Abstract
We introduce a new measure of descriptional complexity on finite automata, called the number of active states. Roughly speaking, the number of active states of an automaton A on input w counts the number of different states visited during the most economic computation of the automaton A for the word w. This concept generalizes to finite automata and regular languages in a straightforward way. We show that the number of active states of both finite automata and regular languages is computable, even with respect to nondeterministic finite automata. We further compare the number of active states to related measures for regular languages. In particular, we show incomparability to the radius of regular languages and that the difference between the number of active states and the total number of states needed in finite automata for a regular language can be of exponential order.
M. Holzer—Part of the work was done while the author was at Institut für Informatik, Technische Universität München, Arcisstraße 21, 80290 München, Germany and at Institut für Informatik, Technische Universität München, Boltzmannstraße 3, 85748 Garching bei München, Germany.
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Notes
- 1.
A set \(S=\{\,(x_i,y_i) \mid 1 \le i \le n\,\}\) is an extended fooling set of size n for the regular language \(L\subseteq \varSigma ^*\), if (i) \(x_iy_i\in L\) for \(1 \le i \le n\), and (ii) \(i\ne j\) implies \(x_i y_j\not \in L\) or \(x_jy_i\not \in L\), for \(1\le i,j\le n\). Then any NFA accepting language L has at least n states [1].
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Bordihn, H., Holzer, M. (2017). On the Number of Active States in Deterministic and Nondeterministic Finite Automata. In: Carayol, A., Nicaud, C. (eds) Implementation and Application of Automata. CIAA 2017. Lecture Notes in Computer Science(), vol 10329. Springer, Cham. https://doi.org/10.1007/978-3-319-60134-2_4
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DOI: https://doi.org/10.1007/978-3-319-60134-2_4
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