# Closure properties of hyper-minimized automata

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2011)

- Volume: 45, Issue: 4, page 459-466
- ISSN: 0988-3754

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topSzepietowski, Andrzej. "Closure properties of hyper-minimized automata." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 45.4 (2011): 459-466. <http://eudml.org/doc/273014>.

@article{Szepietowski2011,

abstract = {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].},

author = {Szepietowski, Andrzej},

journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},

keywords = {finite state automata; regular languages; hyper-minimized automata},

language = {eng},

number = {4},

pages = {459-466},

publisher = {EDP-Sciences},

title = {Closure properties of hyper-minimized automata},

url = {http://eudml.org/doc/273014},

volume = {45},

year = {2011},

}

TY - JOUR

AU - Szepietowski, Andrzej

TI - Closure properties of hyper-minimized automata

JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

PY - 2011

PB - EDP-Sciences

VL - 45

IS - 4

SP - 459

EP - 466

AB - 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].

LA - eng

KW - finite state automata; regular languages; hyper-minimized automata

UR - http://eudml.org/doc/273014

ER -

## References

top- [1] A. Badr, V. Geffert and I. Shipman, Hyper-minimizing minimized deterministic finite state automata. RAIRO-Theor. Inf. Appl. 43 (2009) 69–94. Zbl1170.68023MR2483445
- [2] T.H. Cormen, C.E. Leiserson, R.L. Rivest and C. Stein, Introduction to Algorithms, 2nd edition. MIT Press and McGraw-Hill (2001). Zbl1158.68538MR1848805
- [3] M. Holzer and A. Maletti, An n log n algorithm for hyper-minimizing a (minimized) deterministic automaton. Theoret. Comput. Sci.411 (2010) 3404–3413. Zbl1206.68173MR2723794
- [4] G. Jiraskova and J. Sebej, Note on reversal of binary regular languages, in Proc. of DCFS 2011. Lect. Notes Comput. Sci. 6808 (2011) 212–221. Zbl05934417MR2910378

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