: Consider the piture lnguges L 4 = I m m where I is the identity mtrix is in LOC nd hene in REC. This L 4 n e generted y onstrut = ( [ [ ] ] M M (R P F ) (R P F ) ) with = {0 } M 0 0 M = P = {true in} F notin notout P = {true in} F 0 notin notout Theorem 5. The lss PDCFP m { } is losed under olumn ontention nd row ontention Proof: Consider L 5 = {() n () n / n} nd L 6 = {(x) n (y) n / n} oth n e generted y PDCFPP m ( ). It n e proved tht L 5 L 6 n e generted y the system PDCFPP (empty sustring). V. Conlusion It hs een found tht the power of the new system hs inresed with the puzzle nd onditionl ommunition feture. The new system lso stisfies ertin losure properties. Further investigtions like omprison with Tled 0L systems [] n lso e studied in future work. Aknowledgement The first uthor is thnkful to the University Grnts Commission-SERO: Projet no.f. MRP-454/ Link No.454 under whih this work ws done y her. Referenes [] P. Bottoni A. Lell C. Mrtin-vide Gh. Păun Rewriting P Systems with Conditionl Communition. Leture Notes in Computer Siene Springer Berlin vol [] R. Ceterhi M. Mutym Gh. Păun K.G. Surmnin Arry - Rewriting P Systems. Nturl Computing vol [3] Gh. Păun Memrne Computing (An Introdution. Springer 00). [4] S. Hemlth K.S. Dersnmik K.G. Surmnin C. Sri Hri Ngore Arry-Rewriting P Systems with Conditionl Communition Leture Note Series Rmnujn Mthemtil Soiety vol [5] K.G. Surmnin R. Siromoney V.R. Dre A Soudi Bsi Puzzle Lnguges Int. Journl of Pttern Reognition nd Artifiil Intelligene vol [6] R. Ceterhi M. Mutym Gh. Pun K.G. Surmnin Arry-rewriting P systems Nturl Computing vol [7] M. Nivt A. Soudi K.G. Surmnin R. Siromoney V.R. Dre Puzzle Grmmrs nd Context-free Arry Grmmrs Int. Journl of Pttern Reognition nd Artifiil Intelligene vol [8] Gh. Păun Memrne Computing : An Introdution(Springer-Verlg Berlin Heidelrg 000). [9] Rosenfeld Piture Lnguges (Ademi Press 979). [0] Rosenfeld nd R. Siromoney Piture lnguges - survey Lnguges of design vol [] Slom Forml lnguges( Ademi Press London 973). 0 Pge

6 Pure D Context-free Puzzle P system with Conditionl Communition [] R. Siromoney nd G. Siromoney Extended Controlled Tled L-rrys Informtion nd Control vol. 35 no [3] K.G. Surmnin M. Geethlkshmi P. Helen Chndr Arry rewriting P systems generting retngulr rrys Pper presented t the Ntionl onferene on Intelligent Optimiztion Modeling Gndhigrm Rurl Institute-Deemed University Gndhigrm Indi Mrh 006. [4] K.G. Surmnin R. Srvnn K. Rngrjn Arry P systems nd Bsi puzzle grmmrs Pper presented t the Ntionl onferene on Intelligent Optimiztion Modeling Gndhigrm Rurl Institute-Deemed University Gndhigrm Indi Mrh 006. [5] Y. Ymmoto K. Morit K. Sugt Context-sensitivity of two-dimensionl rry grmmrsin P.S.P. Wng(Ed) Arry grmmrs Ptterns nd reognizers (World Sientifi 989) 74. [6] K.G. Surmnin S. Hemlth C. Sri Hri Ngore M. Mrgernstern On the power of P systems with prllel rewriting nd onditionl ommunition Romnin Journl of Informtion Siene nd Tehnology vol. 0 no [7] H.A. Murer A Slom D. Wood Pure Grmmrs Informtion nd Control vol [8] K.G. Surmnin Atuly K. Ngr M. Geethlkshmi Pure D Piture Grmmrs (PDPG) nd PDPG with Regulr Control. Pge