# 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

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topBianchi, 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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- C. Mereghetti, B. Palano and G. Pighizzini, Note on the succinctness of deterministic, nondeterministic, probabilistic and quantum finite automata. RAIRO–Theor. Inf. Appl.5 (2001) 477–490. Zbl1010.68068
- 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
- A. Paz, Introduction to Probabilistic Automata. Academic Press, New York (1971). Zbl0234.94055
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