Normal forms for unary probabilistic automata

Maria Paola Bianchi; Giovanni Pighizzini

RAIRO - Theoretical Informatics and Applications (2012)

  • Volume: 46, Issue: 4, page 495-510
  • ISSN: 0988-3754

Abstract

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We investigate the possibility of extending Chrobak normal form to the probabilistic case. While in the nondeterministic case a unary automaton can be simulated by an automaton in Chrobak normal form without increasing the number of the states in the cycles, we show that in the probabilistic case the simulation is not possible by keeping the same number of ergodic states. This negative result is proved by considering the natural extension to the probabilistic case of Chrobak normal form, obtained by replacing nondeterministic choices with probabilistic choices. We then propose a different kind of normal form, namely, cyclic normal form, which does not suffer from the same problem: we prove that each unary probabilistic automaton can be simulated by a probabilistic automaton in cyclic normal form, with at most the same number of ergodic states. In the nondeterministic case there are trivial simulations between Chrobak normal form and cyclic normal form, preserving the total number of states in the automata and in their cycles.

How to cite

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Bianchi, Maria Paola, and Pighizzini, Giovanni. "Normal forms for unary probabilistic automata." RAIRO - Theoretical Informatics and Applications 46.4 (2012): 495-510. <http://eudml.org/doc/277831>.

@article{Bianchi2012,
abstract = {We investigate the possibility of extending Chrobak normal form to the probabilistic case. While in the nondeterministic case a unary automaton can be simulated by an automaton in Chrobak normal form without increasing the number of the states in the cycles, we show that in the probabilistic case the simulation is not possible by keeping the same number of ergodic states. This negative result is proved by considering the natural extension to the probabilistic case of Chrobak normal form, obtained by replacing nondeterministic choices with probabilistic choices. We then propose a different kind of normal form, namely, cyclic normal form, which does not suffer from the same problem: we prove that each unary probabilistic automaton can be simulated by a probabilistic automaton in cyclic normal form, with at most the same number of ergodic states. In the nondeterministic case there are trivial simulations between Chrobak normal form and cyclic normal form, preserving the total number of states in the automata and in their cycles.},
author = {Bianchi, Maria Paola, Pighizzini, Giovanni},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Unary languages; normal form; probabilistic automata; unary languages},
language = {eng},
month = {11},
number = {4},
pages = {495-510},
publisher = {EDP Sciences},
title = {Normal forms for unary probabilistic automata},
url = {http://eudml.org/doc/277831},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Bianchi, Maria Paola
AU - Pighizzini, Giovanni
TI - Normal forms for unary probabilistic automata
JO - RAIRO - Theoretical Informatics and Applications
DA - 2012/11//
PB - EDP Sciences
VL - 46
IS - 4
SP - 495
EP - 510
AB - We investigate the possibility of extending Chrobak normal form to the probabilistic case. While in the nondeterministic case a unary automaton can be simulated by an automaton in Chrobak normal form without increasing the number of the states in the cycles, we show that in the probabilistic case the simulation is not possible by keeping the same number of ergodic states. This negative result is proved by considering the natural extension to the probabilistic case of Chrobak normal form, obtained by replacing nondeterministic choices with probabilistic choices. We then propose a different kind of normal form, namely, cyclic normal form, which does not suffer from the same problem: we prove that each unary probabilistic automaton can be simulated by a probabilistic automaton in cyclic normal form, with at most the same number of ergodic states. In the nondeterministic case there are trivial simulations between Chrobak normal form and cyclic normal form, preserving the total number of states in the automata and in their cycles.
LA - eng
KW - Unary languages; normal form; probabilistic automata; unary languages
UR - http://eudml.org/doc/277831
ER -

References

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  10. M. Milani and G. Pighizzini, Tight bounds on the simulation of unary probabilistic automata by deterministic automata. J. Automata, Languages and Combinatorics6 (2001) 481–492.  Zbl1050.68095
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  14. A.W. To, Unary finite automata vs. arithmetic progressions. Inf. Process. Lett.109 (2009) 1010–1014.  Zbl1202.68241

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