Watson-Crick ω-automata. Elena Petre. Turku Centre for Computer Science. TUCS Technical Reports

Transcription

1 Watson-Crick ω-automata Elena Petre Turku Centre for Computer Science TUCS Technical Reports No 475, December 2002

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3 Watson-Crick ω-automata Elena Petre Department of Mathematics, University of Turku and Turku Center for Computer Science Lemminkäisenkatu 14, FIN Turku, Finland Turku Centre for Computer Science TUCS Technical Report No 475 December 2002 ISBN ISSN

4 Abstract Watson-Crick ω-automata are finite automata working with infinite doublestranded words, where the two strands relate to each other through a complementary relation inspired by the DNA complementarity. There are several natural possibilities to define such automata, in particular to define the acceptance condition for such infinite double-stranded words. We present here several results on the equivalence of Büchi, Rabin, Street, and Mullerlike acceptance conditions. We prove that these acceptance conditions are equivalent for nondeterministic Watson-Crick ω-automata. For deterministic automata, we prove that the four conditions are equivalent for non-empty stateless automata, while the general case remains open. We also investigate the relationship between the classes of languages accepted by several types of Watson-Crick ω-automata. Keywords: Watson-Crick, finite automata, ω-automata, decidability TUCS Laboratory Discrete Mathematics for Information Technology

5 1 Introduction Adleman s celebrated experiment [1] on the potential use of biomolecules for computations gave rise and motivation to a large number of studies dealing with DNA computing. Among many others, several models based on Automata Theory and Formal Languages were introduced in this area, see [10]. An important type of automata, inspired by the work on DNA computing is the Watson-Crick finite automata introduced in [4] and then investigated in [6], [8], [9], [12], see also [3] and [5] for surveys. They work on double stranded sequences of letters related by a complementary relation - such a data structure is called a Watson-Crick tape. The automata scan separately each of the two strands, in a correlated manner. They may have a finite number of states controlling the moves and they may have an auxiliary memory which is also a Watson-Crick tape, used in a FIFO-like manner. Combining these possibilities, several types of Watson-Crick automata can be obtained, see [10] (Chapter 5. The notion of ω-automata, i.e., automata working on infinite words, was introduced in the sixties by Büchi [2], McNaughton [7], and Rabin [11]. They made some connections between Automata Theory and other fields, such as Logics and set-theoretical Topology; for some surveys see [13], [14], [15]. In this paper we combine for the first time the notion of Watson-Crick automata and that of ω-automata, defining the notion of Watson-Crick ω- automata. These are automata working on double-stranded words in the same way usual Watson-Crick automata do, except that the input words are now infinite. We prove that several results known for either Watson- Crick automata, or ω-automata hold also for Watson-Crick ω-automata. In particular, this gives a new useful insight on those classical results. In Section 3 we prove that the four acceptance conditions for ω-automata, those of Büchi, Muller, Rabin, and Streett, are equivalent for nondeterministic Watson-Crick ω-automata. In the deterministic case, they are not equivalent in general, except for a special type of automata: the non-empty stateless Watson-Crick ω-automata. We then prove that some relations between the families of languages recognized by classical Watson-Crick finite automata are also valid when working with double-stranded ω-words. In Section 4 we prove that the language accepted by an automaton does not depend on the number of characters read at each step. By modifying the transitions of the automaton, we can recognize the same language regardless of whether we read from the tapes just one character, or a finite word. In Section 5 we extend an undecidability result known for usual Watson-Crick automata, see [6], regarding the position of the two read-only heads of the automaton during a computation. 1

6 2 Definitions We recall some notions of finite automata, Watson-Crick finite automata and ω-automata. For further details we refer to [16], [10] (Chapter 5, and [14]. Definition 1 A (nondeterministic finite automaton is a tuple M = (K,V,s 0,F,δ, where K is the set of states, V is the alphabet of the automaton, s 0 K is the initial state, F K is the set of final states, δ : K V 2 K is the transition mapping. If card(δ(s,a 1 for all s K,a V, then we say that the automaton is deterministic. A relation is defined in the following way on the set K V : for s,s K, a V, x V, we write (s,ax (s,x if s δ(s,a; by definition (s,λ (s,λ. If is the reflexive and transitive closure of the relation, then the language of the strings recognized by the automaton M is defined by: L(M = {x V (s 0,x (s,λ, s F }. We say that an automaton is non-empty if the language recognized by it is non-empty. It is known that both deterministic and nondeterministic finite automata characterize the family of regular languages. The power of finite automata is not increased if we allow λ-transitions or when the input string is scanned in a two-way manner, going along it to the right or to the left, without changing its symbols, see [16]. Let us consider now some automata working with infinite words. An acceptance condition for these automata restricts the occurrences of states in a run under consideration. Usually, it refers to those states which occur infinitely often in the considered run and is fixed by an acceptance component of the ω-automaton. We define an ω-language over an alphabet V as a set of infinite strings x = x 1 x 2... with x i V, i 1. We will use the notation V ω for the set of all ω-words over the alphabet V. Consider a finite automaton M = (K,V,s 0,.,δ without final states and an ω-sequence r = r 1 r 2... for which we denote: In(r = {q K there exists infinitely many i such that r i = q}. The most frequently used acceptance conditions are the following requirements on In(r: 2

7 Büchi condition: A set F K of final states is considered and the acceptance condition In(r F, requiring that some final state occurs infinitely often in the run r. Muller condition: A family F 2 K of final state sets is considered and the acceptance condition F F (In(r = F, requiring that the set of states assumed infinitely often in the run r forms a set in F. Rabin condition ( pairs condition : A sequence Ω of accepting pairs (E 1,F 1,..., (E n,f n with E i,f i K is considered and the acceptance condition n i=1 (In(r E i = In(r F i ; one requires that for some i, all states of E i are visited only finitely often in r, but some state of F i is visited infinitely often. Streett condition ( complemented pairs condition : A sequence Ω of pairs (E 1,F 1,...,(E n,f n, where E i,f i are subsets of K, is considered, and the acceptance condition n i=1 (In(r E i In(r F i = ; it represents a condition that can be read as for each i, if some state of F i is visited infinitely often, then some state of E i is visited infinitely often. Definition 2 An ω-automaton has the form A= (K,A,q 0,Acc, where K is a finite set of states, A is the input alphabet, q 0 is the initial state, K A K is the transition relation, and Acc is an acceptance component depending on the acceptance condition we use: For the Büchi acceptance condition, Acc is a set of states called final; For the Muller acceptance condition, Acc is a family F 2 K of final state sets; For the Rabin condition or Streett condition, Acc is a sequence of pairs (E 1,F 1,..., (E n,f n, where E i,f i K. A run of A on a given input ω-word α = α 0 α 1... with α i A is a sequence r = r 0 r 1... K ω such that r 0 = q 0 and (r i,α i,r i+1 for i 0. In deterministic automata the transition relation is replaced by the transition function δ : K A K, and a run has to satisfy r i+1 = δ(r i,α i for i 0. It is known that nondeterministic Büchi, Muller, Rabin and Streett automata all recognize the same class of ω-languages. A short proof of this result can be found in [15]. The Watson-Crick finite automata were originally introduced in [4]. They use a double-stranded tape, whose strands are separately scanned from left to 3

8 right, by read-only heads controlled by a common state; the symbols placed in corresponding cells of the two strands are linked by a complementarity relation. Definition 3 A Watson-Crick finite automaton is a tuple M = (V,ρ,K,s 0,F,δ, where V is the alphabet of the automaton, K is a set of final states, ρ V V is a symmetric relation (the complementarity relation, ( s 0 K is the initial V state, F K is the set of final states and δ : K V 2 K is a ( x mapping such that δ(s, only for finitely many triples (s,x,y y K V V. s We can( also write the transitions of M as rewriting rules of the ( form x1 s, such a rule having the same meaning as s x1 δ(s,. ( x1 x 2 x 2 If is the reflexive and transitive closure of the relation, then the language accepted by a Watson-Crick automaton is: ( ( ( x x x L(M = { V V xρy, s 0 s, s F }. y y y Definition 4 We say that a Watson-Crick automaton M = (V,ρ,K,s 0,F,δ is: stateless, if K = F = {s 0 }; all-final, if F = K; simple, if for all transitions: s or x 2 = λ; 1-limited, if for all transitions: s ( x1 x 2 ( x1 x 2 ( x1 x 2 x 2 s we have either x 1 = λ ( x1 x 2 s we have x 1 x 2 = 1. We can introduce now the Watson-Crick automata working on ω-words (infinite double-stranded sequences. The acceptance conditions are the same as in classic ω-automata; we use Büchi, Muller, Rabin, and Streett acceptance conditions. 4

9 Definition 5 A Watson-Crick ω-automaton is a tuple A= (V,ρ,K,s 0,F,δ, where V is the input alphabet, ρ V V is a symmetric relation, called the complementary relation( on V, K is a finite set, called the set of states, ( s 0 is V x the initial state, δ : K V 2 K is a mapping such that δ(s, y only for a finite number of triples (s,x,y K V V, called the transition mapping, and F is an acceptance component which depends on the acceptance condition used: For the Büchi acceptance condition, F consists of a set of states called final; For the Muller acceptance condition, F consists of a family F 2 K of final state sets; For the Rabin condition or Streett condition, F consists of a sequence of pairs (E 1,F 1,...,(E n,f n, where E i,f i K. ( V A run of A on a given ω-word α is a sequence r = r 0 r 1... K ω such that r 0 = s 0 ( and r i δ(r i 1,α i, for i 1, where α i is a subsequence V of α (i.e., β i,γ i such that α = β i α i γ i and α = α 1 α 2... So r is V the sequence of states automaton A passes through when scanning the input α. We can extend in the natural way Definition 4 to Watson-Crick ω-automata as follows. Definition 6 We say that a Watson-Crick ω-automaton M = (V,ρ,K,s 0,F,δ is: stateless, if K = {s 0 }; simple, if for all transitions: s or x 2 = λ; 1-limited, if for all transitions: s all-final if: ( x1 - F = K for Büchi automata; x 2 5 V ( x1 x 2 ( x1 x 2 ( x1 s we have either x 1 = λ x 2 s we have x 1 x 2 = 1;

10 - F= 2 K for Muller automata; - the acceptance component has only one pair (,K for Rabin automata; - the acceptance component has only one pair (K, for Streett automata. Depending on the acceptance component of the automata, we consider Watson-Crick Büchi, Watson-Crick Muller, Watson-Crick Rabin and Watson-Crick Streett automata respectively. A double stranded ω-language is called Büchi-, Muller-, Rabin-, Streett-recognizable if it consists of ω-words over the considered alphabet which are accepted by some Watson-Crick Büchi-, Muller-, Rabin-, Streett-automata, respectively. 3 The equivalence between acceptance conditions for Watson-Crick ω-automata We present here some cases when the four acceptance conditions for Watson-Crick ω- automata defined in Section 2 are equivalent, meaning that if a word is accepted by an automaton using one of these acceptance conditions, then it is accepted also by an equivalent automaton using one of the other conditions. The first result does not have a counterpart for usual ω-automata. Theorem 1 For a non-empty stateless Watson-Crick ω-automaton, the acceptance conditions Büchi, Muller, Rabin and Streett are equivalent. Proof:Let us consider A = (V,ρ, {s 0 },s 0,F,δ to be a non-empty stateless Watson-Crick ω-automaton. Büchi Muller: Assume first that the Büchi acceptance condition is used as the acceptance component of A. Consider an accepted word w and let r be an infinite run for this input. This means that In(r F. But F K = {s 0 } and In(r K, so In(r = F = {s 0 }. If we consider a Watson-Crick ω-automaton with Muller acceptance condition A M = (V,ρ, {s 0 },s 0, F,δ with F = {{s 0 }}, clearly L(A = L(A M. Assume now that A is a Watson-Crick Muller automaton. Since K = {s 0 }, we do not have so many choices for the family F of sets used with this condition of acceptance; F {, {s 0 }}. For accepted words we have infinite runs, r, such that In(r F, and In(r. So, a word is accepted by this automaton if and only if In(r = {s 0 }. If we consider the Watson-Crick ω- automaton with Büchi acceptance condition A B = (V,ρ, {s 0 },s 0,F B,δ, with F B = {s 0 }, clearly L(A = L(A B. 6

11 Consequently, Büchi and Muller acceptance conditions are equivalent. Büchi Rabin: If A is a Watson-Crick Büchi automaton with F = {s 0 }, then consider the Watson-Crick ω-automaton using Rabin acceptance condition A R = (V,ρ, {s 0 },s 0, (, {s 0 },δ. Clearly, L(A = L(A R. Assume that A is a Watson-Crick Rabin automaton, with the acceptance component consisting of the sequence of pairs (E 1,F 1,...,(E n,f n, E i,f i K. For any accepted word w and any infinite run r for w in A, there is 0 i n such that In(r E i = In(r F i. Thus, without loss of generality, we can assume that F 1 =... = F n = {s 0 } and E 1 =... = E n =, i.e., the whole sequence reduces to the pair (, {s 0 }. Consider now a Watson-Crick ω-automaton using Büchi acceptance condition defined as follows: A B = (V,ρ, {s 0 },s 0, {s 0 },δ. Clearly, L(A = L(A B. Thus, also Büchi and Rabin conditions are equivalent. Büchi Streett: If A is a Watson-Crick Büchi automaton with F = {s 0 }, then consider the Watson-Crick ω-automaton using Streett acceptance condition A S = (V,ρ, {s 0 },s 0, ({s 0 },,δ. Clearly, L(A = L(A S. Assume that A is a Watson-Crick Streett automaton, with the acceptance component consisting of the sequence of pairs (E 1,F 1,...,(E n,f n, E i,f i K = {s 0 }. This means that a word w is accepted if for the corresponding run r we have that for all 1 i n In(r E i In(r F i =. But, for an infinite run r on A we have that In(r = {s 0 }. So, we obtain that an ω-word w is accepted by A if for the corresponding run r we have for all 1 i n In(r = E i = {s 0 } or F i =. Consider now the Watson-Crick ω- automaton using Büchi acceptance condition A B = (V,ρ, {s 0 },s 0, {s 0 },δ. Clearly, L(A = L(A B. This means that Büchi and Streett conditions are equivalent. Thus, all the four acceptance conditions are equivalent for a stateless nonempty Watson-Crick ω-automaton. The next result demonstrates that this equivalence holds for all nondeterministic Watson-Crick ω-automata. This is similar to the results known for usual ω-automata, see [7] and [15]. Theorem 2 Nondeterministic Watson-Crick ω-automata using the acceptance conditions Büchi, Muller, Rabin and Streett recognize the same class of ω-languages. 7

12 Proof:Büchi Muller: Consider a Watson-Crick ω-automaton B= (V,ρ,K,s 0,F,δ using Büchi acceptance condition. This means that for an accepted word w and a run r for this input, we have In(r F. If we construct the family F of all sets containing at least one state from F, then we make sure that In(r appears as a proper set in this family. With this family of sets we obtain a Watson-Crick ω-automaton with Muller acceptance condition, equivalent with the initial automaton B. Rabin Muller: Let B be a Watson-Crick ω-automaton using Rabin acceptance condition. This means that for an accepted word w and a run r on this input, there exists i {1,...,n} such that In(r E i = In(r F i. For all such pairs (E i,f i with In(r E i = In(r F i we construct the family F of all sets which contain states from F i and do not contain states from E i. This way, we make sure that In(r is an element in this family. So, we obtain a Watson-Crick ω-automaton with Muller acceptance condition equivalent with the initial automaton B. Streett Muller: Let B be a Watson-Crick ω-automaton using Streett acceptance condition, which means that for an accepted word w and a run r on this input we have n i=1 (In(r E i In(r F i =. In this case, we construct a family F of all sets of states C such that if C contains states form F i then C contains states from E i, with 1 i n. With this family we obtain a Watson-Crick ω-automaton with Muller acceptance condition equivalent with the initial automaton B. Büchi Rabin: Let B = (V,ρ,K,s 0,F,δ be a Watson-Crick ω-automaton using Büchi acceptance condition. A word w is accepted by this automaton if for the corresponding run r we have In(r F. Consider now the following ω-automaton using Rabin acceptance condition B R = (V,ρ,K,s 0, (,F,δ. Clearly, L(B = L(B R. Büchi Streett: Let B = (V,ρ,K,s 0,F,δ be a Watson-Crick ω-automaton using Büchi acceptance condition, and define B S = (V,ρ,K,s 0, (F,K F, δ to be a Watson-Crick ω-automaton using Streett acceptance condition. Clearly L(B = L(B S. Muller Büchi: Let us consider now that B = (V,ρ,K,s 0, F,δ is a Watson-Crick ω-automaton which uses Muller acceptance condition, where F is a family of sets of final states and we will construct a Watson-Crick ω- automaton with Büchi acceptance condition equivalent with B. Let it be C = (V,ρ,K,s 0,F,δ. The automaton C has to guess in advance the set F F of states to be visited infinitely often, and also the point on its input from which onwards only states in F will be visited. From this point it suffices to check that the set of visited states is the set F. For every set F F we construct a branch in automaton C supposing this 8

13 F is the one with the property In(r = F. At the beginning, the transitions on this branch are the same with the ones in the initial automaton B. But, after some point, only states from F will be visited. To underline this fact, let us make a copy of each state in F. If we consider that F = {s 1,...,s n }, then let F be the set of all the copies created: F = {s 1,...,s n}. Consider now between these states some transitions similar to those existent between the states in F, i.e., if there is a transition from s i to s j, with 1 i,j n, then we put a similar transition from s i to s j. So, at some point with a λ-transition we enter a state in F and afterwards we visit only states from this set. This means that in the initial automaton we visit only states from the set F. To check the fact that all the states in F are visited infinitely often we need a further construction. Actually, we will check that all the states in F are visited infinitely often, this being equivalent with the fact that all the states in F are visited infinitely often. Let us suppose first that F has only two elements s 1 and s 2. This mean that F = {s 1,s 2}. Consider now a new state s 2 which will be used to construct an automaton with a final state, visited infinitely often, this being equivalent with the fact that in the initial automaton both s 1 and s 2 were visited infinitely often. The transitions from s 1 to s 2 are replaced by transitions from s 1 to s 2, and all the transitions from s 2 to any other states are duplicated in such a way that if there is a transition from s 2 to a state q then there is also a transition from s 2 to q. This new state is considered final; it can be reached from state s 1 and it has a behavior similar with state s 2. So, if in the initial automaton states s 1 and s 2 were visited infinitely often, then in this automaton state s 2 is visited infinitely often, and both automata accept the same language. If F has more than two elements we make an inductive construction: we consider the first two states of F and introduce a new one, then we consider this new state and the third state of F, etc. The last introduced state is declared final, and will be visited infinitely often if all the elements in F were visited infinitely often. The initial guess of set F F of states to be visited infinitely often is simulated by a λ-transition from the initial state s 0 to every one of these branches described previously. The final states set in automaton C contains the final states from all these branches. We prove now that automata B and C accept the same language. Let w be a word accepted by automaton B, and r the proper run for this input. So, there is a set F F such that In(r = F, meaning that after some point only states in this set F will be visited. Automaton C has to make a nondeterministic guess in the initial state and enters the correspondent branch for this set F. First it has a similar behavior with automaton B, 9

14 but then it has to guess a point from which onwards only states from F will be visited. At this point, with a λ-transition it enters a state from set F and moves on only with transitions between states from this set. The additional construction described before makes sure that the final state is visited infinitely often (because all the states in F are visited infinitely often. So, Büchi acceptance condition is verified, and thus w is accepted also by automaton C. Consider now a word w accepted by automaton C. This means that, from the initial state s 0 it moves to one of its branches and at some point it will enter a state from set F. From this point onwards it will visit only states from set F, which means that in automaton B only states from F will be visited. The word w is accepted by C, so the final state is visited infinitely often. Accordingly to the additional construction, this means that all the states from F are visited infinitely often or In(r = F. So, Muller acceptance condition is verified and thus w is accepted also by automaton B. Directly from Definition 6 we have the following result: Theorem 3 If B = (V,ρ,K,s 0,F,δ is a deterministic all-final Watson-Crick ω- automaton, and w is an ω-word such that there is an infinite run in B for this input, then w is accepted by B using any acceptance condition. Proof:Let r be an infinite run of B on w. If we use Büchi acceptance condition, then we have K = F, and because In(r K, we obtain that In(r F, i.e., w is accepted with Büchi acceptance condition. For the Muller acceptance condition, we have an acceptance component consisting of the family F = 2 K. Thus, clearly w is accepted with Muller acceptance condition. If B is an all-final Rabin automaton, then the acceptance component is the pair (,K. Because In(r K, it is obvious that w is accepted by this automaton. If B is an all-final Streett automaton, i.e., the acceptance component is the pair (K,, then w is also accepted by this automaton. 10

15 4 Relationships between the Watson-Crick families In this section we investigate the relationship between the families of languages accepted by various types of Watson-Crick ω-automata. All the results in this section are also valid for the classic Watson-Crick finite automata, and the proofs in that case can be found in Chapter 5 from [10]. Directly from Definition 6 we obtain the extension of a result in [10]: Lemma 4 The language recognized by a Watson-Crick ω-automaton which is either stateless, or all-final, or simple, or 1-limited, or stateless and simple, or stateless and 1-limited, or all-final and simple, or all-final and 1-limited using any of the acceptance conditions can be recognized by an arbitrary Watson-Crick ω-automaton using the same acceptance condition. Lemma 5 The language accepted by a nondeterministic arbitrary Watson-Crick ω- automaton using any of the acceptance conditions can be also accepted by a nondeterministic 1-limited Watson-Crick ω-automaton using the same acceptance condition. Proof:We have already proved that all the acceptance conditions are equivalent in the nondeterministic case, so we have to prove the assertion for one condition only. Let it be the Büchi acceptance condition. Consider an unrestricted Watson-Crick ω-automaton M = (V,ρ,K,s 0,F,δ and construct an 1-limited Watson-Crick ω-automaton M = (V,ρ,K,s 0,F,δ as follows. For each rule ( ( a1...a n a1...a n t : s s, b 1...b m b 1...b m in M, n,m 0, n + m 2 we introduce in M the transitions: ( a1 ( a1 s s t,1, λ λ ( ai+1 ( ai+1 s t,i s t,i+1, 1 i n 1, λ λ ( ( λ λ s t,n s b t,1, 1 11 b 1

16 s t,i ( λ b i+1 ( λ s t,m 1 b i+1 s t,i+1, 1 i m 2, ( ( λ λ b m b m s, where all states s t,i, s t,i are introduced in K, together with all states from K. If for a word w there is an infinite run in one automaton, then there is also an infinite run in the other automaton; instead of a single transition in M we have a linear sequence of transitions in M. Also all the states visited when running an input on M are also visited when running the same input on M. Automaton M is obviously a 1-limited Watson-Crick ω-automaton and we will prove that it accepts the same ω-language using Büchi acceptance condition. Let w be an ω-word accepted by the automaton M using Büchi acceptance condition, and r the corresponding run for this input. So, In(r F in M. In M, the corresponding run for w, say r, contains also some intermediary states. Thus, if a state s K is visited infinitely often in M, it will also be visited infinitely often in M. So, we have In(r In(r. But In(r F and thus, In(r F, which means that w is recognized by M using Büchi acceptance condition. If w is an ω-word accepted by M using Büchi acceptance condition, and r is a run for this input, then we have that In(r F. Because F K we have that at least one state from K, say s, is visited infinitely often in the run r. But, if r is the run associated for the input w in M, then it is obvious that we have {q K i, r(i = q} = {q K i, r (i = q}, because in a run in M we visit all the states visited in M for the same input and some other intermediary states from the set K K. So, the fact that s In(r F implies that s In(r F. This means that w is also accepted by M using Büchi acceptance condition. Using the same reasoning we can prove also the next result. Lemma 6 The language recognized by a nondeterministic simple Watson-Crick ω- automaton using any of the acceptance conditions can be also recognized by a nondeterministic 1-limited Watson-Crick ω-automaton using the same acceptance condition. Corollary 7 Nondeterministic 1-limited, simple and arbitrary Watson-Crick ω- automata recognize the same class of ω-languages. 12

17 5 The relative position of the two heads We now address a question dealt with for usual Watson-Crick automata in [6]. At the beginning of a computation, the two heads of a Watson-Crick ω- automaton are placed in front of the first symbols of each strand, and then they advance to the right. Related to the position of the two heads during a computation we can prove the following result. Theorem 8 Given a simple Watson-Crick ω-automaton A, it is undecidable whether or not there are computations in A where the two heads are ever placed in the same position, except for the start position. Proof:Consider an instance of Post Correspondence Problem (PCP over the alphabet {a,b}, x = (x 1,...,x n, y = (y 1,...,y n, n 1, x i,y i non-empty strings for all i, and construct the simple Watson-Crick ω-automaton: A = ({a,b,c}, {(a,a, (b,b, (c,c},k,s 0, {s },δ, with K = {s 0,s 1,...,s n,s a,s b,s c,s,s }, and the following rules: ( xi ( xi 1. s 0 s i for all 1 i n; λ λ ( ( λ λ 2. s i s 0 for all 1 i n; y i y i ( ( λ λ 3. s i y i c y i c ( c ( c 4. s s λ λ ; ( ( t t 5. s λ λ ( ( λ λ 6. s t t t s ; s t for all t {a,b,c}; s for all t {a,b,c}. We can reach the final state s only if there are two identical sequences x i1...x ik c and y i1...y ik c with 1 i j n for 1 j k, which means that x i1...x ik = y i1...y ik is a solution for PCP. After we read the first character c (by applying Rule 3, we enter state s and from here the only possible move is to read character c on the upper strad by apply Rule 4. At this moment the two read-only heads are in the same position because we have 13

18 the same characters on both upper and bottom strings (the complementarity relation ρ is the identity relation, and c cannot appear in the sequence x i1...x ik {a,b} +. After reaching state s we will read words of the form [ w w] with w {a,b,c} ω, by applying Rules 5 and 6. Every time s is visited, the two heads of the automaton A are in the same position. Since, PCP is not decidable, it follows that it is also not decidable if the two heads are in the same position at least once, except for the initial configuration. If there [ is a] solution for PCP, then automaton A will accept spcp cw words of the form where s pcp is a solution of the PCP for n-tuples s pcp cw x,y, and w {a,b,c} ω. If the two heads are in the same position at least once (this means that we reached the final state s, then they will be in the same position infinitely many times (every time the automaton enters state s. 6 Final remarks We have considered here Watson-Crick automata working on double-stranded infinite tapes, and we have defined for them the classic acceptance conditions. We have investigated whether or not results known for usual (single stranded ω-automata or for Watson-Crick automata working with finite tapes can be extended to Watson-Crick ω-automata. As it turned out, this is the case with the equivalence of the four main acceptance conditions (Büchi, Muller, Rabin and Streett for nondeterministic Watson-Crick ω-automata, as well as with some relations among the generated families of ω-languages. References [1] L. M. Adleman, Molecular computation of solutions to combinatorial problems, Science, 266, (1994, [2] J. R. Büchi, On a decision method in restricted second-order arithmetic, Proc. Int. Congr. on Logic, Methodology and Philosophy of Science, Stanford Univ. Press, (1962, [3] C. S. Calude, Gh. Păun, Computing with Cells and Atoms. An Introduction to Quantum, DNA, and Membrane Computing, Taylor & Francis, London, (

19 [4] R. Freund, Gh. Păun, G. Rozenberg, A. Salomaa, Watson-Crick finite automata, Proc 3rd DIMACS Workshop on DNA Based Computers, Philadelphia, (1997, [5] C. Martin-Vide, Gh. Păun, From the old to the new in theoretical computer science: the theory of formal languages and automata vs. molecular computation, Gac. R. Soc. Mat. Esp. 5, (2002, [6] C. Martin-Vide, Gh. Păun, Normal forms for Watson-Crick finite automata, submitted, (2000. [7] R. McNaughton, Testing and generating infinite sequences by a finite automaton, Inform. Control 9, (1966, [8] V. Mihalache, A. Salomaa, Lindenmayer and DNA: Watson-Crick DOL systems, Current Trends in Theoretical Computer Science, World Sci., (2001, [9] A Păun, M. Păun, State and transition complexity of Watson-Crick finite automata, Proc. Fundamentals of Computation Theory, FCT 99, LNCS 1684, Springer-Verlag, (1999, [10] Gh. Păun, G. Rozenberg, A. Salomaa, DNA Computing. New Computing Paradigms, Springer-Verlag, Berlin, (1998. [11] M. O. Rabin, Decidability of second-order theories and automata on infinite trees, Trans. Amer. Math. Soc. 141, (1969, [12] A. Salomaa, Uni-transitional Watson-Crick DOL systems, Theoret. Comput. Sci. 281, (2002, [13] L. Staiger, ω-languages. In: G. Rozenberg, A. Salomaa (eds., Handbook of Formal Languages, Vol. 3, Springer-Verlag, Berlin, (1997, [14] W. Thomas, Automata on infinite objects. In: J. van Leeuwen (eds., Handbook of Theoretical Computer Science, Vol. B, Elsevier, (1990, [15] W. Thomas, Languages, automata, and logic. In: G. Rozenberg, A. Salomaa (eds., Handbook of Formal Languages, Vol. 3, Springer- Verlag, Berlin, (1997, [16] S. Yu, Regular languages. In: G. Rozenberg, A. Salomaa (eds., Handbook of Formal Languages Vol. 1, Springer-Verlag, Berlin, (1997,

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22 Turku Centre for Computer Science Lemminkäisenkatu 14 FIN Turku Finland University of Turku Department of Information Technology Department of Mathematics Åbo Akademi University Department of Computer Science Institute for Advanced Management Systems Research Turku School of Economics and Business Administration Institute of Information Systems Science