Timed Watson-Crick Online Tessellation Automaton

Transcription

1 Annals of Pure and Applied Mathematics Vol. 8, No. 2, 204, ISSN: X (P), (online) Published on 7 December Annals of Timed Watson-Cric Online Tessellation Automaton Mar Jemima Samuel, V.R. Dare 2 and D.G. Thomas 3 Department of Mathematics, Anna Adarsh College for Women, Anna Nagar Chennai , India. dr_pjaa@ahoo.com 2,3 Department of Mathematics, Madras Christian College, Tambaram East Chennai , India 2 rajumardare@ahoo.com, 3 dgthomasmcc@ahoo.com Received 3 September 204; accepted 2 November 204 Abstract. Watson-Cric finite automata are language recognizing devices similar to finite automata introduced in DNA computing area. We define Watson-Cric online tessellation automata which wor on double-stranded arras where the two strands relate to each other through a complementar relation inspired b the DNA complementarit. Also we define timed Watson-Cric online tessellation automaton and some equivalence results have been established. Kewords: Watson-Cric online tessellation automaton, equivalence AMS Mathematics Subject Classification (200): 03D05. Introduction The remarable progress made b molecular biolog and biotechnolog in the last couple of decades, particularl in sequencing, snthesizing and manipulating DNA molecules, gave rise to the possibilit of using DNA as a support for computation. The computer science communit quicl reacted to this challenge and man computational models were built to exploit the advantages of nano-level biomolecular computing. One of them is Watson-Cric automata. Watson-Cric automata, introduced in [3] represent one instance of mathematical model abstracting biological properties for computational purposes. The are finite automata with two reading heads, woring on double stranded sequences. One of the main features of these automata is that characters on corresponding positions from the two strands of the input are related b a complementarit relation similar with the Watson-Cric complementarit of DNA nucleotides. The two strands of the input are separatel scanned from left to right b read onl heads controlled b a common state. Online tessellation automaton is a recognizing device for accepting finite arras [4]. Online tessellation automaton reads elements diagonalwise. In this paper we define Watson-Cric online tessellation automaton and the language accepted b it. Timed (finite) automaton is defined to accept timed words []. It is a finite automaton with a finite set of real-valued clocs. It accepts timed words which are infinite sequences in which a real valued time of occurrences is associated with each smbol. The infinite word α over an alphabet Σ is represented as a function from the set 75

2 Timed Watson-Cric Online Tessellation Automaton N of all positive integers to Σ. We represent the infinite word α as α = a a 2 where α(i) = a i Σ. The set of all infinite words over Σ is denoted as Σ ω. If L Σ ω then L is called an ω-language. Timed ω-double-stranded sequence and timed Watson-Cric ω- automaton and the language accepted b it are defined. In this paper we extend the concept of timed online tessellation automaton [5] to Watson-Cric online tessellation automaton. 2. Preliminaries Let Σ be a finite alphabet. Σ denotes the set of all finite words over Σ, λ is the empt word and Σ + is the set of all non empt finite words over Σ. i.e., Σ + = Σ \ {λ}. Definition. [7] We now define a complementarit relation on the alphabet Σ (lie the Watson-Cric complementarit relation among the four DNA nucleotides), ρ Σ Σ which is smmetric. Denote WK a ρ(σ) = / a, b Σ,(a,b) ρ. b The set WKρ(Σ) is called the Watson-Cric domain associated to Σ and ρ. The a a 2 a n w elements WKρ ( Σ) b b 2 b are also written in the form for w = a n w 2 a 2 a n and w 2 = b b 2 b n. According to the usual wa of representing DNA molecules as double-stranded sequence, we also write the product monoid Σ Σ in the form Σ and its elements in the form x. Definition 2. [6] A Watson-Cric finite automaton is a 6-tuple M = (Σ, ρ, Q, s 0,, δ) where Σ - a finite alphabet, Q - a finite set of states, ρ Σ Σ - is a smmetric relation (the complementarit relation), s 0 Q is the initial state, Q is the set of final states Q and δ : Q Σ 2 Σ is a mapping such that δ(s, x ) φ onl for finitel man triples (s, x, ) Q Σ Σ. We can also write the transition of M as rewriting rules of the form s x s such a rule having the same meaning as s δ(s, x ). u w or Σ u w, and s, s Q we write u 2 w 2 w s s u iff s u 2 w 2 w 2 u 2 s δ. If is the reflexive and transitive closure of the relation, then the language accepted b a Watson-Cric automaton is 76

3 Mar Jemima Samuel, V.R. Dare and D.G. Thomas w L(M) = {w Σ /s0 sf for some w 2 Σ and sf } w 2 We see that a WK automaton is a finite automaton with a double-stranded tape (and two read heads, one for each strand of the tape). Definition 3. [4] A two dimensional online tessellation automaton A = (Q, Σ, Q 0, δ, ) where Q - is a finite set of states, Σ - is a input alphabet, Q 0 Q - is a set of initial states, Q - is a set of final states and δ : Q Q Σ 2 Q is a transition function. Computation b a two dimensional online tessellation automaton on an infinite arra is as follows. We introduce a special boundar smbol # Σ below the first row and to the left of the first column of p and denote the resulting arra b b(p). At time t = 0, an initial state q 0 Q 0 is associated with all positions of #. At time t =, a state from δ(q 0, q 0, a ) is associated with the position (, ) holding a. At time t = 2, states are associated simultaneousl with positions (, 2) and (2, ) respectivel holding a 2 and a 2. If s is the state associated with the position (, ) then the states associated with the position (2, ) is an element of δ(q 0, s, a 2 ) and to the position (, 2) is an element of δ(s, q 0, a 2 ). We then proceed to the next diagonal and so on. 3. Watson-Cric online tessellation automaton Definition 4. Watson-Cric online tessellation automaton is a 6-tuple M = (Σ, ρ, Q, q 0,, δ), where Σ - is the input alphabet, ρ Σ Σ is a smmetric relation called the complementar relation on Σ, Q - is a set of states, q 0 Q - inital state, Q - is a set of final states δ : Q Q Σ 2 Q is a transition function where Σ = Σ {λ}. i.e., δ(q 0, q 0, x ) = q, such that δ(q, q, x ) φ onl for finitel man (q, q x, ) Q Q Σ Σ. There are two planes. The automaton reads x from the st plane and from the 2 nd plane and enters state q. The st plane and 2 nd plane corresponds to the upper level strand and lower level strand of a double stranded sequence. Computation on the two planes follow the pattern of online tessellation automaton. A special boundar smbol # Σ is below the first row and to the left of the first column. At time t = 0, an initial state q 0 is associated with all positions of #. At time t =, x a state from δ(q 0, q 0, ) is associated with the position (, ) holding x in st plane and in 2 nd plane. At time t = 2, states are associated simultaneousl with positions (, 2) and (2, ) respectivel both in the st plane and 2 nd plane. If q is the state associated with the position (, ) in both the planes then the state associated with the position (2, ) x 2 is an element of δ(q, q 0, ) and to the position (, 2) is an element of δ(q 0, q, 2 77

4 Timed Watson-Cric Online Tessellation Automaton x2 ). We then proceed to the next diagonal and so on. The first plane is the arra 2 denoted b p and the second plane is the arra denoted b p 2. The language accepted b Watson-Cric online tessellation automaton M is p defined as L WC OTA (M) = {p Σ /δ(q 0, ) = qf p for some q f where p is the 2 arra from the st plane and p 2 is the arra from the 2 nd plane}. 4. Timed automaton Definition 5. [2] A non-deterministic finite automaton is a 5-tuple of the form M = (Q, Σ, s 0,, δ) where Q is the set of states Σ is the alphabet s 0 Q is the initial state Q is the set of final states δ : Q Σ 2 Q is the transition mapping written as δ(s, a) = s where s, s Q and a Σ. If card(δ(s, a)) for all s Q and a Σ, then we sa that the automaton is deterministic. A relation is defined in the following wa on the set Q Σ for s, s Q, a Σ, x Σ, we write (s, ax) (s, x) if s δ(s, a), b definition (s, λ) (s, λ). If is the reflexive and transitive closure of the relation, then the language of the strings recognized b the automaton M is defined b L(M) = {x Σ / (s 0, x) (s, λ), s }. The language accepted b an automaton is called a regular language. a a 2 a3 0 2 Let x = a a 2 Σ ω. We sa that r : s s s... is a run of M over x if s 0 = q 0 and δ(s i, a i ) = s i for i =, 2,... and s i Q. or a run r the set Inf(r) consists of the states s Q such that s = s i for infinitel man i 0. The acceptance condition on Inf(r) are given below:. Büchi condition: Inf(r) φ. 2. Muller condition: A famil 2 Q of final state sets is considered and Inf(r). Then L ω (M) is the ω-language consisting of the words x Σ ω such that M has an accepting run over x. An ω-language L is called ω-regular if there exists an automaton M such that L = L ω (M). We now recall the definition of timed automata and timed languages []. We define timed words b coupling a real-valued time with each smbol in a word. The set of nonnegative real numbers R is chosen as the time domain. Definition 6. [] A timed sequence τ = τ τ 2... is an infinite sequence of time values τ i R (set of real numbers) with τ i > 0 satisfing the following conditions.. monotonicit: τ increases strictl monotonicall i.e., τ i < τ i+ for all i. 2. for ever t R, there is some i such that τ i > t. 78

5 Mar Jemima Samuel, V.R. Dare and D.G. Thomas A timed word over Σ is a pair (σ, τ) where σ = σ σ 2... is an infinite word over Σ and τ is a time sequence. A timed language over Σ is a sequence of timed words over Σ. If a timed word (σ, τ) is viewed as an input of an automaton, it presents the smbol σ i at time τ i. If each smbol σ i is interpreted to denote an event occurrence then the corresponding component τ i is interpreted as the time of occurrence of σ i. In an automaton the choice of the next state depends upon the input smbol read. Similarl in the timed automaton defined in [], the choice also depends upon the time of the input smbol relative to the times of the previousl read smbols. or this purpose a finite set of real valued clocs are associated with each transition table. A cloc can be set to zero simultaneousl with an transition. At an instant, the reading of a cloc equals the time elapsed since the last time it was reset. With each transition a cloc constraint is associated such that the transition is considered onl if the current values of the clocs satisf this constraint. Definition 7. [] A timed Büchi automaton M = (Q, Σ, Q 0, C, E, ) where Q is a finite set of states Σ is a finite alphabet Q 0 is a set of initial states C is a finite set of clocs is a set of final states with Büchi acceptance condition E Q Q Σ 2 c φ(c) gives the set of transitions and φ(c) are cloc constraints over C. An edge (s, s, a, µ, δ) represents a transition from state s to state s on input smbol a. The set µ C gives the clocs to be reset with this transition and δ is a cloc constraint over C defined inductivel b δ = x c / c x / δ / δ δ 2 where x is a cloc in C and c is a constant in the set of rationals. A cloc interpretation v for a set C of clocs assigns a real value to each cloc i.e., it is a mapping from C to R. We sa that a cloc interpretation v for C satisfies a cloc constraint δ over C iff δ evaluates to true using the values given b v. Given a timed word (σ, τ), the transition of M starts in its start states at time 0 with all its cloc initialized to 0. As time advances the values of all clocs change reflecting the elapsed time. At time τ i, M changes state from s to s using the transition of the form (s, s, σ i, µ, δ) reading input σ i if the current values of clocs satisf δ. With this transition the clocs in µ are reset to 0 and thus start counting time with respect to the time of occurrence of its transition. A run of timed automaton is defined in the following wa. A run records the state and the values of all the clocs at the transition points. A run r denoted b ( s, v) of a timed automaton over a timed word (σ, τ) is an σ σ2 σ τ τ2 τ3 infinite sequences of the form r : < s, v > < s, v > < s, v >... where s i Q and v i (C R] for all i 0. A run r = ( s, v) of a timed Büchi automaton over a timed word (σ, τ) is called an accepting run if Inf(r) φ. or a timed Büchi automaton M, the language L(M) of timed word it accepts is defined to be the set {(σ, τ) / 79

6 Timed Watson-Cric Online Tessellation Automaton M has an accepting run over (σ, τ)}. A timed language L is a timed regular language if L = L(M) for some timed Büchi automaton M. 5. Timed Watson-Cric online tessellation automaton Watson-Cric online tessellation automaton can be extended to define infinite arras and thereb we define timed Watson-Cric online tessellation automaton on ω-arras. Definition 8. A timed Watson-Cric online tessellation automaton is given as M = (Σ, ρ, Q, q 0,, δ, C, φ(c)) where Σ is the input alphabet, ρ Σ Σ is a smmetric relation, Q is a set of states, q 0 Q is the initial state, C = {C, C 2,..., C n } is a finite set of clocs δ : Q Q Σ 2 Q C is a transition function where Σ = Σ {λ}. xij i.e., δq,q, 2 = (q3,c ) ij (C) {v : Σ φ = i Q {C }, i,...,n} Σ i = R + x ij R + is the set of positive real numbers and v,q,c = ij and ij Q is set of final states. The language accepted b the timed Watson-Cric online tessellation automaton is defined as L ωω TWC OTA (M) = { (p, τ) / p is accepted b Büchi acceptance condition and τ is a time sequence and τ Σ i+j = + ij }. Example. Let M = (Σ, ρ,q,q 0,,δ,C,φ(C)) where Σ = {a, b, c}, ρ = ((a, a), (b, b), (c, c)), Q = {q 0, q, q 2, q 3, q 4 }, = {q 4 }, C = {C, C 2, C 3, C 4 }, φ(c) : {v i : Σ Q Ci R +, i =,, n} and δ : Q Q Σ 2 Q C given b δ(q 0, q 0, ( λ a )) = (q, C ), δ(q 0, q 2, ( b c )) = (q3, C 3 ), δ(q 0, q, ( λ a )) = (q, C ), δ(q 2, q 2, ( a b )) = (q2, C 2 ), δ(q, q 0, ( λ a )) = (q, C ), δ(q 0, q 3, ( b c )) = (q3, C 3 ), δ(q 0, q, ( a b )) = (q2, C 2 ), δ(q 3, q 2, ( b c )) = (q3, C 3 ), δ(q, q, ( λ a )) = (q, C ), δ(q 0, q 3, ( c λ )) = (q4, C 4 ), δ(q 0, q 2, ( a b )) = (q2, C 2 ), δ(q 3, q 3, ( b c )) = (q3, C 3 ) δ(q 2, q, ( a b )) = (q2, C 2 ), δ(q 0, q 4, ( c λ )) = (q4, C 4 ), δ(q 4, q 4, ( c λ )) = (q4, C 4 ), δ(q 4, q 3, ( c λ )) = (q4, C 4 ) 80

7 Mar Jemima Samuel, V.R. Dare and D.G. Thomas v (( λ a ), q0, C ) = and 0 elsewhere, v 2 (( a b ), q, C 2 ) = 2 and 0 elsewhere v 2 (( a b ), q, C 2 ) = 2 and 0 elsewhere, v 3 (( b c ), q2, C 3 ) = 3 and 0 elsewhere v 3 (( b c ), q2, C 3 ) = 3 and 0 elsewhere, v 4 (( c λ ), q4, C 4 ) = 4 and 0 elsewhere. L ωω TWC OTA (M) = {(p, τ) Σ ω / number of consecutive columns containing the letter `a = number of consecutive columns containing the letter `b followed b infinite finite number of columns containing the letter `c in p and for n, t i = 2i for i = 3n, t i for i 3n. Definition 9. A timed online tessellation Muller automaton is a tuple M = (Σ, ρ, Q, q 0, δ, C, φ(c), ) where Σ, ρ, Q, q 0, δ, C, φ(c) are as in Definition 8 and 2 Q is a famil of final states. The collection of all timed arras (p, τ) accepted b M is denoted as (M) L ωω TWC OTA. Theorem. or a timed Watson-Cric online tessellation automaton, the acceptance conditions Büchi and Muller are equivalent. Proof: Let M = (Σ, ρ, Q, q 0,, δ, C, φ(c)) be a timed Watson-Cric online tessellation automaton accepted b the Büchi acceptance mode. Consider the timed online tessellation automaton M where the acceptance is in the Muller mode with the same timed transition table as that of M and with the acceptance famil = {Q Q, Q φ}. Then L ωω (M) = L ωω (M ). Conversel given the timed Watson-Cric OTA in Muller mode, we construct a timed Watson-Cric OTA in Büchi mode accepting the same arra language. Let the timed Watson-Cric OTA be given b M = (Σ, ρ, Q, q 0, δ, C, φ(c), ). ωω ωω We see that L (M) L (A where A = (Σ, ρ, Q, q 0, δ, TWC OTA Muller = TWC OTA Muller ) C, φ(c), {}). Hence it is enough to construct for each acceptance set, a timed Watson-Cric ωω OTA in Büchi mode as A accepting L (A ). TWC OTABuchi ɺ Assume = {q, q 2,..., q }. States of A are of the form q i where q Q and i {0,, 2,..., }. The initial state is {q 0 0 }. The automaton simulates the transition of A. xij or ever transition in A of the form δ q,q2, = qij and ij xij i v,qij,c = ij the automaton A has a transition j x ij i δ q,q2, = qij ij, i = ij xij 0,, 2,..., and v,qij,c = ij. ij 8

8 Timed Watson-Cric Online Tessellation Automaton xij or ever transition δ q,q2, where q, q 2 for each i =,..., there ij i j x ij is an A transition δ q, q2, where ij i j = (i + ) mod if q = q j (i + ) mod if j = i. The onl final state is q. Hence we have seen the equivalence of Büchi and Muller of timed Watson-Cric OTA. 6. Conclusion Watson-Cric online tessellation automaton has been defined. Timed Watson-Cric online tessellation automaton is defined and the equivalence of Büchi and Muller timed Watson-Cric online tessellation automaton has been proved. REERENCES. R.Alur and D.L.Dill, A theor of timed automata, Theoretical Computer Science, 26 (994) E.Petre, Watson-Cric ω-automata, Journal of Automata, Languages and Combinatorics, 8 (2003) R. reund, Gh. Păun, G. Rozenberg and A. Salomaa, Watson-Cric finite automata, Proceedings of 3 rd DIMACS Worshop on DNA based computers, Philadelphia (997) K. Inoue and A. Naamura, Some properties of two dimensional on-line tessellation acceptor, Information Sciences, 3 (977) M.J.Samuel, V.R.Dare and D.G.Thomas, A stud on timed online tessellation automata, Proceedings of the 2 nd National Conference on Mathematical and Computational Models (NCMCM 2003), (2003) A.Păun and M.Păun, State and transition complexit of watson-cric finite automata, Proceedings of undamentals of Computation Theor, CT '99, LNCS, 684, Springer-Verlag (999), W.Thomas, Languages, Automata and Logic, In G. Rozenberg and A. Salomaa, editors, Handboo of ormal Language, 3 (997) i 82