A. R. Smith III, Real-time language recognition by one-dimensional cellular automata, J. Comput. System Sci. 6 (1972) 233-253.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in this section deal with Boolean operations. Since the OCAs are space bounded deterministic devices the positive properties are natural. 15 Lemma 17 Let r : N N be a function, then L n r(n) OCA) is closed under union and intersection. Proof. Using the same two chanel technique of [4] and [8] the assertion can easily be seen. Each cell consists of two registers in which acceptors for both languages are simulated in parallel. 2 The closure under complement is expected for deterministic devices. But here an input is rejected by not entering an accepting state. So in order to accept the ....

A. R. Smith III, Real-time language recognition by one-dimensional cellular automata, J. Comput. System Sci. 6 (1972) 233-253.

....the cells is : ffl ffi i ; 1 i n Gamma 1, if 0 t 2 L i . ffl ffi n , if 0 t = 2 S i=n Gamma1 i=1 L i . 2.5 Language recognition The power of the different models is studied through the languages that can be accepted by the CA. For a better definition of language recognition by CA, see [3]. A given CA accepts a word by having a special cell, called the distinguished cell, going in a special acceptance state 2 , being given the word as input (i.e. embedded in the initial configuration) In a similar way, it rejects a word by having its distinguished cell going in a refusal state. ....

Alvy Ray Smith III. Real-time language recognition by one-dimensional cellular automata. Journal of Computer and System Sciences, 6:233--253, 1972.

....needed by the lower cells to react to operations applied to the top of the stack. 2 The next step of positioning the family L 2 would be to prove or disprove the inclusion L 2 ae L rt (CA) The problem whether or not the real time CA languages are containing the context free ones was raised in [13] and is still open. It is related to the open question whether or not sequential one tape Turing machines are able to accept the context free languages in square time. A proof for the inclusion would imply the existence of square time Turing machines. 4 Iterative arrays Now we turn to the ....

....in terms of context free languages. Although L rt (OCA) does not contain the 2 linear languages, real time OCAs are powerful devices. For example, the non Parikh linear language f(0 i 1) j i 2 N 0 g [11] and the inherently ambiguous language f0 i 1 j 2 k j i = j j = k; i; j; k 2 Ng [13] are belonging to L rt (OCA) 11 6 Two way cellular arrays For structural reasons L rt (IA) L rt (CA) and L rt (OCA) L rt (CA) holds. Since L rt (IA) and L rt (OCA) are incomparable both inclusions must be strict. Unfortunately, this proof of the strictness, although given in terms of ....

Smith III, A. R. Real-time language recognition by one-dimensional cellular automata. J. Comput. System Sci. 6 (1972), 233--253.

....for a CA to be simulated by an OCA based on the same architecture and we present some bounds of the simulation time of this mimic. 2 Some recalls on OCA One of the simplest models of parallel computation is the one way cellular automaton (OCA) which has already been studied by several authors [6, 8, 10]. They define this notion as an array of n identical finite state machines (cells) that operate synchronously at discrete time steps. The input a 1 : a n where a i is in the finite alphabet Sigma, is applied to the array in parallel at time 0 by setting the states of the cells to a 1 ; ....

A. R. Smith III. Real-time language recognition by one-dimensional cellular automata. J. Comput. System Sci., 6:233--253, 1972.

....of the single automata (these are in almost all cases finite or pushdown automata) the interconnection scheme (which sometimes implies a dimension of the system) a local and or global transition function and the input and output modes. Various types have been studied for a long time (e.g. [1, 2, 4, 5, 7, 8, 14, 20, 21, 23, 24, 25, 27]) One kind of system is of particular interest: the cellular automata. In this wellinvestigated model homogeneously connected deterministic or nondeterministic finite automata work synchronously at discrete time steps. Here we are investigating linear arrays with very simple interconnection ....

....two way or nondeterministic one way cellular automata (NCA or NOCA) Although deterministic and nondeterministic finite automata have the same computing capability, nondeterminism can strengthen the power of the deterministic parallel devices. Nondeterministic arrays have been investigated e.g. in [24], where it was proved that NCAs can exactly accept the context sensitive languages, in [8] where the equivalence of NCAs and NOCAs without time restrictions has been shown, and in [16] where it was shown in terms of homogeneous trellis automata that the real time NOCA languages contain the ....

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Smith III, A. R. Real-time language recognition by one-dimensional cellular automata. Journal of Computer and System Sciences 6 (1972), 233--253.

....especially speed up theorems are given in [10, 11, 12] Various generalizations of IAs have been considered. In [15] IAs are studied in which all the finite automata are additionally connected to the communication cell. Several more results concerning formal languages can be found (e.g. in [16, 17, 18]) In the field of computational complexity there is a particular interest in infinite hierarchies of complexity classes defined by bounding some resources. In order to obtain dense hierarchies one has to show that only a slight increase in the growth rate of the bounding function yields a new ....

Smith III, A. R. Real-time language recognition by one-dimensional cellular automata. J. Comput. System Sci. 6 (1972), 233--253.

....theoretical perspective. In [34] Wolfram attempted to describe cellular automata behavior in terms of computation theory, viewing cellular automata as computers whose time evolution processes the information specified by their initial configurations. CA have been studied as language recognizers [20] and as continuous functions on compact topological spaces [13, 14, 15, 16, 17, 18, 22] For the latter characterization, let S be the finite set of cell states of a CA. S with the discrete topology is a compact space. Let d be the dimension of the CA and let Z denote the integers. Let S = S Z ....

A. R. Smith III. Real-time language recognition by one-dimensional cellular automata. Journal of Computer and System Sciences, 6:233--253, 1972.

....and so di ers considerably from the notions used in most other work on designing CAs for computation. For example, propagating particle like signals were used in the solution to the Firing Squad Synchronization Problem [52, 58, 77] in Smith s work on CAs for parallel formal language recognition [68], and in Mazoyer s work on computation in one dimensional CAs [53] However, in all these cases, the particles and their interactions were designed by hand to be the explicit behavior of the CA. That is, the particles are explicitly coded in each cell s local state and their dynamics and their ....

A. R. Smith. Real-time language recognition by one-dimensional cellular automata. J. Comput. System Sci., 6:233, 1972.

....understand it, it contains an error discussed in Section 3.3. Third, the author did not explain precisely what he means by universality . It could be at first sight considered as a minor problem since it is also the case for instance in the works of Banks [1] Nourai and Kashef [11] and Smith [7]. As a matter of fact, all authors we could read on this topic did not defined what they meant by Turing universality. This point may seem of low interest since the definition is usually straightforward, but in the case of simulation by cellular automata, it is a very delicate point because ....

....bounded by a recursive function of w) We are not sure that this definition is the most general. But all the simulations we met in the literature fit these requirements. Concerning the halting condition, the first idea is that a special state should appear somewhere in the configuration (see [7]) This is not good because a new specially dedicated state should be used. For instance in Life the halting condition is different. We could also ask that the dynamics of the cellular automaton becomes stable (i.e. enters a time periodic loop) This definition seems too restrictive as it excludes ....

A. R. Smith III. Real-time language recognition by one-dimensional cellular automata. Journal of Computer and System Sciences, 6:233-- 253, 1971.

....CS the one of context sensitive languages (see [12] Moreover, d] means that the result is a consequence of two facts: PCA = DSPACE(n) that can be deduce from simulations in [26] and CS = NSPACE(n) 34] for example. We can add that the unary languages in RPOCA are the unary rational languages [27]. Rat RPOCA Rat RSCA AlgL RPOCA [27] Alg ae SOCA [15] Alg 6ae RSCA [3] RSCA 6ae Alg [3] RPOCA 6ae Alg [3] Alg 6ae RPOCA [32] PCA = SCA CS So, connections between the Chomsky s hierarchy and the interesting classes via cellular automata are not quite easy to interpret. According to ....

....(see [12] Moreover, d] means that the result is a consequence of two facts: PCA = DSPACE(n) that can be deduce from simulations in [26] and CS = NSPACE(n) 34] for example. We can add that the unary languages in RPOCA are the unary rational languages [27] Rat RPOCA Rat RSCA AlgL RPOCA [27] Alg ae SOCA [15] Alg 6ae RSCA [3] RSCA 6ae Alg [3] RPOCA 6ae Alg [3] Alg 6ae RPOCA [32] PCA = SCA CS So, connections between the Chomsky s hierarchy and the interesting classes via cellular automata are not quite easy to interpret. According to Figure 7, algebraic languages could to be low ....

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Smith III A. Real time language recognition by one-dimensional cellular automata. Journal of Computer and System Science. Vol. no. 4: 299318, 1971.

....ratio n 7 2 n and signals needed to build it. 24 if the number of different signals which can have to be superposed is finite. Actually, would this problem be decidable, would the problem to know whether a state emerges in the evolution of a cellular automaton be decidable, which is not (see [68]) So quite each case is a singular one, and it is not always easy to design a cellular automaton (assuming that such an automaton exists) even governed under an expressive cellular geometric diagram. It is not the case for our example, and a space time diagram of a suitable automaton is to be ....

Smith III A. Real-time language recognition by one-dimensional cellular automata. Journal of Computer and System Sciences Vol. no. 6: 233--253, 1972.

....results, especially speed up theorems, are given in [8, 9, 10] Various generalizations of IAs have been considered. In [11] IAs are studied in which all the finite automata are additionally connected to the communication cell. Several more results concerning formal languages can be found e.g. in [12, 13, 14]. Sometimes completely nondeterministic arrays have been studied. In [3] arrays with restricted nondeterminism have been introduced. There it has been shown that the number of nondeterministic transitions can be reduced by a constant factor and that there exists an infinite hierarchy of properly ....

Smith III, A. R. Real-time language recognition by one-dimensional cellular automata. Journal of Computer and System Sciences 6 (1972), 233--253.

....by an iterative array (IA) An iterative array is simply a cellular space where the input is supplied sequentially to the cell at the origin. All other cells are initially quiescent. It is known that in case of real time language recognition cellular spaces are more powerful than iterative arrays [20]. Nevertheless in a first phase an IA can read the input whereby the symbols are stored in consecutive cells. After a subsequent fssp synchronization the preprocessing is done and the IA simulates the cellular space one to one. The following example is due to [17] Example 4 An one way cellular ....

Smith III, A. R. Real-time language recognition by one-dimensional cellular automata. Journal of Computer and System Sciences 6 (1972), 233--253.

....Copyright c fl 1998 by the authors 1 Introduction Linear arrays of finite automata can be regarded as models for massively parallel computers. Mainly they differ in how the automata are interconnected and in how the input is supplied. Various types have been studied for a long time [1, 2, 3, 4, 5, 8, 11, 14, 15, 16, 17, 19, 21]. Here we are investigating arrays with a very simple interconnection pattern. Each node is connected to its right immediate neighbor only. They are usually called one way cellular automata (OCA) Although deterministic and nondeterministic finite automata have the same computing capability, ....

....when the border cell containing the most significant digit generates a carry over. 2 5 Closure properties The family L rt (1G OCA) has strong closure properties. Lemma 13 L rt (1G OCA) is closed under union, intersection and set difference. Proof. Using the same two channel technique of [17] and [8] the assertion is easily seen. Each cell consists of two registers in which acceptors for both languages are simulated in parallel. 2 Theorem 14 L rt (1G OCA) is an AFL (i.e. is closed under intersection with regular sets, inverse homomorphism, free homomorphism, union, concatenation and ....

Smith III, A. R. Real-time language recognition by one-dimensional cellular automata. J. Comput. System Sci. 6 (1972), 233--253.

....of IAs by restricted Turing machines and several results, especially speed up theorems, are given in [12, 13, 14] In [16] IAs are studied in which all the finite automata are additionally connected to the communication cell. Several more results concerning formal languages can be found e.g. in [17, 18, 19]. In some cases fully nondeterministic arrays have been studied, but up to now it is not known how the amount of nondeterminism influences the capabilities of the model. Here we introduce arrays with restricted nondeterminism. We restrict the ability to perform nondeterministic transformations to ....

.... is a natural example for a hierarchy: Example 10 Let i 1 be a constant and f(n) log i (n) and g(n) log i 1 (n) log i denotes the i fold composition) Then by Theorem 9 we have L rt (gG IA) ae L rt (fG IA) Since L lt (IA) is identical to the linear time cellular automata languages [18] and fa n b 2 n Gamman j n 2 Ng is acceptable by such devices fa g(n) b f(n) Gammag(n) j n 2 Ng 2 L lt (IA) holds. Moreover, from g 2 log(f) follows 8 m;n 2 N : f(m) f(n) g(m) g(n) Thus, the conditions of Theorem 8 are met. Trivially, g is of order o(f ) e.g. for i = 2 we ....

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Smith III, A. R. Real-time language recognition by one-dimensional cellular automata. Journal of Computer and System Sciences 6 (1972), 233--253.

....does not either if we restrict the computations to the uniformly universal mode. A corresponding result does not hold for deterministic cellular automata. Instead, L rt (OCA) ae L lt (OCA) has been shown [2, 13, 5] The relationship is a famous open problem for deterministic two way devices (e.g. [4, 12]) Uniform ACAs have been introduced in [7, 8] The main difference between uniform ACAs and (nonuniform) ACAs is the induction of the global transition. Whereas in an ACA at every time step each cell chooses independently one local transition from the set in an uniform ACA at every time step one ....

Smith III, A. R. Real-time language recognition by one-dimensional cellular automata. J. Comput. System Sci. 6 (1972), 233--253.

....and generation has been a cornerstone of theoretical computer science, and the investigation of computational architectures such as CAs often begins with studies of their formal language recognition abilities. Several researchers have looked into this question for CAs (e.g. Cole, 1969; Smith, 1972; Pecht, 1983; Seiferas, time 0 3 6 9 a b c d c b a cells Figure 10: Schematic illustration of Smith s r = 1 bounded CA that recognizes the context free language L 1 (consisting of palindromes) in real time. The lightly shaded rightmost cell at t = 7 indicates acceptance of the input. 1977; ....

....Terrier, 1994) both for theoretical reasons and for the prospect of using CAs as parallel pattern recognizers. In this section I will review some of the work by Alvy Ray Smith, whose interest was in understanding how to exploit the parallelism of CAs for formal language and pattern recognition. Smith (1972) studied one dimensional r = 1 bounded CAs (BCAs) CAs with two special boundary cells. The rightmost non boundary cell was designated to be the accept cell. A BCA is said to accept language L if, for all words x in L, there is a time t such that the accept cell goes into an accept state, ....

Smith, A. R. III (1972). Real-time language recognition by one-dimensional cellular automata.

....lt (CA) There are several important open problems concerning the relations between the various families. The tally languages are playing an important role in investigations on that field. Under unary restriction several essentially different families are identical (e.g. L rt (IA) ae L rt (CA) [6, 20] but L u rt (IA) L u rt (CA) 18] or L 3 ae L rt (OCA) 15] but L u 3 = L u rt (OCA) 19] where L 3 denotes the regular languages) from which follows that the capabilities of some devices become noticable not for tally languages. There are incomparable families that become comparable ....

Smith III, A. R. Real-time language recognition by one-dimensional cellular automata. Journal of Computer and System Sciences 6 (1972), 233--253.

....languages recognizable by iterative pushdown arrays. 1 Introduction The language recognition capabilities of some classes of interacting automata are studied. Classical one dimensional cellular spaces and various modifications and generalizations have been investigated for a long time (e.g. [1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14]) We restrict our considerations on some of such classes Smith [13] collectively Appeared in Proceedings of 5th Theoryday Automata and Formal Languages, AG Informatik, University of Giessen 1995, 149 158. 150 called polyautomata, their iterative variants and a real generalization obtained by ....

Smith III, A. R. Real-time language recognition by onedimensional cellular automata. Journal of Computer and System Sciences 6 (1972), 233--253.

.... OCA languages [4] On the other hand it is a long standing open problem whether the real time CAs are less powerful than the linear time CAs or not [2] Closely related to that problem are the open problems of whether the real time CA languages are closed under reversal or concatenation [21]. In this coherency Ibarra and Jiang [14] have shown that the closure under reversal would imply the closure under concatenation and that the concatenation of two real time CA languages is a linear time language. In case of one way information flow the closure of real time OCA languages under ....

....CAs [14] and the linear time IA languages are equal to the linear time CA languages. Cole [6] proved that the real time IA languages are neither closed under reversal nor under concatenation. Additionally, it is known that the real time IAs are incomparable to the real time OCAs [20, 4] Smith [21] raised the still open problem whether the context free languages are contained in the real time CA languages. The question was answered for several subsets of the context free languages. e.g. it was shown in [13] that every semi linear language is real time acceptable by CAs. Furthermore it is ....

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Smith III, A. R. Real-time language recognition by one-dimensional cellular automata. Journal of Computer and System Sciences 6 (1972), 233--253.

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A. R. Smith. Real-time language recognition by one-dimensional cellular automata. J. Comput. System Sci., 6:233, 1972.

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A. R. Smith. Real-time language recognition by one-dimensional cellular automata. J. Comput. System Sci., 6:233, 1972.

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