Note on the succinctness of deterministic, nondeterministic, probabilistic and quantum finite automata
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 35 (2001) no. 5, pp. 477-490.

We investigate the succinctness of several kinds of unary automata by studying their state complexity in accepting the family $\left\{{L}_{m}\right\}$ of cyclic languages, where ${L}_{m}=\left\{{a}^{km}\phantom{\rule{0.277778em}{0ex}}|\phantom{\rule{0.277778em}{0ex}}k\in ℕ\right\}$. In particular, we show that, for any $m$, the number of states necessary and sufficient for accepting the unary language ${L}_{m}$ with isolated cut point on one-way probabilistic finite automata is ${p}_{1}^{{\alpha }_{1}}+{p}_{2}^{{\alpha }_{2}}+\cdots +{p}_{s}^{{\alpha }_{s}}$, with ${p}_{1}^{{\alpha }_{1}}{p}_{2}^{{\alpha }_{2}}\cdots {p}_{s}^{{\alpha }_{s}}$ being the factorization of $m$. To prove this result, we give a general state lower bound for accepting unary languages with isolated cut point on the one-way probabilistic model. Moreover, we exhibit one-way quantum finite automata that, for any $m$, accept ${L}_{m}$ with isolated cut point and only two states. These results are settled within a survey on unary automata aiming to compare the descriptional power of deterministic, nondeterministic, probabilistic and quantum paradigms.

Classification: 68Q10,  68Q19,  68Q45
Keywords: deterministic, nondeterministic, probabilistic, quantum unary finite automata
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author = {Mereghetti, Carlo and Palano, Beatrice and Pighizzini, Giovanni},
title = {Note on the succinctness of deterministic, nondeterministic, probabilistic and quantum finite automata},
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Mereghetti, Carlo; Palano, Beatrice; Pighizzini, Giovanni. Note on the succinctness of deterministic, nondeterministic, probabilistic and quantum finite automata. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 35 (2001) no. 5, pp. 477-490. http://www.numdam.org/item/ITA_2001__35_5_477_0/

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