Watson-Crick local languages and Watson-Crick two dimensional local languages

Transcription

1 Interntionl Journl of Mthemtics nd Soft Comuting Vol.5 No.. (5) ISSN Print : ISSN Online: Wtson-Crick locl lnguges nd Wtson-Crick two dimensionl locl lnguges Mry Jemim Smuel nd V.R. Dre Dertment of Mthemtics Ann Adrsh College for Women Ann Ngr Chenni 6 4 Indi. dr_jy@yhoo.com Dertment of Mthemtics Mdrs Christin College Tmrm Est Chenni 6 59 Indi. rjkumrdre@yhoo.com Astrct Wtson-Crick finite utomt re lnguge recognizing devices similr to finite utomt introduced in DNA comuting re. Locl lnguges re of gret interest in the study of forml lnguges using fctors of length two or more. We define Wtson-Crick locl lnguges using doule strnded seuences where the two strnds relte to ech other through comlementry reltion insired y the DNA comlementrity. We lso define tiling recognizle Wtson-Crick locl lnguges nd rove some closure roerties. We lso extend Wtson-Crick locl lnguges to finite rrys nd rove some closure roerties. Keywords: Wtson-Crick locl lnguges Wtson-Crick tiling system. AMS Suject Clssifiction (): 68Q45. Introduction The remrkle rogress mde y moleculr iology nd iotechnology in the lst coule of decdes rticulrly in seuencing synthesizing nd mniulting DNA molecules gve rise to the ossiility of using DNA s suort for comuttion. The comuter science community uickly rected to this chllenge nd mny comuttionl models were uilt to exloit the dvntges of nnolevel iomoleculr comuting. One of them is Wtson-Crick utomt. Wtson-Crick utomt introduced in [] reresent one instnce of mthemticl model strcting iologicl roerties for comuttionl uroses. They re finite utomt with two reding heds working on doule strnded seuences. One of the min fetures of these utomt is tht chrcters on corresonding ositions from the two strnds of the inut re relted y comlementrity reltion similr with the Wtson-Crick comlementrity of DNA nucleotides. The two strnds of the inut re sertely scnned from left to right y red only heds controlled y common stte. Locl lnguges re descried y the fctors of length two or more. Secil clsses of utomt clled scnners re considered s model for comuttions tht reuire only locl informtion. Informlly scnner [] is n utomton euied with finite memory nd sliding window of 65

2 66 Mry Jemim Smue nd V.R. Dre fixed length. In tyicl comuttion the sliding window is moved from left to right on the inut so tht the scnner cn rememer the fctors of length two or more. We use this concet of locl on lnguges nd define Wtson-Crick locl lnguges. We lso define Wtson-Crick tiling system nd lso recognizility of Wtson-Crick locl lnguges. We rove some closure roerties of Wtson-Crick tiling system lnguges. We extend this concet to finite rrys nd define Wtson-Crick two-dimensionl tiling system nd rove some closure roerties. Preliminries Let e finite lhet. * denotes the set of ll fiinite words over is the emty word nd + is the set of ll non emty finite words over. i.e. + = * \ {}. Definition.. [4] We now define comlementrity reltion on the lhet (like the Wtson- Crick comlementrity reltion mong the four DNA nucleotides) which is symmetric. Denote { / ( ) } * y WK (). The set WK () is clled the Wtson-Crick domin ssocited to nd. The elements n WK () re lso written in the form n w = n. We cll such elements w w w w for w = n nd WK () molecules. According to the usul wy of reresenting DNA molecules s doule-strnded seuences we lso write the roduct monoid ( * * ) in the form nd its elements in the form * * x. y Definition.. [3] A two-dimensionl string (or icture ) over is two-dimensionl rectngulr rry of elements of. The set of ll two-dimensionl strings over is denoted s **. A twodimensionl lnguge over is suset of **. The oundry of icture is identified y secil symol which surrounds. Wtson- Crick domin is lso extended to two-dimensionl rrys. We define ij ij WK -D() = {( ) / re m n rrys nd ( ) }. We now define some conctention oertions etween ictures nd two-dimensionl lnguges. Definition.3. [3] Let nd e two ictures over n lhet of size (m n) nd (m n) m n m n > resectively. m n mn The column ctention of nd denoted y is rtil oertion defined only if m = m given y m n mn

3 Wtson-Crick locl lnguges nd Wtson-Crick two dimensionl locl lnguges 67 = m Similrly the row conctention of nd denoted y is defined only if n = n. Definition.4. [3] Let L nd L e two two-dimensionl lnguges over n lhet the column n mn m conctention of L nd L denoted y L L is defined y n mn L L = { / L nd L Similrly the row conctention of L nd L is defined. Definition.5. [3] Let L e icture lnguge. The column closure of L denoted s L * is defined s L * = i L i where L = L = L L n = L L (n). Similrly the row closure of L is defined. 3 Wtson-Crick Locl Lnguges Definition 3.. Let e finite lhet. If w WK () then L WK () is locl if there exist finite set of tiles over the lhet {} such tht L = {w WK () / B (w) } where B (w) is the set of ll locks or tiles of size ( ). We write it s L = L(). The fmily of Wtson- Crick locl lnguges is denoted s WK (LOC). Exmle 3.. Let = { } e n lhet nd e the following set of tiles over. = {( ) ( )} The Wtson-Crick locl lnguge is L = L() = 4 Tiling Recongizle Wtson-Crick Lnguges n n m m /n m. Definition 4.. A Wtson-Crick tiling system is 4-tule T = ( ) where nd re two finite lhets is finite set of tiles over the lhet {} nd : is rojection. The tiling system defines (recognizes) lnguge L over the lhet s L = (L) where L = L() is the locl lnguge over corresonding to the set of tiles written s L = L(T). We sy tht L is the lnguge recognized y T. We sy tht lnguge L WK () is recognizle y tiling systems if there exists tiling system T = ( ) such tht L = L(T). We denote y L (WK -TS) the fmily of ll Wtson-Crick recognizle tiling systems. Exmle 4.. Considering Exmle 3. if we ly the rojection : such tht () = () = then we see tht L = (L). We now rove some closure roerties. Theorem 4.. The fmily L(WK -TS) is closed under rojection. Proof: Let e two finite lhets nd let : e rojection.

4 68 Mry Jemim Smue nd V.R. Dre We hve to rove if L recognizle y tiling system. * is recognizle y Wtson-Crick tiling systems then L = (L ) is Let T = ( ) e Wtson-Crick tiling system for L. i.e. L = L(T ). Then L = (L) where L is the Wtson-Crick locl lnguge reresented y. We see tht L is Wtson-Crick locl lnguge. Also L = (L) where =. :. Hence T = ( ) is Wtson-Crick tiling system for L. Theorem 4.. The fmily L(WK -TS) is closed under ctention. Proof: Let L nd L e lnguges over the lhet nd let L = L. L e the lnguge otined y ctenting L with L. By definition of ctention word w L is comosed of ir of words w L nd w L such tht the rightmost letter of w is glued to the leftmost letter of w. Let ( ) nd ( ) e two Wtson-Crick tiling systems for L nd L resectively. Without loss of generlity we ssume =. We define Wtson-Crick tiling system for L s ( ) where = nd hs to contin ll the elements from set excet those corresonding to the right orders nd ll elements from set excet those corresonding to the left orders. Hence we define s follows: c c if nd d d The rojection : is given s Theorem 4.3. The fmily L(WK -TS) is closed under. π () if Γ π(). π() if Γ Proof: Let L nd L e two lnguges over n lhet nd let ( ) nd ( ) e two WK -TS to recognize L nd L resectively. We ssume =. A WK -TS ( ) for L L = L is defined where = is the locl lhet nd rojection is defined so tht its restrictions to lhets nd coincide with nd s in Theorem 4.. Then =. Wtson-Crick locl lnguges cn lso e extended to two dimensionl finite rrys. Insted of the doule strnds in the Wtson-Crick utomton here we hve two finite lnes. The st lne corresonds to the uer level strnd nd the nd lne corresonds to the lower level strnd of doule strnd seuence. 5 Wtson-Crick Two-Dimensionl Locl Lnguges Definition 5.. Let e finite lhet. If WK -D() then = ( ) nd L WK -D() is locl if there exist finite set of tiles such tht = ( ) nd L = { WK -D() / B ( ) nd B ( ) } where B - reresent tiles of size. We write it s L = L().

5 Wtson-Crick locl lnguges nd Wtson-Crick two dimensionl locl lnguges 69 The fmily of Wtson-Crick two-dimensionl locl lnguges is denoted s WK -D(LOC). Exmle 5.. Let = { } e n lhet nd = ( ) e the following set of tiles over. The lnguge L = L() = ( ) is the lnguge of sures of ictures in which ll min digonl osition re the remining ositions re. 6 Tiling Recognizle WK-D(LOC) Lnguges Definition 6.. A Wtson-Crick two-dimensionl tiling system (WK -D-TS) is 4-tule T = ( ) where nd re two finite lhets = ( ) is finite set of tiles over the lhet {} nd : is rojection. The tiling system T defines (recognizes) lnguge L over the lhet s L = (L) where L = L() = ( ) is the locl lnguge over corresonding to the set of tiles = ( ) written s L = L(T). We sy tht L is the lnguge recognized y T. The WK -D locl lnguge L ** is clled s WK -D locl lnguge for L. We sy tht lnguge L ** is recognizle y tiling system if there exists tiling system T = ( ) where = ( ) such tht L = L(T).

6 7 Mry Jemim Smue nd V.R. Dre Exmle 6.. The lnguge of sures over the lhets = { } with s in the min digonl nd s in the other ositions nd if we ly the rojection : such tht () = () = then we see tht L = (L). We now rove some closure roerties. Theorem 6.. The fmily L(WK -D-TS) is closed under rojection. Proof: Let e two finite lhets nd let : e rojection. To rove: If L systems. ** is recognizle y tiling systems then L = (L ) is recognizle y tiling Let T = ( ) = ( ) e tiling system for L i.e. L = L(T ). Then L = (L) where L is WK -D locl lnguges reresented y = ( ). We see tht L is the WK -D locl lnguge. Also L = (L) where = :. Hence T = ( ) = ( ) is tiling system for L. Theorem 6.. The fmily L(WK -D-TS) is closed under row nd column conctention oertions. Proof: Let L nd L e icture lnguges over n lhet nd let L = L L e the lnguge corresonding to the column conctention of L nd L. By definition of column conctention icture L is comosed y ir of ictures L nd L with the sme numer of rows such tht the rightmost column of is glued to the leftmost column of. Let ( ) where ( ) nd ( ) where ( ) e two tiling systems for L nd L. We ssume tht the locl lhets nd re disjoint. We define WK - D tiling system for L s ( ). Here =. hs to contin ll the elements from set where ( ) excet those corresonding to the right orders nd ll elements from set where ( ) excet those corresonding to the left orders. Some middle tiles corresonding to the two columns must e dded where the gluing is done. Such tiles contin ieces of the right order of ictures in L in the left side nd ieces of the left order of ictures in L in the right side. i.e. the following three set of tiles re defined. i nd d c d c d i nd c c d c d i i c d c d c d c d

7 Wtson-Crick locl lnguges nd Wtson-Crick two dimensionl locl lnguges 7 Then i. i i i i π () The rojection : is π() π() if if Γ Γ. Similrly we cn otin WK -D tiling system for the row conctention. Theorem 6.3. The fmily L(WK -D-TS) is closed under row nd column closure oertions. Proof: Let L e icture lnguge over n lhet which is recognizle y tiling systems. By definition of the column closure L of L is given y successions of column conctention oertions etween ictures in L. Then we cn find WK -D tiling system ( ) for L using the sme rocedure s in Theorem 6.. Consider two different WK -D tiling systems for L s ( ) where ( ) nd ( ) where ( ) where the locl lhets nd re disjoint. Let = nd construct i i s in Theorem 6.. Then i. The rojection : i is sme s in Theorem 6.. Similrly the row closure L cn e done. Theorem 6.4. The fmily L(WK -D-TS) is closed under nd. Proof: Let L nd L e two icture lnguge over n lhet nd let ( ) where ( ) nd ( ) where ( ) e two WK -D tiling systems to recognize L nd L resectively. Assume =. A WK -D tiling system ( ) for L L = L is defined. Let = e the locl lhet nd define the rojection so tht its restrictions to lhets nd coincide with nd resectively (sme s in Theorem 6.). Then = where ( ) nd ( ). For L = L L we construct WK -D tiling system ( ) we find locl lnguge L whose ictures elong to locl lnguges L nd L. For this let such tht ( ) ( ) = ( ). Then which reresents is defined s ( ) ( ) nd (cc ) (dd) c d c d The rojection : is well defined s (( )) = ( ) = ( ) ( ). Theorem 6.5. The fmily L(WK -D-TS) is closed under rottion. Proof: Let L ** nd let T = ( ) = ( ) e WK -D tiling system to recognize L. Rottion of L is recognized y WK -D tiling system T R = ( R ) where R is the rottion of set = ( ).

8 7 Mry Jemim Smue nd V.R. Dre Conclusion: We used the concet of locl lnguges on Wtson-Crick domin nd roved some closure results. We hve extended these concets to rrys nd some results hve een estlished. References [] D. Beuuier nd J.E. Pin Scnners nd lnguges Theoreticl Comuter Science 84 (99) 3. [] R. Freund Gh. Păun G. Rozenerg nd A. Slom Wtson-Crick finite utomt Proceedings of 3rd DIMACS Worksho on DNA sed comuters Phildelhi (997) [3] D. Gimmrresi nd A. Restivo Two dimensionl lnguges Hndook of Forml Lnguges Sringer-Verlg Volume 3(997) [4] W. Thoms Lnguges utomt nd logic In G. Rozenerg nd A. Slom editors Hndook of Forml Lnguge Volume 3 (997)