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Stochastic Semantics

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Bozapalidis S. , Rahonis G.

Fundamenta Informaticae

2017

tom Vol. 150, nr 2

143--177

We consider stochastic systems of equations of tree series, i.e., systems of equations whose right-hand sides are stochastic tree polynomials. We obtain their least solutions in arbitrary substochastic algebras, using both the [IO]- and OI-substitution mode. In the term algebra, we show that the consistency problem of the least [IO]- and OI-solutions is decidable, by reducing it to the consistency problem of stochastic context-free grammars. We prove a Kleene type theorem for the components of the least OI-solutions. The folklore Mezei-Wright result stating the coincidence of the components of least OI-solutions and behaviors of tree automata fails in the stochastic setup. As an application of our theory, we prove a Kleene theorem for the class of series generated by stochastic context-free grammars.