Language Recognition Power of Watson-Crick Automata

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1 01 Third International Conference on Netorking and Computing Language Recognition Poer of Watson-Crick Automata - Multiheads and Sensing - Kunio Aizaa and Masayuki Aoyama Dept. of Mathematics and Computer Science Shimane University Matsue, Shimane Japan aizaa@cis.shimane-u.ac.jp Kazuki Murakami TIS Inc. Tokyo, Japan Abstract A Watson-Crick automaton is introduced as a computational model of DNA computing frameork. In this paper, e introduce a 3-head reverse Watson-Crick finite automaton having three read-only heads but no sensing poer. We sho that the class of 3-head reverse Watson-Crick finite automata includes the class of simple sticker language and full reading sensing 5' 3' Watson-Crick finite automata. Keyords-DNA computing; Watson-Crick Automata; sticker systems; multi-head automata; sensing I. INTRODUCTION DNA computing provides ne paradigms of computation [1, ]. The Watson-Crick automaton, introduced in [3], provides a computational model of the DNA computing frameork. It uses a double-stranded tape called Watson- Crick tape, hose strands are scanned separately by readonly heads. The symbols in the corresponding cells of the double-stranded tapes are related by complimentary relations similar to the actual DNA molecules. The relationships beteen the classes of the Watson-Crick automata are investigated in [3, 4, 5, 6]. The to strands of an actual DNA molecule have opposite 5' 3' orientation. This suggests considering a variant of Watson-Crick finite automaton that reads the to strands of its input tape in opposite directions. Such automata are called reverse Watson-Crick automaton and introduced in [3]. Some variations of the reverse Watson-Crick automaton ith sensing poer hich tells the upper and the loer heads are ithin a fixed distance (or meet at the same position) are discussed in [7, 8, 9]. In this paper, e introduce a 3-head reverse Watson- Crick finite automaton having three read-only heads but no sensing poer. We sho that the class of 3-head reverse Watson-Crick finite automata called 3RWK includes the class of simple sticker language [] and Full reading sensing 5' 3' Watson-Crick finite automata [7]. In Section II, basic definitions of the Watson-Crick finite automaton and the sticker system are revieed. In Section III, e introduce the 3-head reverse Watson-Crick finite automaton. The relationships among the 3-head reverse Watson-Crick automata, sticker systems, and full reading sensing 5' 3' Watson-Crick finite automata are discussed in Section IV, V, and VI. Finally a brief conclusion is given in the last section. II. BASIC DEFINITIONS DNA strands consist of polymer chains that are composed of nucleotides. There are four types of nucleotides: A (Adenine), G (Guanine), C (Cytosine), and T (Thymine). The double helix of DNA arises by the bonding of to separate strands. Bonding occurs by the pairise attraction of types: A alays bonds ith T, and G ith C. This phenomenon is knon as Watson-Crick complementarity. Consider an alphabet V and a symmetric relation ρ V V over V. We consider the complementarity relation as a generalization of Watson- Crick complementarity. A. Watson-Crick Domain We associate ith V the monoid V * V * of pair of strings. In accordance ith the ay of representing DNA molecules, here one considers the to strands placed one over the other, e rite the elements (x 1, x ) V * V * in the form x 1 x. The concatenation of to pairs x 1 x y 1 and y is x 1 y 1 V *. We also rite x y V * instead of V * V *. We denote V a = { a, b V,(a,b) ρ} and V b ρ * V WK ρ (V ) =. The set WK ρ (V ) is called the Watson- V ρ Crick domain associated to the alphabet V and the complementarity relation ρ. For 1 = a 1 a a n, = b 1 b b n the element WK ρ (V ) is ritten in the form 1 a 1 b 1 a b a n b n. We call the /1 $ IEEE DOI /ICNC

2 element double stranded sequence. The string 1 is called the upper strand and is called the loer strand. B. Watson-Crick Finite Automaton A Watson-Crick automaton [] is a construct here V is an alphabet, ρ V V is a complementarity relation, K is a set of states, s 0 K is the initial state, F K is the set of final states, and V * δ : K V * (K) is a transition function. A transition of a Watson-Crick finite automaton is defined as follos: u 1 x 1 v 1 For u x v WK ρ (V ) and s, s ' K, e rite u 1 u s x 1 v 1 x v u 1 x 1 u x s ' v 1 iff v x 1 s ' δ(s, ). We denote the reflexive and transitive x closure of the relation by *. The language L(M ) of M is defined as follos: 1 L(M ) = { 1 s 0 * 1 s f, s f F}. C. Simple Sticker System A sticker system [] is a construct γ = (V, ρ, A, D), here V and ρ are same as those in the Watson-Crick automaton, A is a finite subset of LR ρ (V ), and D is a finite subset of W ρ (V ) W ρ (V ). The elements of A are called axioms. Starting from these axioms and using the pairs (u, v) of bricks in D, e obtain a set of double stranded sequences in WK ρ (V ) by using the sticking operations μ r, μ l. For a given sticker system γ and sequences x, y LR ρ (V ), e rite x y iff y = μ l (u,(μ r (x, v)), for some (u, v) D. The language L(γ ) of γ is defined as follos: 1 L(γ ) = { 1 x *, x A}. A sticker system γ is said to be simple, if for all pairs (u, v) D, u, v V * or u, v V * the class of simple sticker systems SSL.. We denote III. 3-HEAD REVERSE WATSON-CRICK FINITE AUTOMATA A 3-head reverse Watson-Crick finite automaton has three heads. We call the head on upper left of the input tape first-head, on loer right second-head, and the head under first-head third-head, respectively. In the initial configuration, these heads are on positions shon in Fig. 1, and they can move only the direction pointed by the bold arros. This automaton has no ability of sensing (this means automata cannot detect the position of three heads on input tape). Figure 1. A 3-head reverse Watson-Crick finite automaton A 3-head reverse Watson-Crick finite automaton is defined as follos: here V, ρ, K, s 0, and F are same as those in the Watson- Crick automaton. Transition function δ is defined by the folloing: δ : K V * V * V* (K). The transition function s ' δ(s, x, y, z) means that in state s, first-head scans string x in the upper strand, second-head scans string y in the loer strand, and third-head scans string z in the loer strand, respectively, then the state is changed to the next state s. Input tape is accepted by a 3-head reverse Watson-Crick finite automaton, if and only if each head reached opposite end of tape in final state. Therefore the language of this automaton is defined in similar ay as that of the Watson- Crick finite automaton. We denote the class of these automata 3RWK. IV. RELATIONSHIP BETWEEN 3RWK AND SSL Sensing 5' 3' Watson-Crick finite automata and s t heads one ay multi-head finite automata [10] are knon as the automata that recognize the languages generated by simple sticker systems. These automata have more than or equal to to heads and ability of sensing. It seems that the sensing ability gives these automata the poer to recognize sticker languages. 3-head reverse Watson-Crick finite automata can also recognize the languages generated by simple sticker systems 358

3 even though they have no sensing ability. Instead of the sensing ability, they use the third-head as counter to recognize the languages in SSL. Theorem 1. SSL 3RWK. Consider a given simple sticker system γ = (V, ρ, A, D), here e assume that the axiom A of γ is in the form as follos ithout loss of generality: x p y p z p A = { x ' p y ' p z ' p y p y ' p V +, V ρ x p x ' p, z p z ' p V * V *,1 p m, m = A } We construct a 3-head reverse Watson-Crick finite automaton here K = {s 0, s f } {s p, s ' p 1 p m}, F = {s f }, and the transition function δ is constructed as follos: s 0 δ(s 0,u, ρ(v), ) u (for (, v ) D, V*, = uv ) (1) s p δ(s 0, x p, ρ(z p ), ) x p y p (for x ' p y ' p s ' p δ(s p, y p, y ' p, ) z p z ' p A, V*, = x p z p )() x p y p z p (for x ' p y ' p z ' p A, V*, = y p ) (3) s f δ(s ' p, ρ(z ' p ), x ' p, ) x p y p (for x ' p y ' p s f δ(s f, ρ(v), u, ) (for ( u, v z p A ) (4) z ' p ) D ) (5) M simulates the behavior of γ as in folloing three stages: Stage 1: At first, M starts to simulate by rules in (1). These rules check hether or not the pairs of dominos having only upper string in D are introduced by correct ay in generation of simple sticker system γ. (Fig. ) With the state s 0, the first-head scans left element of pair of dominos in the upper strand, hile the second-head scans a complementary string of right element of pair of dominos in the loer strand, and, the third-head scans an arbitrary string that length is equal to the length of concatenation of strings scanned by the firsthead and the second head. Within this stage, M remains in the same state s 0. Rules in (1) are used several times. In this stage, if the first-head is placed at the right of the secondhead, third-head cannot scan loer strand any more and M terminates ithout the final state. Figure. Heads movements in Stage 1. Stage : Next, M checks hether or not the dominos in axiom are introduced by correct ay in generation of simple sticker system γ by using rules in ()~(4). When rules of () are applied, M s state is s 0. Then the first-head scans a string x p in p-th domino (labeled by p) in axiom A, hile the second-head scans a string z p from loer strand, and third-head scans an arbitrary string length of x p z p. After they move, M changes its state to s p. In this stage, if the first-head is placed at the right of the second-head, third-head cannot scan loer strand any more and M terminates ithout the final state. When rules of (3) are applied, M s state is s p. Then the first-head scans the string y p included in the upper part of p-th domino ( y p is upper string of complete double string in the axiom), hile the second-head scans the loer string y ' p, and the third-head scans an arbitrary string length of y p ( = y ' p ). After they move, M changes its state to s ' p. Hoever, if the distance beteen the first and the second-head is smaller than y p before using rules of (3), the third-head cannot move any farther. And, if and only if the third-head is positioned at the right-end of tape, the first and the second-head scan same string on the tape. (After the stage, the thirdhead doesn t move any more.) (Fig. 3) 359

4 When rules of (4) are applied, M state is s ' p. Then the first-head scans the complementarity string of z ' p in p-th domino in axiom, and the second-head scans a string x ' p. After they move, M changes its state to s f, the final state. Figure 3. Checking the axiom. Stage 3: Finally, M checks hether or not the pairs of dominos having only loer string in D are introduced by correct ay in generation of simple sticker system γ by using rules in (5). At state s f, the first-head scans a complementarity string of loer string of right part of dominos pair, and the second-head scans a string of loer string of left part of dominos pair. After rules in (5) are used several times, tape is accepted by M if only if the first and the second-head are finished to scan tape and third-head is positioned at the right-end of tape. From the arguments in above three stages, it is clear that L(γ ) = L(M ). V. RELATIONSHIP BETWEEN 3RWK AND FULL READING SENSING 5' 3' WK In this section, e sho that the class of 3-head reverse Watson-Crick finite automata includes the class of full reading sensing 5' 3' Watson-Crick finite automata [7]. It means that the sensing ability ith -head (each head moves opposite direction, respectively) can be simulated by 3-head WK. Theorem. fs5' 3'WK 3RWK. We consider a given full reading 5' 3' Watson-Crick finite automata M = (V, ρ, K, s 0, F, δ ) ith K = {s 0, s 1,, s k } ( k 0 ) and D = {,0,1,, r, + }. 3-head reverse Watson-Crick finite automaton M ' = (V, ρ, K ', s 0, F ', δ ') hich simulates the behavior of M is constructed as follos: K ' = K + K sen sin g K, K + = {s 0, s 1,, s k }, K sen sin g = {s ' 0, s ' 1,, s ' k }, K = {s" 0, s" 1,, s" k }, F ' = {s" i s i F,0 i k}, and transition function δ ' is defined as follos: s ' i δ '(s 0,,, ) x (for s i δ(s 0, y, d), 0 i k,0 d r ) (6) s j δ '(s i, x, y, ), ( V*, = xy ) x x ' (for s j δ(s i, y, + ), s h δ(s j, y ', + ) 0 i, j, h k ) (7) s ' j δ '(s i, x, y, ), ( V*, = xy ) x x ' (for s j δ(s i, y, + ), s h δ(s j, y ', d) 0 d r,0 i, j, h k ) (8) s ' j δ '(s ' i, x, y, ), ( V*, = xy ) x (for s j δ(s i, y, d), 0 d r, xy d,0 i, j k ) (9) s" j δ '(s ' i, x, y, ), ( V*, = d) x (for s j δ(s i, y, d), 0 d r, xy > d,0 i, j k ) (10) s" j δ '(s" i, x, y, ) (for s j δ(s i, x y, ), 0 i, j k ) (11) From the above construction, it is not so difficult to see that L(M ) = L(M '). VI. RECOGNITION OF LANGUAGE NOT INCLUDED IN CF Here, e sho that 3-head reverse Watson-Crick finite automata can recognize a language L = {x x {a,b, c}*, x a = x b = x c } CS CF that cannot be recognized by sensing 5' 3' Watson-Crick finite automata [7]. This suggest that the control ith 3-head has more poer than sensing ith -head. Theorem 3. 3RWK can recognize the language L hich is not included in CF. For the language L = {x x {a,b, c}*, x a = x b = x c }, 360

5 e construct 3-head reverse Watson-Click finite automaton here V = {a,b, c}, ρ = {(a, a) a V}, K = F = {s 0 }, and the transition function δ is defined as follos: s 0 δ(s 0, a,, ) (1) s 0 δ(s 0,b,, ), (for V *, = 3 ) (13) s 0 δ(s 0, c,, ), (for V *, = 3 ) (14) From above construction, it is easy to see L = L(M ). VII. COCLUSION We have investigated some relationships among 3-head reverse Watson-Crick finite automata. These relations are summarized as shon in Fig. 4. ACKNOWLEDGMENT Kunio Aizaa thanks Prof. Akira Nakamura for his valuable suggestions. REFERENCES [1] L. M. Adleman, Molecular computation of solutions to combinational problems, Science, vol. 6, 1994, pp [] G. Păun, G. Rozenberg, and A. Salomaa, DNA Computing - Ne Computing Paradigms, Springer-Verlag, 189. [3] R. Freund, G. Păun, G. Rozenberg, and A. Salomaa, Watson-Crick finite automata, Proc. Of the Third Annual DIMACS Symp. On DNA Based Computers, Philadelphia, 1997, pp [4] S. Hirose, K. Tsuda, and Y. Ogoshi, Some relations beteen Watson-Crick finite automata and Chomsky hierarchy, IEICE Trans. Inf. & Syst., vol. E87-D, 004, pp [5] S. Okaa and S. Hirose, The relations among Watson-Crick automata and their relations ith context-free languages, IEICE Technical Report, vol. 105, 005, pp [6] D. Kuske and P. Weigel, The role of the complementarity relation in Watson-Crick automata and sticker systems, DLT 004, LNCS 3340, 004, pp [7] B. Nagy, On 5 ->3 sensing Watson-Crick finite automata, DNA 13, LNCS 4848, 008, pp [8] B. Nagy, On a hierarchy of 5 ->3 sensing WK finite automata languages, mathematical theory and computational practice CiE 009, Abstract Booklet, 009, pp [9] P. Leupold and B. Nagy, 5 ->3 Watson-Crick automata ith several runs, Fundamenta Informaticae, vol. 104, 010, pp [10] P. Weigel, Complexity analysis of sticker systems by means of comparison ith multihead finite automata, 007. Figure 4. Relationships of Language classes. 361