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Normal forms for unary probabilistic automata

Published online by Cambridge University Press:  02 August 2012

Maria Paola Bianchi  and
Giovanni Pighizzini
Maria Paola Bianchi
Affiliation:
Dipartimento di Informatica, Università degli Studi di Milano, Via Comelico 39, 20135 Milano, Italy. bianchi@di.unimi.it, pighizzini@di.unimi.it
Giovanni Pighizzini
Affiliation:
Dipartimento di Informatica, Università degli Studi di Milano, Via Comelico 39, 20135 Milano, Italy. bianchi@di.unimi.it, pighizzini@di.unimi.it
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Abstract

We investigate the possibility of extending Chrobak normal form to the probabilisticcase. While in the nondeterministic case a unary automaton can be simulated by anautomaton in Chrobak normal form without increasing the number of the states in thecycles, we show that in the probabilistic case the simulation is not possible by keepingthe same number of ergodic states. This negative result is proved by considering thenatural extension to the probabilistic case of Chrobak normal form, obtained by replacingnondeterministic choices with probabilistic choices. We then propose a different kind ofnormal form, namely, cyclic normal form, which does not suffer from the same problem: weprove that each unary probabilistic automaton can be simulated by a probabilisticautomaton in cyclic normal form, with at most the same number of ergodic states. In thenondeterministic case there are trivial simulations between Chrobak normal form and cyclicnormal form, preserving the total number of states in the automata and in theircycles.

Keywords

Unary languagesnormal formprobabilistic automata

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References

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