Stateless multicounter 5������ 3��� Watson-Crick Automata

Stateless Multicounter 5′ → 3′ Watson-Crick Automata Ömer Eğecioğlu1 , László Hegedüs2 and Benedek Nagy2 1: Department of Computer Science, University of California, Santa Barbara, CA 93106, USA Email: [email protected] 2: Department of Computer Science, Faculty of Informatics, University of Debrecen, Debrecen, 4032 Hungary Email: [email protected], [email protected] Abstract—The model we consider are stateless counter machines which mix the features of one-head counter machines and special two-head Watson-Crick automata (WK-automata). These biologically motivated machines have reading heads that read the input starting from the two extremes. The reading process is finished when the heads meet. The machine is realtime or non-realtime depending on whether the heads are required to advance at each move. A counter machine is k-reversal if each counter makes at most k alternations between increasing mode and decreasing mode on any computation, and reversal bounded if it is k-reversal for some k. In this paper we concentrate on the properties of deterministic stateless realtime WK-automata with counters that are reversal bounded. We give examples and establish hierarchies with respect to counters and reversals. Even 1-counter stateless WK-automata turn out to be quite powerful. I. I NTRODUCTION Watson-Crick automata ([3], [15]) are finite state machines motivated by nature. They work on DNA molecules, i.e., on double stranded sequences of bases. The strands of a DNA molecule have directions as a resut of the underlying chemical bonds, determining the 5′ and 3′ ends of a strand. The two strands of a molecule are paired by hydrogen bonds if they have opposite directions and the sequence of bases match each-other, i.e., there is a one-toone correspondence given by the so-called WatsonCrick complementarity relation. In this way a strand of the molecule uniquely defines the other, and therefore the DNA molecules can be described by ordinary strings (as, for instance, in [9], [10]). In 5′ → 3′ Watson-Crick automata the reading heads start from the 5′ ends of their strands, i.e. from opposite ends of the molecule regarding its physical body or mathematical description. These automata have been used to characterize linear context-free languages in [12]. In this paper we consider only 5′ → 3′ Watson-Crick automata, and consequently use the terminology WK-automata and omit the qualification 5′ → 3′ . Stateless machines (i.e. machines with only one state) have been the subject of recent investigation because of their connection to to certain aspects of membrane computing and P systems, a subarea of molecular computing that was introduced by Gheorge Păun [13], [14]. A membrane in a P system consists of a multiset of objects drawn from a given finite type set {a1 , . . . , am }. The system has no global state and works on the evolution of objects in a massively parallel way. Thus, the membrane can be modeled as having counters c1 , . . . , cm to represent the multiplicities of objects of types a1 , . . . , am , respectively. The P system can then be thought of as a counter machine in a nontraditional form: without states, and with parallel counter increments/decrements. It is therefore natural to consider the model of computation which has no states but is equipped with counters. These are the two features that motivate the study of stateless multicounter WK-automata. Stateless machines have no states to store information, and the move of such a machine depends only on the symbol(s) scanned by the input head(s) and the local portion of the memory unit(s). Since there are no final states, acceptance of an input string has to be defined in a different way. It is well known that nondeterministic PDA with states are equivalent to stateless nondeterministic PDA (where acceptance is by “null” stack) although this is not true for the deterministic case [4], [7]. In [6], [16] the computing power of stateless multihead automata with respect to decision problems and head hierarchies were investigated. The machine can be deterministic, nondeterministic, one-way, twoway, etc. In [8], various types of stateless restarting automata and two-pushdown automata were compared to the corresponding machines with states. If the machine is not allowed to make moves without moving at the read head, then the model is called realtime. Otherwise the machine is nonrealtime and can make moves that depend only on the contents of the counters without moving the head. A counter machine is k-reversal if each counter makes at most k “full” alternations between increasing mode and decreasing mode and viceversa on any computation (accepting or not), and is reversal bounded if it is k-reversal for some k. Deterministic stateless (one-way) m-counter machines were investigated in [1], where hierarchies with respect to the number of counters and number of reversals were studied. Similar hierarchy results and characterizations are reported in [5] for the non-realtime versions. Hierarchies of the accepted language families by WK-automata are presented in [11] including stateless versions without counters. In this paper we concentrate on deterministic stateless realtime WK-automata with counters. We give examples and establish hierarchies of WK-automata with respect to counters and reversals. II. S TATELESS MULTICOUNTER WK- AUTOMATA The version of stateless WK-automata with counters has the following components. The input is of the form cw$ with w ∈ Σ∗ and c and $ are endmarkers that are not in Σ. The machine has two read-only heads H1 , H2 . H1 moves from left to right and H2 moves from right to left. Originally, H1 is on c and H2 is on $. The machine is equipped with m counters, that are initially all zero. A move of the machine depends on the symbols under the heads and the signs of the counters (zero or positive), and consists of moving the heads and at the same time incrementing or decrementing each of the counters. Depending on the types of head movements allowed, we obtain different classes of machines. Since there are no states, the acceptance of the input cannot be defined by final state. Instead, the input w is accepted by M if the counters are again zero when the heads meet. The essence of when the heads H1 and H2 meet is captured best by making use of a function φ which indicates whether the heads are close or far apart in processing the input. This locality requirement can be justified in part by biological properties that give rise to WK-automata. For the model it suffices to know if there are zero, one, two, or more than two letters between the heads. Define  p     if there are p letters between the two heads of M and p ≤ 2, φ(M ) =  ∞ if there are more than two letters    between the heads of M . We use the notation φ(M ) although φ is actually a function of the current positions of the heads of M . For a deterministic stateless multicounter WKautomaton M , a move (x, y; s1 , s2 , . . . , sm ; p) → (d1 , d2 ; e1 , e2 , . . . , em ) (1) has the following parameters: x, y ∈ Σ ∪ {c, $} are the symbols under the heads H1 and H2 , respectively; si is the sign of counter Ci : si = 0 if the i-th counter is zero, si = 1 if it is positive. s1 s2 · · · sm is referred to as a sign vector; d1 , d2 ∈ {0, 1} indicate the direction of move of the heads with d1 +d2 ≤ p. A 0 value signifies that the head stays where it is. d1 = 1 means that H1 moves one cell to the right, and d2 = 1 means that H2 moves one cell to the left; ei = +, −, or 0, corresponding to the operations of increment, decrement, or leave unchanged the contents of the i-th counter. Here ei = − applicable only if si = 1. A move (1) is possible if and only if φ(M ) = p. It should be noted that φ(M ) is not part of the system, nor it is a counter, just a technical parameter. M is nondeterministic if multiple choices are allowed for the right hand side of (1). The machine is realtime if not both d1 and d2 are zero for any move of the machine. Otherwise it is non-realtime. Thus in a realtime machine at least one of the heads must move at every step of the computation. In this paper we focus on deterministic realtime WK-automata with counters. The machine is k-reversal if for a specified k, no counter makes more than k alternations between increasing mode and decreasing mode (i.e. k pairs of increase followed by decrease stages) in any computation, accepting or not. The machine is reversal bounded if it is k-reversal for some k. We denote the set of all realtime (deterministic) k-reversal m counter non-realtime WK-automata by WKCkm , and the realtime versions by RWKCkm . The reversal bounded versions are given by WKC∗m = ∞ ∪ k=0 WKCkm , RWKC∗m = ∞ ∪ RWKCkm ; k=0 while the unbounded reversal versions are denoted ∞ by WKC∞ m and RWKCm . This notation is also used for the corresponding language classes. The formal definition of deterministic stateless multicounter WK automata is as follows. Definition 1: A deterministic stateless multicounter WK-automaton is a quadruple M = (Σ, δ, c, $) where Σ is a nonempty alphabet, δ is a mapping from (Σ ∪ {c, $})2 ×{1, 0}m ×{0, 1, 2, ∞} to {0, 1}2 ×{0, +, −}m and c, $ ̸∈ Σ are two specia symbols called endmarkers. Note that in the nondeterministic case, δ is a mapping from (Σ ∪ {c, $})2 ×{1, 0}m ×{0, 1, 2, ∞} to P ({0, 1}2 ×{0, +, −}m ) i.e., to the subsets of {0, 1}2 ×{0, +, −}m . (d1 , d2 ; e1 , e2 , . . . , em ), x must be equal to y. This is because both heads would read the same symbol, so x ̸= y is not possible. An instantaneous description (ID) of M with input cw$ is a tuple (s1 , s2 , , . . . , sm , cx⌈y⌋z$) where si is the sign of the i-th counter and w = xyz with the left head reading the first letter of y and the right head reading the last letter of y. The initial ID of M is (0, 0, . . . , 0, ⌈cw$⌋). We use ID1 ⊢ ID2 to indicate the change in the ID after a single move of M . ⊢∗ denotes the reflexive, transitive closure of ⊢. The language accepted by M is {w ∈ Σ∗ | (0, 0, . . . , 0, ⌈cw$⌋) ⊢∗ (0, 0, . . . , 0, cx⌋⌈z$) with w = xz} . III. E XAMPLES AND THE 1- REVERSAL CASE We start with the following examples. Example The language of palindromes L = {w|w = wR } over an alphabet Σ is accepted by a stateless deterministic realtime WK-automaton without counters. The machine only makes a move when the symbols under both heads are equal. If the length of the input word is odd, the last symbol can be read by either head. This way the machine accepts exactly the palindromes over Σ. Remark The language family det-2Lin (see [11], for more In the examples, the ⌈ and ⌋ symbols are used to details on this family) is accepted by deterministic indicate the read heads H1 and H2 respectively. 2-head automata in such a way that the heads start Thus, while an automaton is reading some word from the opposite ends of the input and proceed a1 a2 . . . an over Σ∗ , the string until they meet. a1 a2 . . . ak−1 ⌈ak ak+1 . . . al−1 al ⌋al+1 . . . an By simulating deterministic finite automata, or deterministic 2-head (i.e., deterministic 5′ → 3′ with w = a1 a2 · · · an signifies that the left head is WK) automata, it is easy to show that all regular and reading the symbol ak and the right head is reading all det-2Lin languages can be accepted by determinthe symbol al . istic non-realtime multicounter WK-automata with Note that if one of the heads never moves, then unbounded reversals. The simulation goes by binary the machine is of the type already considered in [1], counters, i.e., every counter refers for a unique [5]. If both heads of M move, then ⌈, H1 and ⌋, H2 state of the simulated automaton, noting that the can be used synonymously. initial configuration refers also for the starting state. Remark When the machine senses that the input is fully read If there is only one symbol between the two heads, (i.e., in the last step), the counters used to simulate in any move of the form (x, y; s1 , s2 , . . . , sm ; 1) → accepting states can be emptied. Stateless deterministic realtime WK-automata of the word, at least one of the counters must be with counters that are 1-reversal are already quite increased/decreased for otherwise a word with an powerful as the following examples show: additional a or b would also be accepted. It follows that after at most 3m increasing steps there must Example be a one in which at least one counter decreases. Let Σ = {a, b, c, d}. Consider the stateless WKFollowing this, after at most 3m decreasing steps automaton M with two counters whose moves are there must be a step when a counter increases again. Then after at most 3m steps a counter must decrease (c, $; 0, 0; ∞) → (1, 1; 0, 0) again, and so on. Therefore for enough long word (a, d; 0, 0; ∞) → (1, 1; +, +) 2 ((ab)7m k works) we can find a counter which makes (a, d; 1, 1; ∞) → (1, 1; +, 0) more than k reversals. This contradiction gives the (b, c; 1, 1; ∞) → (1, 1; −, 0) proof of the theorem for this case. If only the second head moves in M , the proof is similar. (b, c; 1, 1; 2) → (1, 1; −, −) Now we consider the case in which both heads Then M is 1-reversal and accepts the language move in the accepting computation on (ab)n . Then {an bn cn dn | n ≥ 0}. there are two possibilities: if the machine M senses that their heads are close to each other The well known mildly context-sensitive, non- when only one letter is being read to finish the context-free language {an bn cn | n ≥ 0} can also input, then at this point every counter must be be accepted in a similar way. 0 or 1. Before this configuration both of the Therefore non context-free languages can be ac- heads moved. There were three letters between cepted by a small number of counters and with only them (configurations (c , c , . . . , c , cu⌈aba⌋v$) or 1 2 m one reversal. (c1 , c2 , . . . , cm , cu⌈bab⌋v$)) and every counter value IV. R ESTRICTION OF REVERSAL BOUNDEDNESS ci was 0, 1 or 2. Since M has no knowledge that the For w ∈ Σ∗ and a ∈ Σ, define |w|a as the input will be processed in two steps, the last but one number of occurrences of a in w and consider the configuration is reached in a deterministic manner, language L = {w | w ∈ {a, b}∗ , |w|a = |w|b }. It is even if the input is rlonger. Thus,r working on longer v (or ub(ab) v) (with r > 1) the known that L can be accepted by a stateless non- input words ua(ba) r−1 ′ ′ ′ ⌋av$) (or configuration (c , c , 1 2 . . . , cm , cua⌈b(ab) realtime 1-reversal 2-counter machine M but not by ′ ′ r−1 ′ a stateless realtime k-reversal m-counter machine (c1 , c2 , . . . , cm , cub⌈a(ba) ⌋bv$) is obtained with for any k, m ≥ 1. We show that this result also counters having possible values only 0’s and 1’s. Then if r = 2, then the input must be accepted in holds for WK-automata. at most 3 steps, therefore not any counters can have Theorem 1: The language L = {w | w ∈ value more than 3 in this computation. Moreover {a, b}∗ , |w|a = |w|b } cannot be accepted by a state- at least one counter is changed in every step of the less realtime k-reversal m-counter WK-automaton computation. When there are two letters between the for any k, m ≥ 1. heads when they sense their distance is small, then Proof: Suppose L is accepted by some realtime at this configuration the value of every counter is 0, k-reversal m-counter WK-automaton M . Then ev- 1 or 2. For longer input words the same holds if the ery word of the regular language (ab)∗ is accepted. heads read the same prefix and suffix of the input If a word (ab)n is accepted by the machine in a way word. Therefore in this computation the possible that only one of the heads moves while the other values of counters cannot exceed 3. stays put, then the value of every counter is among In both cases for long enough words the accepting 0, 1, 2, 3 at any time. This is because arbitrarily long computation has more than k reversals, by a similar words in (ab)∗ are accepted and when the machine argument as we showed for the case when only one senses that there are at most 2 letters to finish the head moves during the computation. input, it must be possible to decrease all counters to zero in two steps. Moreover reading any letter reversals. Remark ∗ The language L = {w ∈ {a, b} | |w|a = |w|b } given in Theorem 1 can be accepted with a stateWe briefly consider the following combinatorial less deterministic realtime WK-automaton with un- question: For 1-reversal stateless WK-automata with bounded reversals. m counters over Σ = {a} that accept singletons, what is the length of the longest string an that can be accepted as a function of m? A similar consideration was used to establish the counter hierarchy in V. T HE UNARY CASE the case of stateless deterministic realtime reversal A number of interesting combinatorial issues bounded multicounter machines in [1]. We omit the arise in the case of the unary alphabet Σ = {a}. It is proof of the following result that holds for the WKknown that the language L = {a2n | n ≥ 1} cannot automata case. be accepted by any stateless deterministic nonTheorem 3: The maximal length of the word realtime reversal bounded multicounter machine. n a accepted by stateless deterministic 1-reversal mHowever this language can be accepted by a stateless realtime WK-automaton without counters, with counter WK-automaton that accepts a singleton is the following three deterministic rules: n = (m − 1)2m+1 + 2m. (c, $; ; ∞) → (1, 1; ) (a, a; ; ∞) → (1, 1; ) (a, a; ; 2) → (1, 1; ) On the other hand if the length is required to be a multiple of an integer i ≥ 3, then the WK-automata can no longer accept this language as the following theorem shows. Theorem 2: For fixed i ≥ 3, languages in the form L = {ain | n ≥ 0}, cannot be accepted by any stateless deterministic realtime reversal bounded WK-automaton. Proof: Let i be fixed. The language L is infinite. Since the machine is deterministic realtime and it senses that the input is nearly processed only when the heads are close enough (their distance is not greater than 2), in any accepting run the value of a counter cannot be more than i. This can be shown by the same method as used in the proof of Theorem 1. At every step of the machine some (at least one) counters change their values in order not to accept words with one or two additional a’s. Then by similar argument as in the proof of Theorem 1 one can show that for long enough words of L the machine makes more than k reversals for any given k. Note that the languages in Theorem 2 are regular, therefore they can be accepted by stateless deterministic realtime machine with unbounded In fact the case of k reversals can be treated similarly following the proof sketch in [1]. The maximum value of n in the case of stateless deterministic k-reversal m-counter WK-automata is found to be n = ((2k − 1)m − 1)2(2k−1)m+1 + 2(2k − 1)m. This dependence on m and k can be exploited to prove the hierarchy results with respect to realtime stateless multicounter WK-automata as given in Theorem 9. VI. N ON - REALTIME MACHINES We give a number of results for the non-realtime case mostly without proofs. The language L = {w ∈ {a, b}∗ | |w|a = |w|b } given in Theorem 1 can trivially be accepted by a 1-reversal non-realtime WK-automaton by accumulating the number of a’s and b’s in two counters, then simultaneously decreasing the counters after the heads meet. Theorem 4: The language L = {an |n ≥ 0} is accepted by a non-realtime deterministic stateless WK-automaton with four counters and unbounded reversals. Proof: We use the fact that n2 is the sum of the first n odd numbers. The main task is to read consecutive odd number of a’s with a certain sequence of moves called an iteration. We will 2 use a parity counter denoted by Cp . We need 3 more counters C1 , C2 , Cm respectively. The counter values Cm = 0 and Cm = 1 indicate whether C1 , or C2 is to be used in the iteration. Each odd number is of the form 2k + 1 where k is 0, 1, 2, . . .. In the first iteration, only the Cp counter is incremented and we have the following sign vector: (1, 0, 0, 0). If there are no more a’s to be read, the machine can now behave non-realtime and decrement Cp . Since 1 is a square, the word of only one a is accepted by the machine. If there are more a’s to be read, first both heads move simultaneously each reading an a, decrementing Cp , while incrementing C1 . Then we get to the (0, 1, 0, 0) sign vector. Since Cp is zero, which indicates, that the machine has read an even number of a’s in the iteration, one additional a must be read by one of the heads, while incrementing Cp and incrementing Cm . Now we have read the word aaaa and got the sign vector (1, 1, 0, 1). Then, like before, if there are no a’s left, the counters are decremented in nonrealtime mode, else the operation is continued. A general move can be described as follows. The parity counter is always 1 at the beginning of the iteration except the first one, but that case is specified above. At the first step, Cp is decremented, while reading two a’s and incrementing C1 if Cm is 0 or C2 if Cm is 1. Then we get two similar cases: 1) Cm = 0: in each step both heads move simultaneously, reading a’s, while C2 is decremented and C1 is incremented. Thus the content of C2 is moved to C1 . When C2 is zero, C1 contains the former contents of C2 plus 1. 2) Cm = 1: Exactly the same as the Cm = 0 case, with the roles of C1 and C2 changed. To finish the iteration, one additional a must be read, while Cp is incremented and Cm is decremented if it’s value is 1, incremented if it’s value is 0. During the computation, at the end of any iteration, if no a’s can be read, the machine can behave in non-realtime mode and decrements its counters. (0, 0, 0, 0, ⌈caaaaaaaaaaaaaaaa$⌋) ⊢ (0, 0, 0, 0, c⌈aaaaaaaaaaaaaaaa⌋$) ⊢ (1, 0, 0, 0, ca⌈aaaaaaaaaaaaaaa⌋$) ⊢ (0, 1, 0, 0, caa⌈aaaaaaaaaaaaa⌋a$) ⊢ (1, 1, 0, 1, caaa⌈aaaaaaaaaaaa⌋a$) ⊢ (0, 1, 1, 1, caaaa⌈aaaaaaaaaa⌋aa$) ⊢ (0, 0, 2, 1, caaaaa⌈aaaaaaaa⌋aaa$) ⊢ (1, 0, 2, 0, caaaaaa⌈aaaaaaa⌋aaa$) ⊢ (0, 1, 2, 0, caaaaaaa⌈aaaaa⌋aaaa$) ⊢ (0, 2, 1, 0, caaaaaaaa⌈aaa⌋aaaaa$) ⊢ (0, 3, 0, 0, caaaaaaaaa⌈a⌋aaaaaa$) ⊢ (1, 3, 0, 1, caaaaaaaaaa⌋⌈aaaaaa$) ⊢∗ (0, 0, 0, 0, caaaaaaaaaa⌋⌈aaaaaa$) Note that each time, if Cp is 1, a square number of a’s have been read. Since the machine can only decrease it’s counters when Cp is 1, only words consisting of square number of a’s are accepted by it. Thus the machine works in the expected way. It would be interesting to see if this language can be accepted with fewer than four counters. Similar to the above result, we can prove: Theorem 5: L = {a2 −1 | n ≥ 0} can be accepted by a non-realtime deterministic stateless WK-automaton with four counters and unbounded reversals. n The machine has four counters, C1 , C2 , C3 , Cm respectively. In the case of i = 0, one step is made by one of the heads, reading an a, while the counters C1 and Cm increased. Now, the sign vector is (1, 0, 0, 1). The other steps are made as follows: If C1 and Cm are both not zero, then decrease C1 , while increasing C2 and C3 simultaneously and reading a’s with both heads. If C1 is zero and Cm is one, that means the machine has read the next power of two number of a’s, which may be called j and can start setting C1 to 2j. This is done, when C1 is zero, Cm is one and C2 , C3 both contain some number. So the sign vector is (0, 1, 1, 1). Then in non-realtime Example mode, C2 is decreased, while C1 is increased. Note, We give an example of an execution of the machine that Cm has to be decreased to zero in the first step described in the proof of Theorem 4 Suppose the too. After a few steps, the sign vector is (1, 0, 1, 0). input is a16 . Then the machine makes the following Now, the contents of C3 must be moved to C1 in sequence of moves: a similar way, giving the sign vector (1, 0, 0, 0) and C1 now contains the sum of the values of C2 and C3 . Then Cm is increased by one, so that the following iteration can be started. We specified, that in the first iteration, one a is read and C1 ’s value is set to one. By induction, if C1 ’s value is j and C2 , C3 are both zero, a2j is read, while C2 , C3 are set to j. In the next iteration C1 will contain j +j and a4j is read and so on. . . . Thus in each iteration the machine reads twice the a’s as in the previous iteration. The computation can be ended using non-realtime mode, when C1 is zero and no more a’s can be read. Example The machine does the following steps in the case of a15 : (0, 0, 0, 0, ⌈caaaaaaaaaaaaaaa$⌋) ⊢ (0, 0, 0, 0, c⌈aaaaaaaaaaaaaaa⌋$) ⊢ (1, 0, 0, 1, ca⌈aaaaaaaaaaaaaa⌋$) ⊢ (0, 1, 1, 1, caa⌈aaaaaaaaaaaa⌋a$) ⊢ (1, 0, 1, 0, caa⌈aaaaaaaaaaaa⌋a$) ⊢ (2, 0, 0, 0, caa⌈aaaaaaaaaaaa⌋a$) ⊢ (2, 0, 0, 1, caa⌈aaaaaaaaaaaa⌋a$) ⊢ (1, 1, 1, 1, caaa⌈aaaaaaaaaa⌋aa$) ⊢ (0, 2, 2, 1, caaaa⌈aaaaaaaa⌋aaa$) ⊢ (1, 1, 2, 0, caaaa⌈aaaaaaaa⌋aaa$) ⊢ (2, 0, 2, 0, caaaa⌈aaaaaaaa⌋aaa$) ⊢∗ (4, 0, 0, 1, caaaa⌈aaaaaaaa⌋aaa$) ⊢ (3, 1, 1, 1, caaaaa⌈aaaaaa⌋aaaa$) ⊢∗ (0, 4, 4, 1, caaaaaaaa⌋⌈aaaaaaa$) ⊢∗ (0, 0, 0, 0, caaaaaaaa⌋⌈aaaaaaa$) As our previous examples show, there are nonsemilinear languages that are accepted by nonrealtime WK-automata. The following theorem holds both for realtime and non-realtime machines and analogous with the results of [1], [5]. Theorem 6: For every stateless deterministic (realtime/non-realtime) WK-automaton with k reversals and m counters, there exists a 1-reversal stateless (realtime/non-realtime) WKautomaton with (2k − 1)m counters that accepts the same language. The proof is given in [1] in the proof of Theorem 4 of that paper for stateless multicounter automata, which can be applied to WK-automata as well. The main idea is that if a counter does one reversal and would increase again, some other counter can take its place thus avoiding multiple reversals. This way one counter with k reversals can be substituted by k counters with one reversal. Further (k − 1) counters are needed to indicate which additional counter is being used. Thus to simulate m counters, k + (k − 1) = 2k − 1 times m counters are needed. Theorem 7: There exist infinitely many regular languages that are not accepted by any realtime, or non-realtime k-reversal m-counter stateless WKautomaton for any fixed m, k ≥ 0. Proof: By Theorem 6 we can assume that k = 1. Consider a finite language L over some alphabet V , which can be accepted by a stateless WK-automaton with m counters and 1 reversal. Select some word s ∈ L, which is accepted by using the most counters. Then s can be written in the form xyz, where y is some subword, which is processed by using l counters, x, z are some (possibly empty) words over V . Then the language L′ = {xy i z|i ≥ 0} is regular and any word in the form xy i z can only be accepted by m + l(i − 1) counters. Since there are infinitely many non negative integers, there is always a word w ∈ L′ that can not be accepted by m counters for any fixed m ≥ 0. Example The language (ab)∗ is a simple regular language, but it cannot be accepted by any (realtime or non-realtime) stateless multicounter WK-automaton which is reversal bounded. (See the proof of Theorem 1 also.) Theorem 8: The language L = {ww | w ∈ {a, b}∗ } cannot be accepted by any stateless deterministic non-realtime multicounter k-reversal WKautomaton with m counters for any fixed k, m ≥ 0. Proof: Suppose that some stateless WKautomaton M accepts L. Then there exists some x ∈ {0, 1}∗ such that xx ∈ L and is accepted by M doing k reversals and using all m counters. We can assume, that there is a sequence 01 (or 10) in x. As seen in Theorem 7, these sequences can only be processed while using some (at least one) counters because we don’t have any states to indicate that there is a 1 to be read in the next step. Then x can be written in the form x′ 01x′′ . If the machine accepts the language L, it should also accept x′ (01)i x′′ x′ (01)i x′′ for all i ≥ 1, thus making more than k reversals, which is a contradiction. We have the following hierarchy results: Theorem 9: The following hierarchy results hold: ∞ 1) RWKC∞ ∗ ⊂ WKC∗ , ∗ ∗ 2) RWKC∗ ⊂ WKC∗ , 3) WKCkm ⊆ WKC1(2k−1)m , 4) WKCkm ⊂ WKCkm+1 . 5) WKCkm ⊂ WKCk+1 (for k < 2m−1 /m). m It can also be shown that stateless WK-automata without counters can only accept a proper subset of linear context-free languages. We omit this discussion. VII. F UTURE WORK The power of WK-automata lies somewhere between stateless one-head counter machines and stateless two-head counter machines. There remain a number of interesting aspects of the hierarchy of languages accepted by stateless multicounter WKautomata: non-realtime, nondeterministic versions and the case of unbounded reversals are among these. It is also of interest to identify the nature of separating languages for each of these classes. Work is in progress on various extensions of the properties considered in the present paper, as well as closure properties of the corresponding language classes. 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