# On the computing power of one-way cellular arrays

## Abstract

There are two simple models of parallel language recognizer: one-way cellular array (OCA) and one-way iterative array (OIA). For inputs of length n, both arrays consist of n identical finite-state machines (cells). The communication between cells is one-way, from left to right. The difference in the two models is in the manner in which the input is applied. For the OCA, the input is applied to the cells in parallel. For the OIA, the input is applied serially to the leftmost processor. An input string is accepted if the rightmost cell eventually enters an accepting state. We show that OCA's accept exactly the same class of languages as OIA's. It is relatively easy to show that OIA's can simulate OCA's. The difficult part is the converse, i.e., that OCA's can simulate OIA's. This is rather surprising since in an OIA, *every* cell of the array has access to each symbol of the input string, whereas in an OCA, the i-th cell can only access the first i symbols of the input. This result when combined with known results concerning OIA's answers some open questions concerning the computational complexity of OCA's. We also prove some new results concerning linear-time OCA's and OIA's. For example, we show: (1) linear-time OCA's are equivalent to 2n-time OIA's (note that 2n-time is optimal for OIA's); (2) the concatenation of a linear-time OCA language with a real-time (i.e., n-time) OCA language is a linear-time OCA language; (3) every semilinear language is a linear-time OCA language.

## Keywords

Cellular Automaton Input Pattern Sequential Machine Input String Input Symbol## Preview

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## References

- [CHAN81]Chandra, A., D. Kozen and L. Stockmeyer, Alternation,
*J. ACM*28-1, 1981, pp. 114–133.Google Scholar - [CHAN83]Chan, T. and O. Ibarra, On the space and time complexity of functions computable by simple programs,
*SICOMP*12-4, 1983, pp. 708–716.Google Scholar - [CHAN86a]Chang, J., O. Ibarra and M. Palis, Efficient simulations of simple models of parallel computation by space-bounded TMs and time-bounded alternating TMs, Revision of
*Tech. Rep. 85-47, Department of Computer Science, University of Minnesota*, Nov. 1985; submitted to*Theoretical Computer Science*.Google Scholar - [CHAN86b]Chang, J., O. Ibarra and A. Vergis, On the power of one-way communication,
*Proceedings of the 27th IEEE Annual Symposium on Foundations of Computer Science*, 1986, pp. 455–464.Google Scholar - [CHOF84]Choffrut, C. and K. Culik II, On real-time cellular automata and trellis automata,
*Acta Inform.*21, 1984, pp. 393–409.CrossRefGoogle Scholar - [COOK71]Cook, S., Characterizations of pushdown machines in terms of time-bounded computers,
*J. ACM*18-1, 1971, pp. 4–18.CrossRefGoogle Scholar - [CULI84]Culik II, K., J. Gruska and A. Salomaa, Systolic trellis aotumata; Part I,
*Internat. J. Comput. Math.*15, 1984, pp. 195–212.Google Scholar - [DYER80]Dyer, C., One-way bounded cellular automata,
*Information and Control*44, 1980, pp. 54–69.CrossRefGoogle Scholar - [GURA79]Gurari, E. and O. Ibarra, The complexity of the equivalence problem for two characterizations of Presburger sets,
*Theoretical Computer Science*13, pp. 295–314.Google Scholar - [HARR68]Harrison, M. and O. Ibarra, Multitape and Multihead pushdown automata,
*Information and Control*13, 1968, pp. 433–470.CrossRefGoogle Scholar - [HARR78]Harrison, M., Introduction to formal language theory,
*Addison-Wesley*, 1978.Google Scholar - [HENN61]Hennie, F., Iterative arrays of logical circuits,
*MIT Press, Cambridge, Mass.*, 1961.Google Scholar - [HOPC79]Hopcroft, J. and J. Ullman, Introduction to automata theory, languages, and computation,
*Addison-Wesley*, 1979.Google Scholar - [IBAR74]Ibarra, O., A note on semilinear sets and bounded-reversal multihead pushdown automata,
*Inform. Process. lett.*3, 1974, pp. 25–28.CrossRefGoogle Scholar - [IBAR85a]Ibarra O., and M. Palis, Two-dimensional systolic arrays: characterizations and applications,
*Tech. Rep. 85-1, Department of Computer Science, University of Minnesota*, Jan. 1985; to appear in*Theoretical Computer Science*.Google Scholar - [IBAR85b]Ibarra, O., M. Palis and S. Kim, Some results concerning linear iterative (systolic) arrays,
*J. of Parallel and Distributed Computing*2, 1985, pp. 182–218.CrossRefGoogle Scholar - [IBAR86]Ibarra, O., S. Kim and M. Palis, Designing Systolic Algorithms using sequential machines,
*IEEE Trans. on Computers*C35-6, June 1986, pp. 31–42; extended abstract in*Proc. 25th IEEE Symposium on Foundations of Computer Science*, 1984, pp. 46–55.Google Scholar - [SEID79]Seidel, S., Language recognition and the synchronization of cellular automata,
*Tech. Rep. 79-02, Department of Computer Science, University of Iowa*, 1979.Google Scholar - [SMIT70]Smith, A., III, Cellular automata and formal languages,
*Proc. 11th IEEE Ann. Symp. on Switching and Automata Theory*, 1970, pp. 216–224.Google Scholar - [SMIT71]Smith, A., III, Cellular automata complexity trade-offs,
*Information and Control*18, 1971, pp. 466–482.CrossRefGoogle Scholar - [UMEO82]Umeo, H., K. Morita, and K. Sugata, Deterministic one-way simulation of two-way real-time cellular automata and its related problems,
*Inform. Process. Lett.*14, 1982, pp. 159–161.CrossRefGoogle Scholar