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Automaticity. II: Descriptional complexity in the unary case. (English) Zbl 0959.11015
Summary: Let $$\Sigma$$ and $$\Delta$$ be finite alphabets, and let $$f$$ be a map from $$\Sigma^{*}$$ to $$\Delta$$. Then the deterministic automaticity of $$f$$, $$A_{f}(n)$$, is defined to be the size of the minimum finite-state machine that correctly computes $$f$$ on all inputs of size $$<n$$. A similar definition applies to languages $$L$$. We denote the nondeterministic analogue (for languages $$L)$$ of automaticity by $$N_{L}(n)$$. In a previous paper, J. Shallit and Yu. Breitbart [J. Comput. Syst. Sci. 51, 10-25 (1996; Zbl 0859.68059)] examined the properties of this measure of descriptional complexity in the case $$|\Sigma |\geqslant 2$$.
In this paper, we continue the study of automaticity, focusing on the case where $$|\Sigma |=1$$. We prove that $$A_{f}(n)<n+1-\lfloor \log_{\ell} n\rfloor$$, where $$\ell =|\Delta |$$. We also prove that $$A_{f}(n)>n-2 \log_{\ell} n-2 \log_{\ell}\log_{\ell} n$$ for almost all functions $$f$$. In the nondeterministic case, we show that there exists a $$c$$ such that for almost all unary languages $$L$$, we have $$N_{L}(n)>cn/\log n$$ for all sufficiently large $$n$$. The proof is based on a new enumeration method for languages accepted by unary $$q$$-state NFAs. If $$L$$ is not a regular language, then it follows from a result of Karp that $$\limsup_{n\rightarrow \infty}A_{L}(n)/n\geqslant \frac{1}{2}$$. We conjecture that if $$L\subseteq 0^{*}$$, then this bound can be improved to $$(5-1)/2$$. Finally, we give some lower bounds for nondeterministic automaticity for nonregular languages. For Parts III and IV, cf. Comput. Complexity 7, 371-387 (1998); J. Théor. Nombres Bordx. 8, 347-367 (1996; Zbl 0876.11010).

##### MSC:
 11B85 Automata sequences 68Q45 Formal languages and automata 68R15 Combinatorics on words
##### Keywords:
deterministic automaticity; languages
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##### References:
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