Relating the power of cellular arrays to their closure properties.

*(English)*Zbl 0646.68071Summary: There are many fundamental open problems concerning cellular arrays (CA’s). For example:

(1) Is the class of real-time CA languages closed under reversal (concatenation)?

(2) Are linear-time CA’s more powerful than real-time CA’s?

(3) Are nonlinear-time CA’s more powerful than linear-time CA’s?

(4) Does one-way communication reduce the computing power of a CA? Although some of these problems appear to be easier to resolve than the others, e.g., problem (1) seems easier than (2), no solution to any of these problems is forthcoming. In this paper, we investigate the relationships among these problems as well as prove some positive results concerning CA’s. We show:

(a) the class of real-time CA languages is closed under reversal if and only if linear-time CA’s are equivalent to real-time CA’s; (b) if CA’s are more powerful than CA’s restricted to one-way data communication (i.e., one-way CA’s), then nonlinear-time CA’s are more powerful than linear-time CA’s;

(c) if the class of real-time CA languages is closed under reversal, then it is also closed under concatenation. In the case of unary CA languages, we show that the class is closed under concatenation.

We also show that the language \(L=\{0^ n1^ m |\) \(m,n>0\), m divides \(n\}\) is a real-time CA language, disproving a conjecture of Bucher and Culik.

(1) Is the class of real-time CA languages closed under reversal (concatenation)?

(2) Are linear-time CA’s more powerful than real-time CA’s?

(3) Are nonlinear-time CA’s more powerful than linear-time CA’s?

(4) Does one-way communication reduce the computing power of a CA? Although some of these problems appear to be easier to resolve than the others, e.g., problem (1) seems easier than (2), no solution to any of these problems is forthcoming. In this paper, we investigate the relationships among these problems as well as prove some positive results concerning CA’s. We show:

(a) the class of real-time CA languages is closed under reversal if and only if linear-time CA’s are equivalent to real-time CA’s; (b) if CA’s are more powerful than CA’s restricted to one-way data communication (i.e., one-way CA’s), then nonlinear-time CA’s are more powerful than linear-time CA’s;

(c) if the class of real-time CA languages is closed under reversal, then it is also closed under concatenation. In the case of unary CA languages, we show that the class is closed under concatenation.

We also show that the language \(L=\{0^ n1^ m |\) \(m,n>0\), m divides \(n\}\) is a real-time CA language, disproving a conjecture of Bucher and Culik.

##### Keywords:

linear-time cellular arrays; real-time cellular array; cellular array language; language recognition power; cellular arrays; concatenation
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\textit{O. H. Ibarra} and \textit{T. Jiang}, Theor. Comput. Sci. 57, No. 2--3, 225--238 (1988; Zbl 0646.68071)

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##### References:

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