Two deterministic finite automata are almost equivalent if they disagree in acceptance only for finitely many inputs. An automaton A is hyper-minimized if no automaton with fewer states is almost equivalent to A. A regular language L is canonical if the minimal automaton accepting L is hyper-minimized. The asymptotic state complexity s∗(L) of a regular language L is the number of states of a hyper-minimized automaton for a language finitely different from L. In this paper we show that: (1) the class of canonical regular languages is not closed under: intersection, union, concatenation, Kleene closure, difference, symmetric difference, reversal, homomorphism, and inverse homomorphism; (2) for any regular languages L1 and L2 the asymptotic state complexity of their sum L1 ∪ L2, intersection L1 ∩ L2, difference L1 - L2, and symmetric difference L1 ⊕ L2 can be bounded by s∗(L1)·s∗(L2). This bound is tight in binary case and in unary case can be met in infinitely many cases. (3) For any regular language L the asymptotic state complexity of its reversal LR can be bounded by 2s∗(L). This bound is tight in binary case. (4) The asymptotic state complexity of Kleene closure and concatenation cannot be bounded. Namely, for every k ≥ 3, there exist languages K, L, and M such that s∗(K) = s∗(L) = s∗(M) = 1 and s∗(K∗) = s∗(L·M) = k. These are answers to open problems formulated by Badr et al. [RAIRO-Theor. Inf. Appl. 43 (2009) 69-94].
Mots clés : finite state automata, regular languages, hyper-minimized automata
@article{ITA_2011__45_4_459_0,
author = {Szepietowski, Andrzej},
title = {Closure properties of hyper-minimized automata},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {459--466},
publisher = {EDP-Sciences},
volume = {45},
number = {4},
year = {2011},
doi = {10.1051/ita/2011128},
mrnumber = {2876117},
zbl = {1232.68087},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ita/2011128/}
}
TY - JOUR AU - Szepietowski, Andrzej TI - Closure properties of hyper-minimized automata JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2011 SP - 459 EP - 466 VL - 45 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ita/2011128/ DO - 10.1051/ita/2011128 LA - en ID - ITA_2011__45_4_459_0 ER -
%0 Journal Article %A Szepietowski, Andrzej %T Closure properties of hyper-minimized automata %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2011 %P 459-466 %V 45 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ita/2011128/ %R 10.1051/ita/2011128 %G en %F ITA_2011__45_4_459_0
Szepietowski, Andrzej. Closure properties of hyper-minimized automata. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 45 (2011) no. 4, pp. 459-466. doi : 10.1051/ita/2011128. http://www.numdam.org/articles/10.1051/ita/2011128/
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