Abstract
We investigate unary regular languages and compare deterministic finite automata (DFA’s), nondeterministic finite automata (NFA’s) and probabilistic finite automata (PFA’s) with respect to their size.
Given a unary PFA with n states and an ε-isolated cutpoint, we show that the minimal equivalent DFA has at most \(n^{\frac{1}{2\epsilon}}\) states in its cycle. This result is almost optimal, since for any α< 1 a family of PFA’s can be constructed such that every equivalent DFA has at least \(n^{\frac{\alpha}{2\epsilon}}\) states. Thus we show that for the model of probabilistic automata with a constant error bound, there is only a polynomial blowup for cyclic languages.
Given a unary NFA with n states, we show that efficiently approximating the size of a minimal equivalent NFA within the factor \(\frac{\sqrt{n}}{\ln n}\) is impossible unless P=NP. This result even holds under the promise that the accepted language is cyclic. On the other hand we show that we can approximate a minimal NFA within the factor ln n, if we are given a cyclic unary n-state DFA.
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© 2003 Springer-Verlag Berlin Heidelberg
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Gramlich, G. (2003). Probabilistic and Nondeterministic Unary Automata. In: Rovan, B., Vojtáš, P. (eds) Mathematical Foundations of Computer Science 2003. MFCS 2003. Lecture Notes in Computer Science, vol 2747. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45138-9_40
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DOI: https://doi.org/10.1007/978-3-540-45138-9_40
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40671-6
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