J. Comp. & Math. Sci. Vol.2 (1), 47-50 (2011)
Regular Property of Bicenter Marker Languages J. PADMASHREE and S. JEYA BHARATHI Department of Mathematics, Thiagarajar College of Engineering, Madurai-15. India E-mail id: shreepadma@tce.edu, sjbmat@tce.edu. ABSTRACT Head introduced the structure of the languages of DNA molecules (strings) which could be produced through the splicing operation. Here we study the regularity of the spliced DNA molecule with the help of bicenter marker language. This bicenter marker language form the syntactic congruence class with the form L=L1[x][y]L2 where L1,L2 are regular language and [x][y] is a bicenter marker. Keywords: Automata, Regular Languages, Bicenter marker.
1. INTRODUCTION In 1987 Head5 introduced a generative formalism called the splicing system to analyse the generative capacity of the recombinant behaviour of DNA molecules in terms of formal languages. Later, in 1995, splicing systems were suggested to represent DNA computations. For performing computations they use only the splicing operation, which is a combination of cutting and pasting and can be viewed as a mathematical generalization of the DNA recombination. P. Bonizzoni introduced the marker language in the linear splicing of DNA molecule using splicing system(section 2.3 of1). 2. PRELIMINARIES Definition 2.1: Syntactic congruence: Syntactic congruence ≡L with respect to [w] is defined as follows:
w ≡L w' [ ∀ x, y ∈A*, xwy ∈ L w'y ∈ L] C(w) = C(w'). where C(w) = {(x, y) ∈ A* × A*| xwy ∈ L}, the set of contexts C(w) of w ∈ A* for L ⊆ A* and [w] = {w'∈ A*|w' ≡L w the class of w modulo ≡L }. Definition 2.2 : Bicenter marker language Let L be regular language and let [x],[y] be two markers for L. Denote L1 = {z∈A*/ δ(q0, z) = qx & δ(q0,z)=qy} L2 = {z∈A*/ δ(px,z) = qx & δ(py,z) ∈F}. If L = L1([x][y])L2 , then L is the bicenter marker language L([x][y]) associated with ([x][y]). L is a bicenter marker language if there exists a markers [x] and [y] for L such that L is the bicenter marker language associated with ([x][y]).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
J. Padmashree, et al., J. Comp. & Math. Sci. Vol.2 (1), 47-50 (2011)
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Definition 2.3: Cycle Class A class [x][y] ∈ L is called cyclic if there exists q ∈Q and z∈ [x][y] such that z is a q-label1 in A. Definition 2.4: Singular Class
A class [x][y] ∈ L is called singular if |Q[x][y]| =1. 3. MODELS Example 3.1 Consider the finite automata
The language L([cd][ef]) = ((bcg)+bc(def)+g)*a is the bicenter marker language associated with ([cd][ef]). we have, L1= ((bcg)+bc(def)+g) * L2= ((bcg)+bc(def)+g) *a 4. RESULTS Proposition 4.1:Let L=L1w[x][y]L2 be a bicenter marker language where wx is a single word and [x][y] ∈ L. Then, we have L=L1[wx] [y]L2. Proof:Since w[x][y] ⊆ [w][x][y] ⊆ [wx][y]. We obviously have L1w[x][y]L2 ⊆ L1[wx][y]L2. By remark 2.1 of1, for each m' ∈ [wx], we have Qm’=Qwx= {q}. Furthermore, in view of theorem 2.1 of1, We have δ(q,wx) = p=δ(q, m’).
Each element in [wx] is a constant for L, consequently, if L=L1w[x][y]L2, we have L1={y1 ∈ A*| δ(q0,y1)=q}, L2= {y2 ∈ A*| δ(p,y2) ∈ F} For the regular languages L1=L(w[x][y]L2)-1, L2=(L1w[x][y])-1L We have L1[wx][y]L2 ⊆ L=L1w[x][y]L2, i.e., L=L1[wx][y]L2. Proposition 4.2: Let [x][y] be a singular and cyclic class and let Q[X][Y] = {q}. For every u,v ∈ [x][y],we have δ(q,u)=δ(q,v)=q . Remark 4.1: We analyzed the existence of more cycles in bicenter marker languages than the marker languages. (By comparing the example 4.1 of1 and example 2 of this paper).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
J. Padmashree, et al., J. Comp. & Math. Sci. Vol.2 (1), 47-50 (2011)
Remark 4.2:Let L=L1([x][y])L2 be marker language associated with a two maker [x]and [y] , Since the minimal automaton recognizing L is trim and Qx≠φ,Qy≠φ there exist a,b∈A∗such that a[x][y]b ⊆ L ,i.e., we have L1≠φ and L2≠φ . In addition, L1 = L([x][y]L2) -1, L = (L1[x][y]) -1L and L1, L2 are regular languages. Theorem 4.1: An algorithm for constructing the set of bicenter markers for L. Proof: Given a regular language L , for each [X],[Y]∈ L, the structure of the regular language [X] and [Y] can be obtained by standard construction algorithms7. Let {x1………,xk ,y1,…yk} be complete set of representatives of the syntactic congruence classes of L. Obviously, for all i ∈{1,2,…k},we can decide whether │Qxi│ =1 and│Qyi│ =1 in the minimal automaton A recognizing L. For this xi and yi we can decide whether ([xi][yi]) is a cyclic class since, in view of proposition 4.2,([xi][yi]) is a cyclic class iff (xi yi )is a q-label. In addition, we can decide whether ([xi][yi]) is a finite class since it is decidable whether a regular set is finite. Therefore, in view of definition 2.2, It is decidable whether ([xi][yi]) is a bicenter marker. Then the above argument allows us to design an algorithm for constructing the set of bicenter marker for L. Theorem 4.2: If L is a regular language. Then L is a bicenter marker language. Proof: Given a regular language L, in above theorem, we have proved that There exists an algorithm for constructing the set of
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bicenter markers ([x][y]) for L. In the proof of this result we have observed that the structure of the regular language ([x][y]) can also be described. Given a bicenter marker ([x][y]) for L, let us consider the regular sets F([x][y])=A∗([x][y]) A∗∩L, suf([x][y])=(A∗([x][y]) -1F[x][y]), pref([x][y])=F([x][y])([x][y]A∗)-1. Then, L is a bicenter marker language if and only if exists a bicenter marker ([x][y]) for L such that L= F ([x][y]) if [x] and [y] is a finite class, L= F([x][y])∪ pref([x][y]), suf([x][y]) otherwise. Thus, the conclusion holds since given two regular languages X, Y, it is decidable whether X=Y. REFERENCES 1. P.Bonizzonia, C. De Felice, G.Mauri, R. Zizza, Linear splicing and syntactic monoid, Discrete Applied Mathematics, 154, pp, 452-470 (2004).’ 2. E. Csuhaj-Varju, L. Freund, L. Kari, Gh. Paun, DNA computing based on splicing: universality results, First Annual Paci_c Symp. on Biocomputing, Hawaii, Jan. 1996 (L. Hunter, T. E. Klein, eds.), World Sci. Publ., Singapore, (1996). 3. T. Head, Formal language theory and DNA: an analysis of the generative capacity of specific recombinant behaviors, Bull. Math. Biology, 49, 737759 (1987). 4. T. Head, Gh. Paun, D. Pixton, Language theory and molecular genetics. Generative mechanisms suggested by DNA recombination, chapter 7 in vol. 2 of [26], 295 - 360.
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5. J. E. Hopcroft, R. Motwani, J.D.Ullman, Introduction to Automata Theory, Languages and Computation, AddisonWesley, Reading, M.A, (2001). 6. Gh. Paun, G. Rozenberg, A. Salomaa, Computing by splicing, Theoretical Computer Sci., 168, 2, 321 – 336 (1996).
7. J.E.Pin, Variétés de langages formels, Masson ,Paris, 1984(English translation: Varietes of formal languages , Plenum, New york,1986). 8. D. Pixton, Regularity of splicing languages, Discrete Appl. Math., 69 101-124, (1996).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)