Equivalence of Subclasses of Two-Way Non-Deterministic Watson Crick Automata

Watson Crick automata are finite automata working on double strands. Extensive research work has already been done on non deterministic Watson Crick automata and on deterministic Watson Crick automata. Parallel Communicating Watson Crick automata systems have been introduced by E. Czeziler et al . In this paper we discuss about a variant of Watson Crick automata known as the two-way Watson Crick automata which are more powerful than non-deterministic Watson Crick automata. We also establish the equivalence of different subclasses of two-way Watson crick automata. We further show that recursively enumerable (RE) languages can be realized by an image of generalized sequential machine (gsm) mapping of two-way Watson-Crick automata.


Introduction
The tremendous progress in biotechnology has resulted in decoding of DNA sequences, synthesizing and manipulating DNA, which lead to its usage in computation by computer scientists.As a result sticker systems, splicing systems and carving systems came into existence [1].Many of the NP-complete problems were solved efficiently using DNA computing.The first, the Adleman experiment was done in 1994 [2].As the interest in using DNA in computation increased so did the need for automata which exploit the properties of DNA.The first such automata which exploited the DNA property were the Watson-Crick automata [3] which are the automata counterpart of the Sticker Systems.Essentially Watson-Crick automata are finite automata having two independent heads working on double strands where the characters on the corresponding positions of the two strands are connected by a complementarity relation similar to the Watson-Crick complementarity relation.
The movement of the heads although independent of each other is controlled by a single state.
Details of several variants of non-deterministic Watson-Crick automata have been explored in [4].
Deterministic Watson-Crick automata and their variants have been explicitly handled in [5,6].Parallel Communicating Watson-Crick automata were introduced in [7] and further investigated in [8].A survey of Watson-Crick automata can be found in [9].The effect of the complementarity relation on the computing power of Watson Crick automata is discussed in [10].
Two-way finite automaton (FA) is an abstract machine, a generalized version of the finite automaton which can revisit characters already processed.As in FA, in twoway FA there are finite number of states with transitions between them based on the current character; but each transition is also labeled with a value indicating whether the machine will move its reading head to the left, right, or stay at the same position.Equivalently, 2FAs can be seen as read-only Turing machines with no work tape; only a read-only input tape.The accepting condition is that when the reading head falls off the right end of the tape and the state in which the machine is at that time is final state then the input word is accepted.A twoway Watson Crick automaton (2AWK) is similar in concept to a two-way finite automaton.The only difference between them is that in two-way Watson Crick automata the input tape is double stranded.The idea of two-way Watson Crick automata were introduced in [4] but no comparison of its power with respect to AWK was discussed.The importance of 2AWK is that unlike two-way FA which is equal in power to a FA, 2AWK are more powerful than AWK which we establish in this paper.
In this paper, we give a general description of non-deterministic Watson Crick automata and its different subclasses in Sections 2 and 3.In Section 4 we describe the twin shuffle language and state the relation of twin shuffle language with RE languages.In the following section we state the rules governing two-way non-deterministic Watson Crick automata.In Section 6 we give the definition of the different classes (variants) of 2AWK and investigate the relationship between classes of 2AWK automata.We show that 2AWK = 2SWK = 21WK = 2FWK = 2FSWK = 2F1WK similar to the case of Watson Crick automata.We further show the family of languages accepted by 2AWK is context sensitive.In Section 7 we show that two-way non-deterministic Watson Crick automata are more powerful than non-deterministic Watson Crick automata.In Section 9 we further show that recursively enumerable (RE) languages can be realized by an image of generalized sequential machine (gsm) mapping of two-way Watson-Crick automata.

Basic Terminology for Watson-Crick Automata
V is a finite alphabet.V * denotes the set of all finite words over V, including the empty word .
denotes the length of w.Let and be two words and if there is some word , then u is the prefix of v, denoted by .Two words, u and v are prefix comparable denoted by u~pv if u is a prefix of v or vice versa.
Given two alphabets V and U a mapping h: otherwise, then we say that h is a projection (associated with ) and we denote it by pr V1 .
1 For V x y V  we define their shuffle by A generalized sequential machine (gsm) is a sequential transducer.Such a device is a system  1 2 0 , , , , , .
where Q is the set of states, 1 2 are the alphabets(input and output alphabets) of the automaton, 0 is the initial state, F Q  is the set of final states and is the transition mapping.
The definitions of morphism, gsm and shuffle are stat-ed in [1].A Watson-Crick automaton is a 6-tuple of the form that the machine in state q parses w 1 in upper strand and w 2 in lower strand and goes to state where q just a pair of strings written in that form instead of the two strands are of same length i.e. 1 2 and the corresponding symbols in two strands are complementarity in the sense of relation ρ.
A transition in a Watson-Crick finite automaton can be defined as follows:


The language accepted by Watson-Crick Automata is where w 1 is any string in V * and where q 0 is the initial state and q f is a final state.Then Another important language associated with Watson-Crick automaton is defined taking into consideration the transitions and not the language recognised.
For a Watson-Crick Automaton  0 , , , , , f by denoted  the control word of  , that is the sequence of labels of transition rules used in  .In this way the language is obtained.
The definition of is stated in [4].

Subclasses of Non-Deterministic Watson-Crick Automata (AWK)
Depending on the type of states and transition rules there are four types or subclasses of Watson-Crick Automata.
; Q F  3) simple (SWK): If at each step the automaton reads either from the upper strand or from the lower strand, i.e. for any transition rule 4) 1-limlited (1WK): If for any transition rule q , we have 1 2 1 w w  .

Theorem 1: Simple and 1 limited Watson-Crick automata accept the same family of languages as the family of languages accepted by Watson-Crick automata with arbitrary transition rules.
The proof of Theorem 1 is in [4].The proof of Theorem 2 and Corollary 1 are given in [4].

Twin-Shuffle Language
Consider an alphabet V and its barred variant, is called the twin-shuffle language over .(For a string denotes the string obtained by replacing each symbol in x with its barred variant).
For the morphism h:   Clearly the equality is , where is an alphabet and R is a regular language.

V
In this representation, the language TS V depends on the language L. This can be avoided in the following way: Let a coding be , where i is the symbol of V in a specified ordering.The language gsm can simulate the intersection with a regular language, the projection T as well as the decoding of elements in pr  .

V f TS
Thus we obtain: Corollary 2: For each recursively enumerable language L there is a gsm g L such that Therefore, by using a sequential transducer which can be a deterministic one, we can obtain all recursively enumerable language, starting from the unique language Proofs of Theorem 3 and Corollary 2 are in [1].

Two-Way Non-Deterministic Watson Crick Automata (2AWK)
Two-way non-deterministic Watson Crick automata system is a 6 tuple, where is a set of alphabet, are the beginning and the end marker respectively; that is the word w to be recognized is provided as an input to the automaton in the form is a set of states, is the complementarity relation and is the initial state and F Q  is the set of final states. is the finite number of transition rules; 1) either of the form , which denotes that the machine in state q parses 1 in upper strand in dir 1 direction and 2 in lower strand in dir 2 direction and goes to state where L signifies that the head is reading the word in the left direction, R signifies that the head is reading the word in right direction and if a head reads the empty word  it remains in its current position denoted by 0.

2) or of the forms
, where ble stranded input tape and the two heads at the left end of The word 1 is rejected if one of the following 3 conditions occurs: w 1) The two-way WK automaton goes into a loop which is identified in a similar way as loops in two-way FAs are identified.
2) When both the heads fall off the right hand side of the input tape and the machine is in a non final state.
3) If the machine comes to a halt (i.e.there are no transition rules that can be applied for that particular state in which the machine is) before the heads fall off the right hand side of the input tape.

Subclasses of Two-Way Non-Deterministic Watson-Crick Automata (2AWK)
Depending on the type of states and transition rules there are four types or subclasses of two-way Watson-Crick Automata similar to Watson Crick automata.
; Q F  3) simple (2SWK): If at each step the automaton reads either from the upper strand or from the lower strand, i.e.
for any transition rule either Many combinations of these classes can also be obtained such as all-final simple two-way WK automata (2FSWK), all final 1 limited two-way WK automata (21FWK), stateless 1 limited two-way WK automata (21NWK) etc.
Theorem 4: Simple and 1 limited two-way Watson Crick automata accept the same family of languages as the family of languages accepted by two-way Watson Crick automata with arbitrary transition rules.
The proof of theorem is similar to the proof done in [4] for Theorem 1. Let  be a non-deterministic two-way Watson Crick automaton.We introduce a 1 limited two-way Watson Crick automaton For each transition rule t of the form   in  where 1  We introduce new rules in   of the form ,0 All the new states are introduced in Q' along with states in Q.From the construction of M' which is obtained from M it is obvious that both M' and M recognize the same language.So 2AWK are subset of 21WK and from the definition of 21WK and AWK we know that 21WK are subset of 2AWK.So 2AWK and 21WK are equivalent i.e. they accept the same family of languages.A similar proof can also be established for 2SWK.Therefore we can say, 2AWK = 2SWK = 21WK.
The transition rules of M are modified as follows to form the transition rules of M  .
q is a final state.In this case the transition rules are kept same in M  .

case : For transition
, where q is a non final state.In this case the transition rules of M are modified as follows for M  .
For each transition ru M belonging to class 2 where q is a non final state, where and  q is a non final state.In this case the transition rules of M are modified as follows for M  .
For each transition rule in M belonging to class 3 where q is a non final state, q denotes that the head on the upper strand has go past the right end marker $ in the original machine ne M on application of the above transition rule.
Only rules having λ on the upper strand are applied to $ u q because in the actual machine M if the above s of class 3 are applied then the u er head would have gone past the right end of the tape.So only rules rule pp having λ on the upper head can be applied to the machine M .As M  replicates M similar thing is done in M  T too.hus, all the transition hat can be applied to rules t q in M with  on the upper strand and 2 1 2 w a a   and $ n  in the lower strand can also Finally for rules with  on the upper strand and and th transition rules from ere are no $ .

ul q
Thes les ensure that when M reaches the end of string on a non final state then the M  goes to $ .
ul q and M  does not accept the string as re is no tra tion $ .
ul q i.e. the above stated rules ensure the heads do no ll off the right end of the tape for be a two-way 1 limited non-deter istic Watson ick automaton.We introduce an all final 1 limited two-way Watson Crick automaton  where 1 2 1 w w  falls under one of the four classes.The classes are derules of the form fined as follows: Class 1: Transition  case 1: For transitio $, q is a final state.In this case the transition rules are kept same in M  .
case 2: For transition , where $, ,0 q is a non final state.In this case the transition rules of M are modified as follows for M  .
For each transition rule $, , 0 in M belonging to class 2 where q is a non final state, Only rules having  on the upper strand can be applied to $ u q (for reasons similar to reasons stated in proof of T orem 5).Thus, all the transition rules that can be applied to q he  with  on the upper strand and 2 $ w  in the lowe trand ar applied to $ u q .For rules r s e having  on the upper strand and 2 $ w  the lower strand w ere the transition goes to a fi tate are applied to $ u q in l s h na  .Finally for rules with  on the upper strand and 2 $ w  in the lower strand where the transition goes to inal state in a non f M , the rules of the form , 0 , where . These rules ensure that when M reaches the end of the string on a non final state, M  does not accept the string as there are no transit from state $ .

WK Automata
, .w w V  We introduce a two-way non-deterministic Watson Crick automaton  where is a set of V he ing and t end marker respective s, the word w to be recognized is provided as an input to the automaton in the form # $. w is the co tate and q f is a final state.

mplementarity relation and 0
q is the initial s   is the fi number of transition rules of the form 1) For ea nite ch rule 3) For each state .
From the construction of .
M  it is evident that all that will be accepted by M wil accepted by l be M  .Theorem 8: One-Way Two headed finite to au mata ar is theorem is in [4].

 e equivalent to AWK An informal proof of th
respectively, that i are the beg d inning and the en marker s, the word w to be recognized is provided as an input to the automaton in the form # $. w Q is a set of states,   , , , , , Q q q q q q   is th the e identity complementarity relatio initial state and n and 0 q is is the set of final states.The transition ru M are as follow Theorem 9: One-way finite automata with 2 heads ca erful than AWK i.e we know that AWK is equivale epted by WK au hine (LBA) can simulate th 3 4 $, $, nnot accept the mirror language.
The above theorem is stated in [11].Theorem 10: 2AWK are more pow .AWK  2AWK.Proof.Fr Theorem 8 om nt to 1-way two headed finite automata and from Theorem 9 we know that 1-way two headed finite automata cannot recognize the mirror language.Thus AWK cannot recognize the mirror language.But in Example 1 we have shown that two-way AWK can accept the mirror language and in theorem 7 we have shown that AWK  2AWK i.e. 2AWK accepts all the family of languag which are accepted by AWK.Moreover it also accepts the mirror language which AWK cannot accept.Thus 2AWK accepts at least one language more than AWK.Hence we conclude that the accepting power of two-way AWK is more than AWK.Mathematically AWK  2AWK, i.e. the subset relation is proper.

Enumerable (RE) Languages in Terms of 2AWK Automata
this section we discuss 2AWK in th languages.We show each language in the family of RE is the image of a gsm mapping of a language in 2 AWK.
Theorem 12: TS V  AWK(ctrl) The proof of this theorem is in [4].
Theorem 13: For each recursively anage L there is a gsm g L such that L = g L (2AWK (ctrl)).

Conclusion
iscuss about the power of a variant o

*
denotes the transitive and reflexive closure of .
state at the same time both the heads fall off the right hand side of the double stranded input tape.

Theorem 5 :
All final two-way Watson Crick automata accept the same family of languages as the family of languages accepted by two-way Watson Crick automata with arbitrary transition rules.-way non-deterministic Watson Crick automaton.We introduce an all final two-way Watson Crick automaton Transition rules of M which fall in class 1 and class 5 are kept same in M  .For transition ru o les f M which belong to class 2

M
 .Rules having  on the upper strand and

Theorem 6 :
heads off M fall off the right end and the state to which M goes is non final.M  that M  accepts the same family of languages as All final 1 ick automata accept the same family of languages as the family of languages accepted by 1 limited twoway Watson Crick automata with arbitrary transition rules.Let

Class 4 :M
Transition rules of the form or transition rules of the form ,are modified as follows to form the transition rules of M  .Transition rules of M which fall in class 1 and class 4 are kept same in M  .For transition ru o les f M which belong to class 2 head on the are introduced in upper strand has g e the right end marker $ in the original machine on M .

7 .
way as class 2. ced in It is obvious from the transition rules introdu M  that M  accepts the same family of languages as M .Thus, 21FWK = 21WK.1 limited two-way Watson C Corollary 3: All final rick automata accept the same family of languages as the family of languages accepted by arbitrary two-way Watson Crick automata with arbitrary transition rules.Proof: From Theorem 4 we know 2AWK = 21WK an that 2A Power of Two-Way Non-Deterministic In t show that AWK are subset of rministic Watson Crick au d from Theorem 6 we obtain 21FWK = 21WK.Thus combining both the results we get 21FWK = AWK.Thus from the above Theorems we can state WK = 21FWK = 21WK = 2SWK = 2FSWK = 2FWK.

Theorem 11 ::
Family of languages acc es tomata is context sensitive.A linear bounded Turing mac e actions of two-way Watson Crick automaton.As the language accepted by LBA is context sensitive so the We have already shown in Theorem 7 AWK family of languages accepted by two-way Watson Crick automaton is also context sensitive.
These rules ensure that the reading heads do not go past the input word on the left side or the heads do not move when it reads empty word.Moreover once a head goes past the right end of the tape it cannot comeback.