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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
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