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.
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 WatsonCrick 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.
2. Basic Terminology for Watson-Crick Automata
V is a finite alphabet. V* denotes the set of all finite words over V, including the empty wordFor denotes the length of w. Let and be two words and if there is some word, such that, 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:
extended to s: by and for is called a morphism. If for each, then h is a λ free morphism.
A morphism h: is called a coding iffor each and a weak coding iffor each If is the morphism defined by for, and otherwise, then we say that h is a projection (associated with) and we denote it by prV1.
For we define their shuffle by
A generalized sequential machine (gsm) is a sequential transducer. Such a device is a systemwhere Q is the set of states, are the alphabets(input and output alphabets) of the automaton, is the initial state, is the set of final states and is the transition mapping.
The definitions of morphism, gsm and shuffle are stated in [1].
A Watson-Crick automaton is a 6-tuple of the form where is an alphabet set, is a set of states, is the complementarity relation similar to Watson Crick complementarity relation and q0 is the initial state and is the set of final states. The function contains a finite number of transition rules of the form, which denotes that the machine in state q parses w1 in upper strand and w2 in lower strand and goes to state where. is different from. is just a pair of strings written in that form instead ofwhereas in the two strands are of same length i.e. and the corresponding symbols in two strands are complementarity in the sense of relation ρ.
and
.
A transition in a Watson-Crick finite automaton can be defined as follows:
For, such that
and
iff there is transition rule in.
denotes the transitive and reflexive closure of.The language accepted by Watson-Crick Automata is
For a Watson-Crick Automaton M, with inputwhere w1 is any string in V* and where q0 is the initial state and qf is a final state. Then is a computation in M denoted by.
Another important language associated with WatsonCrick automaton is defined taking into consideration the transitions and not the language recognised.
For a Watson-Crick Automaton consider a labelingof rules in with elements in a set Lab. For computationdenoted by 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].
3. 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. Watson-Crick Automaton is
1) stateless (NWK): If it has only one state, i.e.
2) all-final (FWK): If all the states are final, i.e.
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.
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].
Theorem 2: Non-deterministic Watson-Crick automata are equivalent with non-deterministic simple Watson-Crick automata.
Corollary 1: Non-deterministic Watson-Crick automata are equivalent with non-deterministic 1-limited Watson-Crick automata.
The proof of Theorem 2 and Corollary 1 are given in [4].
4. Twin-Shuffle Language
Consider an alphabet V and its barred variant,. The language
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: defined by
Clearly the equality is
Theorem 3: Each recursively enumerable language can be written in the form , where is an alphabet and R is a regular language.
In this representation, the language TSV depends on the language L. This can be avoided in the following way:
Let a coding be for instance,, where is the symbol of in a specified ordering. The language is regular. A generalized sequential machine can simulate the intersection with a regular language, the projection as well as the decoding of elements in Thus we obtain:
Corollary 2: For each recursively enumerable language there is a gsm gL 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 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, and and is the complementarity relation and is the initial state and 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 in upper strand in dir1 direction and in lower strand in dir2 direction and goes to state wherewhere 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, whereand with restrictions that when the correspondingand when the corresponding. Moreover, there cannot be transition rules having the form wherewhere and or where and or both. 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.
Accepting conditions
is accepted by if, starting in state (initial state) with and on the double stranded input tape and the two heads at the left end of and respectively, eventually enters a final state at the same time both the heads fall off the right hand side of the double stranded input tape.
The word is rejected if one of the following 3 conditions occurs:
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.
i.e. mathematically
6. 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.
2-way Watson-Crick Automaton
is
1) stateless (2NWK): If it has only one state, i.e.
2) all-final (2FWK): If all the states are final, i.e.
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
4) 1-limlited (21WK): If for any transition rule, we have
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 automatonFor each transition rule t of the form in where where and where and the condition is imposed because rules with is already in the 1-limited form and no further modification is required for them. We introduce new rules in of the form
.
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.
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.
Let be a two-way non-deterministic Watson Crick automaton. We introduce an all final two-way Watson Crick automatonEach transition rule t of the form in δ where whereand where falls under one of the five classes. The classes are defined as follows:
Class 1: Transition rules of the form
in where whereand where and and i.e. w1 and w2 do not have $ at their ends.
Class 2: Transition rules of the form
in where where and
where, and i.e. and both have $ at their ends.
Class 3: Transition rules of the form
in where where
and where and i.e. has at its end and does not have at its end.
Class 4: Transition rules of the form
in where where andwhere and i.e. does not have at its end and has at its end.
Class 5: Either transition rules of the form
in or transition rules of the form
in or transition rules of the form in.
The transition rules of are modified as follows to form the transition rules of.
Transition rules of which fall in class 1 and class 5 are kept same in.
For transition rules of which belong to class 2 two instances can occur;
case 1: For transition, where is a final state. In this case the transition rules are kept same in.
case 2: For transition, where is a non final state. In this case the transition rules of are modified as follows for.
For each transition rule in belonging to class 2 where is a non final state,where and are introduced in and there is no transition from in. These new rules in ensure that if the heads go off the right end of the tape in when is in a non final state then would go to state and would not accept the string as there is no transition from i.e. the above stated rules ensure the heads do not fall off the right end of the tape for when does not accept the word. As is all final if the heads go off the right end of the tape it will accept the given string.
For transition rules of which belong to class 3 the following modifications are needed. Class 3 also has two instances similar to class 2.
case 1: For transition, where is a final state. In this case the transition rules are kept same in.
case 2: For transition, where is a non final state. In this case the transition rules of are modified as follows for.
For each transition rule in belonging to class 3 where is a non final state,where is introduced inwhere denotes that the head on the upper strand has gone past the right end marker in the original machine on application of the above transition rule.
Only rules having λ on the upper strand are applied to because in the actual machine if the above rules of class 3 are applied then the upper head would have gone past the right end of the tape. So only rules having λ on the upper head can be applied to the machine. As replicates similar thing is done in too.
Thus, all the transition rules that can be applied to in with on the upper strand and and in the lower strand can also be applied to in. Rules having on the upper strand and and in the lower strand where the transition goes to a final state are applied to. Finally for rules with on the upper strand andand in the lower strand where the transition goes to a non final state, the rules of the form where are introduced in and there are no transition rules from These rules ensure that when M reaches the end of the string on a non final state then goes to and does not accept the string as there is no transition from i.e. the above stated rules ensure the heads do not fall off the right end of the tape for when heads off M fall off the right end and the state to which M goes is non final.
Class 4 rules are handled in a similar way to class 3 rules.
It is obvious from the transition rules introduced in that accepts the same family of languages as M.
Thus, 2FWK = 2AWK.
Theorem 6: All final 1 limited two-way Watson Crick 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 be a two-way 1 limited non-deterministic Watson Crick automaton. We introduce an all final 1 limited two-way Watson Crick automaton Each transition rule t of the form in where falls under one of the four classes. The classes are defined as follows:
Class 1: Transition rules of the form
in where
Class 2: Transition rules of the form
in.
Class 3: Transition rules of the form
in.
Class 4: Transition rules of the form
in or transition rules of the form in
The transition rules of are modified as follows to form the transition rules of.
Transition rules of which fall in class 1 and class 4 are kept same in.
For transition rules of which belong to class 2 have two instances.
case 1: For transition, where is a final state. In this case the transition rules are kept same in.
case 2: For transition, where is a non final state. In this case the transition rules of are modified as follows for.
For each transition rule in belonging to class 2 where is a non final state,and where are introduced in. denotes that the head on the upper strand has gone past the right end marker in the original machine.
Only rules having on the upper strand can be applied to (for reasons similar to reasons stated in proof of Theorem 5). Thus, all the transition rules that can be applied to with on the upper strand and in the lower strand are applied to. For rules having on the upper strand and in the lower strand where the transition goes to a final state are applied to. Finally for rules with on the upper strand and in the lower strand where the transition goes to a non final state in, the rules of the formand, whereare introduced in. These rules ensure that when reaches the end of the string on a non final state, does not accept the string as there are no transitions from state
Class 3 is handled in a similar way as class 2.
It is obvious from the transition rules introduced in that accepts the same family of languages as.
Thus, 21FWK = 21WK.
Corollary 3: All final 1 limited two-way Watson Crick automata accept the same family of languages as the family of languages accepted by arbitrary twoway Watson Crick automata with arbitrary transition rules.
Proof: From Theorem 4 we know 2AWK = 21WK and from Theorem 6 we obtain 21FWK = 21WK. Thus combining both the results we get 21FWK = AWK.
Thus from the above Theorems we can state that 2AWK = 21FWK = 21WK = 2SWK = 2FSWK = 2FWK.
7. Power of Two-Way Non-Deterministic WK Automata
In this section we first show that AWK are subset of 2AWK. Then we further show that this subset relation is proper i.e. 2AWK are more powerful than AWK.
Theorem 7: AWK 2AWK.
The theorem says that non-deterministic Watson Crick automata are subset of two-way non-deterministic Watson Crick automata.
Proof:
Let be a non-deterministic Watson Crick automaton where is a set of alphabet, is a set of states, is the complementarity relation and is the initial state and is the set of final states. is the finite number of transition rules of the form, where
We introduce a two-way non-deterministic Watson Crick automaton
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 the complementarity relation and is the initial state and qf is a final state. is the finite number of transition rules of the form
1) For each rule in introduce
in.
2).
3) For each state in introduce
in.
From the construction of it is evident that all that will be accepted by will be accepted by.
Theorem 8: One-Way Two headed finite automata are equivalent to AWK
An informal proof of this theorem is in [4].
Example 1
Let be a 2AWK where 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 identity complementarity relation and is the initial state and is the set of final states. is the finite number of transition rules. In this example the mirror language andwhere denotes the reverse of is accepted using two-way Watson Crick automaton.
The transition rules of are as follow
.
Theorem 9: One-way finite automata with 2 heads cannot accept the mirror language.
The above theorem is stated in [11].
Theorem 10: 2AWK are more powerful than AWK i.e. AWK 2AWK.
Proof. From Theorem 8 we know that AWK is equivalent 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 languages 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.
Theorem 11: Family of languages accepted by WK automata is context sensitive.
A linear bounded Turing machine (LBA) can simulate the actions of two-way Watson Crick automaton. As the language accepted by LBA is context sensitive so the family of languages accepted by two-way Watson Crick automaton is also context sensitive.
8. Characterization of Recursively Enumerable (RE) Languages in Terms of 2AWK Automata
In this section we discuss 2AWK in the light of the RE 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: TSV ( AWK(ctrl)
The proof of this theorem is in [4].
Theorem 13: For each recursively enumerable language L there is a gsm gL such that L = gL (2AWK (ctrl)).
Proof: We have already shown in Theorem 7 AWK 2AWK and it is stated in [4] that TSV Î AWK(ctrl) and from corollary 2 we know that each language in the family of RE is the image of a gsm mapping of a language in TS{0,1}. As TSV Î AWK(ctrl) and AWK Î 2AWK, so we can state, each language in the family of RE is the image of a gsm mapping of a language in 2AWK(ctrl).
9. Conclusion
In this paper, we discuss about the power of a variant of non-deterministic Watson Crick automata known as 2AWK. We describe their structure and accepting conditions. We introduce different subclasses of 2AWK similar to AWK and show the equivalence of some of those subclasses. We further establish the fact that 2AWK are more powerful than AWK. Based on the relation between AWK and 2AWK we show that a gsm mapping of 2AWK results in the generation of each language in the family of the recursively enumerable languages.
2AWK: two way non-deterministic Watson-Crick automata.
2NWK: two way stateless non-deterministic WatsonCrick automata.
2FWK: two way all final non-deterministic WatsonCrick automata.
2SWK: two way simple non-deterministic WatsonCrick automata.
21WK: two way 1-limited non-deterministic WatsonCrick automata.
TSV: twin-shuffle language.
RE: recursive enumerable.
gsm: generalized sequential machine.
ReferencesL. C. S. Calude and G. Paun, “Computing with Cells and Atoms: An Introduction to Quantum, DNA and Membrane Computing,” Taylor & Francis Publishers, London, 2001.L. M. Adleman, “Molecular Computation of Solutions to Combinatorial Problems,” Science, New Series, Vol. 226, No. 5187, 1994, pp. 1021-1024.
http://dx.doi.org/10.1126/science.7973651R. Freund, G. Paun, G. Rozenberg and A. Saloma, “A, Watson-Crick Finite Automata,” Proceedings of the 3rd DIMACS Workshop on DNA Based Computers, Philadelphia, 1997, pp. 297-328.G. Paun, G. Rozenberg and A. Salomaa, “DNA Computing: New Computing Paradigms,” Springer-Verlag, Berlin, 1998.E. Czeizler, E. Czeizler, L. Kari and K Salomaa, “WatsonCrick Automata: Determinism and State Complexity,” Proceeding of: 10th International Workshop on Descriptional Complexity of Formal Systems, DCFS, 16-18 July 2008, pp. 121-133.E. Czeizler, E. Czeizler, L. Kari and K. Salomaa, “On the Descriptional Complexity of Watson-Crick Automata,” Theoretical Computer Science, Vol. 410, No. 35, 2009, pp. 3250-3260.
http://dx.doi.org/10.1016/j.tcs.2009.05.001E. Czeizler, E. Czeizler,” Parallel Communicating Watson-Crick Automata Systems,” In: Z. Fulop Z. Esik (Ed.), Proceedings of 11th International Conference, AFL 2005, 2005, pp. 83-96.E. Czeizler, “On the Power of Parallel Communicating Watson-Crick Automata Systems,” Theoretical Computer Science, Vol. 358, No. 1, 2006, pp. 142-147.E. Czeizler, “A Short Survey on Watson-Crick Automata,” Bulletin of the EATCS, Vol. 88, 2006, pp. 104-119.D. Kuske and P. Weigel, “The Role of the Complementarity Relation in Watson-Crick Automata and Sticker Systems,” Developments in Language Theory, Vol. 3340, Lecture Notes in Computer Science, Springer, Berlin, 2004, pp. 272-283.M. Holzer, M. Kutrib and A. Malcher, “Multi-Head Finite Automata: Characterizations, Concepts and Open Problems,” Proceedings International Workshop on the Complexity of Simple Programs, Cork, 6-7 December 2008, pp. 93-107.
For 
denotes the length of w. Let 
and 
be two words and if there is some word
, such that
, 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.
extended to s: 
by 
and 
for 
is called a morphism. If 
for each
, then h is a λ free morphism.
is called a coding if
for each 
and a weak coding if
for each 
If 
is the morphism defined by 
for
, and 
otherwise, then we say that h is a projection (associated with
) and we denote it by prV1.
we define their shuffle by
where Q is the set of states, 
are the alphabets(input and output alphabets) of the automaton, 
is the initial state, 
is the set of final states and 
is the transition mapping.
where 
is an alphabet set, 
is a set of states, 
is the complementarity relation similar to Watson Crick complementarity relation and q0 is the initial state and 
is the set of final states. The function 
contains a finite number of transition rules of the form
, which denotes that the machine in state q parses w1 in upper strand and w2 in lower strand and goes to state 
where
. 
is different from
. 
is just a pair of strings written in that form instead of
whereas in 
the two strands are of same length i.e. 
and the corresponding symbols in two strands are complementarity in the sense of relation ρ.
and
.
such that
and 
iff there is transition rule 
in
.
denotes the transitive and reflexive closure of
.The language accepted by Watson-Crick Automata is
where w1 is any string in V* and 
where q0 is the initial state and qf is a final state. Then 
is a computation in M denoted by
.
consider a labeling
of rules in 
with elements in a set Lab. For computation
denoted by
the control word of
, that is the sequence of labels of transition rules used in
. In this way the language is obtained.
is stated in [
is
, we have
.
. The language
. (For a string 
denotes the string obtained by replacing each symbol in x with its barred variant).
defined by
can be written in the form 
, where 
is an alphabet and R is a regular language.
for instance,
, where 
is the 
symbol of 
in a specified ordering. The language 
is regular. A generalized sequential machine 
can simulate the intersection with a regular language, the projection 
as well as the decoding of elements in 
Thus we obtain:
there is a gsm gL such that
Proofs of Theorem 3 and Corollary 2 are in [
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, 
and 
and 
is the complementarity relation and 
is the initial state and 
is the set of final states. 
is the finite number of transition rules;
, which denotes that the machine in state q parses 
in upper strand in dir1 direction and 
in lower strand in dir2 direction and goes to state 
where
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.
, where
and 
with restrictions that when 
the corresponding
and when 
the corresponding
. Moreover, there cannot be transition rules having the form 
where
where 
and 
or 
where 
and 
or both. 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.
is accepted by 
if, starting in state 
(initial state) with 
and 
on the double stranded input tape and the two heads at the left end of 
and 
respectively, 
eventually enters a final state at the same time both the heads fall off the right hand side of the double stranded input tape.
is rejected if one of the following 3 conditions occurs:
is
either
, we have 
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 
where 
and 
where 
and 
the condition 
is imposed because rules with 
is already in the 1-limited form and no further modification is required for them. We introduce new rules in 
of the form
.
be a two-way non-deterministic Watson Crick automaton. We introduce an all final two-way Watson Crick automaton
Each transition rule t of the form 
in δ where 
where
and 
where 
falls under one of the five classes. The classes are defined as follows:
in 
where 
where
and 
where 
and 
and 
i.e. w1 and w2 do not have $ at their ends.
in 
where 
where 
and
where
, and 
i.e. 
and 
both have $ at their ends.
in 
where 
where
and 
where 
and 
i.e. 
has 
at its end and 
does not have 
at its end.
in 
where 
where 
and
where 
and 
i.e. 
does not have 
at its end and 
has 
at its end.
in 
or transition rules of the form
in 
or transition rules of the form 
in
.
are modified as follows to form the transition rules of
.
which fall in class 1 and class 5 are kept same in
.
which belong to class 2 two instances can occur;
, where 
is a final state. In this case the transition rules are kept same in
.
, where 
is a non final state. In this case the transition rules of 
are modified as follows for
.
in 
belonging to class 2 where 
is a non final state,
where 
and 
are introduced in 
and there is no transition from 
in
. These new rules in 
ensure that if the heads go off the right end of the tape in 
when 
is in a non final state then 
would go to state 
and would not accept the string as there is no transition from 
i.e. the above stated rules ensure the heads do not fall off the right end of the tape for 
when 
does not accept the word. As 
is all final if the heads go off the right end of the tape it will accept the given string.
which belong to class 3 the following modifications are needed. Class 3 also has two instances similar to class 2.
, where 
is a final state. In this case the transition rules are kept same in
.
, where 
is a non final state. In this case the transition rules of 
are modified as follows for
.
in 
belonging to class 3 where 
is a non final state,
where 
is introduced in
where 
denotes that the head on the upper strand has gone past the right end marker 
in the original machine 
on application of the above transition rule.
because in the actual machine 
if the above rules of class 3 are applied then the upper head would have gone past the right end of the tape. So only rules having λ on the upper head can be applied to the machine
. As 
replicates 
similar thing is done in 
too.
in 
with 
on the upper strand and 
and 
in the lower strand can also be applied to 
in
. Rules having 
on the upper strand and 
and 
in the lower strand where the transition goes to a final state are applied to
. Finally for rules with 
on the upper strand and
and 
in the lower strand where the transition goes to a non final state, the rules of the form 
where 
are introduced in 
and there are no transition rules from 
These rules ensure that when M reaches the end of the string on a non final state then 
goes to 
and 
does not accept the string as there is no transition from 
i.e. the above stated rules ensure the heads do not fall off the right end of the tape for 
when heads off M fall off the right end and the state to which M goes is non final.
that 
accepts the same family of languages as M.
be a two-way 1 limited non-deterministic Watson Crick automaton. We introduce an all final 1 limited two-way Watson Crick automaton 
Each transition rule t of the form 
in 
where 
falls under one of the four classes. The classes are defined as follows:
in 
where 
in
.
in
.
in 
or transition rules of the form 
in 
are modified as follows to form the transition rules of
.
which fall in class 1 and class 4 are kept same in
.
which belong to class 2 have two instances.
, where 
is a final state. In this case the transition rules are kept same in
.
, where 
is a non final state. In this case the transition rules of 
are modified as follows for
.
in 
belonging to class 2 where 
is a non final state,
and 
where 
are introduced in
. 
denotes that the head on the upper strand has gone past the right end marker 
in the original machine
.
on the upper strand can be applied to 
(for reasons similar to reasons stated in proof of Theorem 5). Thus, all the transition rules that can be applied to 
with 
on the upper strand and 
in the lower strand are applied to
. For rules having 
on the upper strand and 
in the lower strand where the transition goes to a final state are applied to
. Finally for rules with 
on the upper strand and 
in the lower strand where the transition goes to a non final state in
, the rules of the form
and
, where
are introduced in
. These rules ensure that when 
reaches the end of the string on a non final state, 
does not accept the string as there are no transitions from state 
that 
accepts the same family of languages as
.
2AWK.
be a non-deterministic Watson Crick automaton where 
is a set of alphabet, 
is a set of states, 
is the complementarity relation and 
is the initial state and 
is the set of final states. 
is the finite number of transition rules of the form
, where 
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 the complementarity relation and 
is the initial state and qf is a final state. 
is the finite number of transition rules of the form
in 
introduce
in
.
.
in 
introduce
in
.
it is evident that all that will be accepted by 
will be accepted by
.
be a 2AWK where 
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 identity complementarity relation and 
is the initial state and 
is the set of final states. 
is the finite number of transition rules. In this example the mirror language 
and
where 
denotes the reverse of 
is accepted using two-way Watson Crick automaton.
are as follow
.
2AWK.
2AWK, i.e. the subset relation is proper.
( AWK(ctrl)