Sticker Systems over Permutation Groups

Transcription

1 World Applied Sciences Journl 1 (Specil Issue of Applied Mth): , 01 ISSN ; IDOSI Pulictions, 01 DOI: /idosi.wsj.01.1.m.11 Sticker Systems over Permuttion Groups 1 N.A. Mohd Sery, N.H. Srmin, W.H. Fong nd 4 S. Turev 1 Deprtment of Mthemticl Sciences, Fculty of Science, Universiti Teknologi Mlysi, Mlysi Deprtment of Mthemticl Sciences nd Inu Sin Institute for Fundmentl Science Studies, Fculty of Science, Universiti Teknologi Mlysi, Mlysi Inu Sin Institute for Fundmentl Science Studies, Universiti Teknologi Mlysi, Mlysi 4 Fculty of Computer Science nd Informtion Technology, Universiti Putr Mlysi, Mlysi Astrct: Sticker systems were first introduced s lnguge generting device sed on the opertion of sticking. A new molecule is produced from the strting xiom, which is prolonged to the left or to the right using given single strnded strings or dominoes. For ech sticker opertion, n element of group (clled vlence) will e ssocited with the xioms nd dominoes nd then the vlue of the group opertion of the newly produced string is computed. Vlence grmmrs were introduced s grmmrs with regulted rewriting using permuttion groups nd groups s control mechnisms. The definition for vlence grmmr is lter extended to vlence H systems which ssocite integer numers with strings nd the vlues ssocited with the result of newly produced strings re computed. A complete doule strnded molecule is considered to e vlid if the computtion of the ssocited element produces the identity element. However, the converse is not true. Using the ide from the definition of vlence H system nd extended vlence H system, the concept of vlence sticker system over permuttion groups is introduced in this reserch nd the computtionl power of the lnguge produced is lso investigted. Key words: Vlence grmmr vlence sticker system h system sticker opertion INTRODUCTION PRELIMINARIES Moleculr computing hs gined mny interests mong reserchers since Hed [1] introduced the first theoreticl model for DNA sed computtion using the splicing opertion in Another model for DNA computing ws proposed y Adlemn [] in 1994 using the sticker opertion. He used the opertion in his successful experiment for the computtion of Hmiltonin pths in grph. According to this sticker opertion, lnguge genertive mechnism clled sticker system is defined s set of (incomplete) doule-strnded sequences (xioms) nd set of pirs of single or doule strnded complementry sequences []. The initil sequences re prolonged to the left nd to the right y using sequences from the ltter set, respectively. These processes stop when complete doule strnded sequence is otined. The min im of this reserch is to introduce sticker systems over permuttion groups, in which with ech sticker opertion, n element of the permuttion group is ssocited nd complete doule strnded sequence is considered to e vlid if the computtion of the ssocited elements of the permuttion group produces the identity element of the group tht is (1). Sticker opertion: Consider the lphet V ( finite set of symols) nd symmetric iologicl reltion ρ, where ρ V V Let e specil symol not in V, denoting n empty spce (the lnk symol). The symol V + denotes the set of ll strings of symol over V while V + represents the set of ll strings not including the empty string. The mtching se pirs of doule x1 strnded molecules of DNA cn e represented s x where x 1 is the upper strnd nd x is the lower strnd. Also, x 1 is complement to x. Using the elements of V, the following set of composite symols is constructed: The Wtson-Crick domin is clled well-formed doule strnded sequence (or simply doule strnded sequence) or molecules [4, 5]. The Wtson-Crick Corresponding Author: N.A. Mohd Sery, Deprtment of Mthemticl Sciences, Fculty of Science, Universiti Teknologi Mlysi, Mlysi 119

2 World Appl. Sci. J., 1 (Specil Issue of Applied Mth): , 01 A sequence x1 x... x k, where x1 A, is domin is denoted y WKρ (V), which is ssocited to the lphet V nd the complementry reltion ρ. The concept of incomplete molecules is used in this pper s n strction of DNA molecules with sticky ends [4, 6]. The incomplete molecule (W ρ (V)) is the element of the set: clled computtion in γ with length k-1 (notice tht, we strt from x1 A). If xk W ρ (V), where no sticky end nd hence lnk symol is present in the lst string of composite symol, the ove computtion is considered s complete. Sticker lnguge: The sticker lnguge is lnguge generting device sed on the sticker opertion which is n strct model of the nneling opertion occurring in DNA computing [, 7]. The lnguge generted y sticker system consists of ll strings formed y the set of upper strnds of ll complete molecules derived from the xioms for which n exctly mtching sequence of lower stickers cn e found [7, 8]. The sticker lnguge is the set of ll possile lnguges (from complete molecules) generted y sticker system. Let γ = (V,ρ,A,D) e sticker system, the unrestricted moleculr lnguge generted y γ is defined in [9] s follows: where:. The symol Lρ (V) represents for left incomplete molecule, Rρ (V) represents right incomplete molecule nd LRρ (V) represents left-right incomplete molecule. A sticker opertion µ stimultes the conctention nd the ligtion or nneling opertion mong the elements of Wρ (V). The elements of Wρ (V) cn e prolonged to the left or to the right with rick or dominoes, providing the complementrity etween the corresponding sticky ends of dominoes to produce complete doule strnded molecule. of γ} Furthermore, the unrestricted sticker lnguge generted y is the projection onto the first (upper) component of the moleculr lnguge, ecuse of the complementrity reltion of ρ. The lnguge is denoted s L(γ) nd is defined y: Sticker system: Sticker system is lnguge generting device which ws first introduced y Kri et l. [] in 1998 s forml lnguge model for the self-ssemly phse of Adlemn s experiment. A sticker system is construct of 4-tuple In this reserch, SL will denote sticker lnguge. Only complete computtion will e considered when defining SL. Permuttion group: Permuttion mens of ojects such s permuttion of numers {1,,, 4,, n}. Permuttions cn e written in mtrix form for where V is n lphet, endowed with the symmetric reltion ρ V V in V, A is finite suset of xioms (W ρ (V)) nd D is finite set of pirs (Bd, Bu ) where Bd nd Bu re finite susets of lower nd upper stickers of + V exmple σ = + Definition 1[10] Permuttion: A permuttion of set is function φ: A A tht is oth one to one nd onto. the forms nd, respectively. V For given sticker system γ = ( V, ρ, A, Bd, B u ) nd Definition [10] permuttion group: Let A e finite set {1,,, 4,, n}. The group of ll permuttions of or permuttion group A is the symmetric group of n letters nd is denoted y Sn. two sequences x,y Wρ ( V ), we write, x y if nd only if y = µ ( Bd, µ ( x, Bu ) ). Note tht, µ ( Bd, µ ( x,b u ) ) = µ( µ ( Bd, x ), B u ) ecuse the prolongtion to the left is independent with the prolongtion to the right []. By *, we denote the reflexive nd trnsitive closure of the reltion. Vlence grmmr: Grmmr with regulted rewriting ws first introduced y Pun [11] nd is termed s vlence grmmr. In vlence grmmr, ech 10

3 World Appl. Sci. J., 1 (Specil Issue of Applied Mth): , 01 production of Chomsky grmmr is ssocited with n integer vlue nd computes the vlue of derivtion y dding the vlences nd the vlence of the production is only considered if the vlences of the productions used dd to zero [1]. Definition [11] Vlence Grmmr: A vlence grmmr over permuttion group is quintuple G = (N, T, S, P, M) where N, T, S re specified s in context -free grmmr. M = (M, ) is permuttion group with neutrl element, e nd inry opertion. The set P is finite set of pirs r = (P, m) with context -free rule P nd m M. The lnguge L(G) generted y G is clled vlence lnguge nd is defined s Extended H system over groups ( Z, +,0) nd ( Q,,0) hve een introduced in [1] nd re known + s n extended vlence H system. The definition of extended vlence H system hs een defined in [1] which discusses the computtionl power of H systems with vlences. multiplying the vlences of the strings to produce new vlence which equls to (1). Following is the definition of sticker systems over permuttion groups. Definition 6: Sticker systems over permuttion groups: Let (G,,e) e permuttion group with opertion nd the identity e. A sticker system over permuttion group, G is construct of vlence sticker system over permuttion group, VSS = (V,ρ,A,D,M) where V, ρ, A re s in the usul definition of sticker system nd D is finite suset of ( B G,B G) for nd,v1 Bu M,,v Bd M,v 1 v WK ρ V where v 1,v G nd we write d ( ) µ,v 1,,v =,v1 v u Definition 4 [1] Extended vlence H systems: Let (M,,e) e group with opertion nd the identity e. An extended vlence H system over M is construct γ = (V,T,A,R), where V, T nd R re s in usul extended H system nd A is finite suset of V * M. For ( x,v 1),( y,v ),( w,v) V M nd r R, we write ( ) ( ) ( ) ( x,y) r x,v 1, y,v r w,v V M if nd only if w nd v = v1 v. Then the lnguge generted y the system is defined s ( γ ) = ( ) σ ( ) L x T x,e A The vlence grmmr introduced y Dssow nd Pun in 1989 [14] will e used to introduce the concept of sticker system over permuttion groups. The definition of extended vlence H system termed y Mnc nd Pun in [1] is used to define sticker system over permuttion groups. Vlence sticker systems over permuttion groups: In this study, the set of sticker system with opertion of permuttion group is used where for ech sticker opertion, n element of permuttion group is ssocited nd the vlue of permuttion group is computed from two initil strings to produce new string. The vlue of derivtion is computed y if nd only if (,) ρ. The set of possile lnguges generted y vlence sticker system over permuttion group is clled vlence sticker lnguge over permuttion group denoted y VSL nd is defined in the following: re given. 11 ( γ ) = ρ( ) ( ) ( ( )) VSL w w WK V, A,(1) W, 1 where w WK ρ (V) is complete doule strnd, produced y itertive opertion of sticking on strting xiom, A with elements in D. The vlue of the vlence ssigned to A must e equl to the identity of the permuttion group, which is (1) nd the computtion of sticker systems with the ssocitive elements of the permuttion group produces the identity element, (1). The ide ehind this computtion is with ech sticker opertion, n element of permuttion group is ssocited to ech xiom nd the vlence of permuttion group opertion of the new string from two initil strings is computed. A complete doule strnded sequence produced is considered to e vlid if the computtion of the ssocited elements of the permuttion group produces the identity element of the group. In the following, some exmples of the computtion of sticker system with permuttion group

4 World Appl. Sci. J., 1 (Specil Issue of Applied Mth): , 01 µ, (1),, (1 ) =, (1) Exmple 1: Computtion of Sticker System over Permuttion Group of Length Two Given vlence sticker system VSS = (V,ρ,A,D,G) with µ, (1),, (1 ) = G = ( S,,( 1) ), ρ = {(, ), (,)}, A = nd D ( B d M,Bu M ), (1) µ, (1),, (1) =,(1 ) where 4 µ, (1),, (1 ) =, (1) nd 5 4 µ, (1),, (1) =, (1 ) µ Here, the permuttion group is S = {(1),()}. We strt with A =, (1). Then, the string 5 6, (1),, (1 ) =, (1) Note tht, from the ove computtions, if the string, (1) is used followed y, (1), (1) is joined to the strting xiom A. The repetedly with (k-1)th itertions, string of the form computtions re s follows: k 1 µ, (1),, (1 ) =, (1),(1 ) is otined with the vlence equl to (1) where (k-1) is n even numer. Then the string, (1) is used nd this will µ, (1 ),, (1) =, (1) result in n extr strnd. Thus, the string of the k µ, (1),, (1 ) =, (1 ) form, (1 ) is otined with the vlue of the vlence equl to (1). The incomplete molecule µ, (1 ),, (1) =, (1) otined is then terminted y the string, (1 ) k string to produce the string of the form, (1). The string produced is complete doule strnded molecule nd the multiplied vlence is equl to the identity element of the permuttion group, which is (1). However, to otin lnguge in the generl form, longer computtion is needed. The computtion gin strts with The string produced stisfies the condition needed which is complete doule strnded molecule with the vlue of the vlence equl to the identity of the permuttion group. Hence, the lnguge generted from the computtion is VSL = { k : k }. For k = 1, the A =, (1) nd joined with, (1). The computtion is not vlid nd for k =, the only possile lnguge will e produced is VSL = { }. computtions re shown s in the following: 1

5 World Appl. Sci. J., 1 (Specil Issue of Applied Mth): , 01 D ( B d A, B u A ) Then, we hve to derive grmmr tht genertes the lnguge so tht we cn clssify the grmmr s well s the lnguge to the clss of Chomsky grmmr. Therefore, we will hve the following grmmr: where G = ({S,A}, {,},S,P ) nd with productions Here, the permuttion group A = {(1), (1), (1 )} is n From the solution, the lnguge produced is CFREG lnguge. The next exmple is the computtion of sticker system with the permuttion group of length three. lternting group nd sugroup of S. We strt with A =, (1). Then, the computtions Exmple : Computtion of Sticker System over Permuttion Group of Length Three Given vlence sticker system VSS = (v,ρ,a,d,g) with re done s follows: 1. µ, (1),, (1) =,1 ( ) G = A, ρ = {(, ), (,)}, A = µ, (1),,(1 ) =, (1) nd. µ, (1),, (1) =,1 ( ) µ, (1 ),,(1 ) =, (1) µ, (1),, (1 ) =,1 ( ) µ, (1),,(1 ) =, (1). µ, (1),, (1) =,1 ( ) µ, (1 ),, (1 ) =, (1) µ, (1 ),, (1) =, (1) µ, (1 ),, (1 ) =, (1) µ, (1),,(1) = µ, (1),, (1 ) = 1, (1 ), (1)

6 World Appl. Sci. J., 1 (Specil Issue of Applied Mth): , µ,1 ( ),,1 ( ),1 ( ) = 4 µ,1,,1 =,1 4 4,1,,1,1 = µ ( ) ( ) ( ) Notice tht, t computtion numer 4, the numer of generted is not equl to the numer of. The lnguge produced from the computtion which is n n is only stisfied when n =. However, further computtions re needed to identify the lnguge in generl form s well s the grmmr when the computtions re done with n th itertions. Hence the computtions re continued s in the following; 4 5. µ,1 ( ),,1 ( ),1 ( ) = 4 5 µ,1 ( ),,1 ( ),1 ( ) = 5 5,1,,1 =,1 µ ( ) ( ) ( ) 5 5 µ,1 ( ),,1 ( ),1 ( ) = µ,1 ( ),,1 ( ),1 ( ) = 5 6 µ,1 ( ),,1 ( ),1 ( ) = 6 6 µ,1,,1 =,1 6 6 µ,1 ( ),,1 ( ) =,1 ( ) 6 6 µ,1 ( ),,1 ( ),1 ( ) = µ,1 ( ),,1 ( ),1 ( ) = 6 7 µ,1,,1 =,1 7 7,1,,1,1 = µ ( ) ( ) ( ) 14

7 World Appl. Sci. J., 1 (Specil Issue of Applied Mth): , µ,1 ( ),,1 ( ),1 ( ) = 7 8 µ,1 ( ),,1 ( ),1 ( ) = 8 8,1,,1 =,1 µ ( ) ( ) ( ) 8 8 µ,1 ( ),,1 ( ),1 ( ) = µ,1,,1 =,1 8 9 µ,1,,1 =,1 9 9 µ,1,,1 =,1 9 9 µ,1,,1 =,1 9 9 µ,1,,1 =,1 Continuing the sticking opertion with n th itertions, string of the forms n1 + n+,1,,1 n,1 nd ( ) re otined. Therefore the lnguges produced re; 1 n1 + VSL = ;n 1 with productions with productions G G = = ({ S,A },,,S,P) ({ S,A },,,S,P) n+ VSL = ;n 1 n VSL = ;n The following grmmrs re otined with respect to the lnguges produced ove: with productions G1 = ({ S,A },,,S,P) 15 From the solutions, the lnguges produced re ll CF-REG lnguge. The computtion of sticker system over groups must stisfy two conditions which re: The string from the computtion is only considered vlid if complete doule strnded molecule is produced, The vlence of the complete doule strnded molecule is equl to the identity of the respective groups. It is importnt to know tht oth conditions must e stisfied when deling with computtion of sticker

8 World Appl. Sci. J., 1 (Specil Issue of Applied Mth): , 01 system over groups. Furthermore, the choose of the strting xioms, nd the finite suset of pirs of B d B u ffects the lnguge produced from the computtion of the sticker system. Besides, if we define ρ of complementrity to other thn identity, for exmple (, ), different lnguge will e produced. ACKNOWLEDGMENT The uthors would like to cknowledge Reserch Mngement Center (RMC), UTM for the prtil finncil funding through Vote No. 00H48. Thnks lso to the Applied Alger nd Anlysis Group (AAAG) nd Postgrdute Student Society, Fculty of Science (PGSS), Universiti Teknologi Mlysi for ll the ctivities eneficil to this reserch. REFERENCES 1. Hed, T., Forml lnguge theory nd DNA: An nlysis of the genertive cpcity of specific recominnt ehviors. Bulletin of Mthemticl Biology, 49: Adlemn, L.M., Moleculr computtions of solutions to comintoril prolems. Science, 66: Kri, L., G. Pun, G. Rozenerg, A. Slom nd S. Yu, DNA computing, sticker systems nd universlity. Act Informtic, 5: Pun, G. nd G. Rozenerg, Sticker systems. Theoreticl Computer Science, 04: Rozenerg, G., G. Pun nd A. Slom, DNA computing: A new computing prdigms. New York: Springer-Verlg. 6. Xu, J., Y. Dong nd X. Wei, 004. Sticker DNA computer model, prt 1: Theory. Chinese Science Bulletin, 49 (8): Freund, R., G. Pun, G. Rozenerg nd A. Slom, A Bidirectionl Sticker Systems. In Altmn, R.B., A.K. Dunker, L. Hunter nd T.E. Klein (Eds.). Pcific Symposium on Biocomputing. World Scientific, Singpore, pp: Hoogeoom, H.J. nd N.V. Vugt, 000. Fir sticker lnguges. Act Informtic, 7: Vugt, N.V., 00. Models of Moleculr Computing. University of Leiden, Ph.D. Thesis. 10. Clerk Mxwell, J., 189. A Tretise on Electricity nd Mgnetism, rd Edn., Oxford: Clrendon, : Dssow, J. nd G. Pun, Regulted Rewriting in Forml Lnguge Theory. In EATCS Monogrph in Theoreticl Computer Science, Springer-Verlg, Vol: Fernu, H. nd R. Stiee, Regultion y Vlences. In Rovn, B. (Ed.). Mthemticl Foundtions of Computer Science 1997 nd Interntionl Symposium, MFCS 97 Proceedings, Springer-Verlg. 1. Mnc, V. nd G. Pun, Arithmeticlly controlled H systems. Computer Science of Moldov, 6: