Closure properties of hyper-minimized automata

Andrzej Szepietowski

RAIRO - Theoretical Informatics and Applications (2012)

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

Abstract

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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].

How to cite

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Szepietowski, Andrzej. "Closure properties of hyper-minimized automata." RAIRO - Theoretical Informatics and Applications 45.4 (2012): 459-466. <http://eudml.org/doc/221950>.

@article{Szepietowski2012,
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},
keywords = {Finite state automata; regular languages; hyper-minimized automata; finite state automata},
language = {eng},
month = {1},
number = {4},
pages = {459-466},
publisher = {EDP Sciences},
title = {Closure properties of hyper-minimized automata},
url = {http://eudml.org/doc/221950},
volume = {45},
year = {2012},
}

TY - JOUR
AU - Szepietowski, Andrzej
TI - Closure properties of hyper-minimized automata
JO - RAIRO - Theoretical Informatics and Applications
DA - 2012/1//
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; finite state automata
UR - http://eudml.org/doc/221950
ER -

References

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  1. A. Badr, V. Geffert and I. Shipman, Hyper-minimizing minimized deterministic finite state automata. RAIRO-Theor. Inf. Appl.43 (2009) 69–94.  Zbl1170.68023
  2. T.H. Cormen, C.E. Leiserson, R.L. Rivest and C. Stein, Introduction to Algorithms, 2nd edition. MIT Press and McGraw-Hill (2001).  Zbl1047.68161
  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.  Zbl1206.68173
  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. Zbl05934417

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