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Journal of Automata, Languages and Combinatorics
formerly:
Journal of Information Processing and Cybernetics /
Elektronische Informationsverarbeitung und Kybernetik
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@article{jalc010301,
author = {Symeon Bozapalidis},
title = {Convex Algebras, Convex Modules and Formal Power Series
on Trees},
journal = jalc,
year = 1996,
volume = 1,
number = 3,
pages = {165--180},
keywords = {convex set, convex algebra, convex module,
stochstic treeseries},
abstract = {We obtain behaviors of stochastic treeautomata by means of
convex algebras. The following minimal realization result is
established: for each convexly reachable algebra ${\cal A}$,
there exists a unique convex epimorphism
${\cal A}\to \mbox{\rm CSynt}(S_{\cal A})$, where
$\mbox{\rm CSynt}(S_{\cal A})$ denotes the syntactical
convex algebra associated with the treeseries $S_{\cal A}$
computed by ${\cal A}$. Treeseries whose syntactical convex
algebra is convexly finite, are studied. Finally, the
following conditions are shown to be equivalent:\par
i) $S : T_\Sigma\to [0,1]$ is computed by a
convexly finite $\Sigma$-module,\par
ii) all right derivatives of $S$ are included into the
same convexly finite sub-$\Sigma$-module of
$[0,1]^{T_\Sigma}$,\par
iii) there exist a function $\psi : \Sigma_0\to [0,1]^n$,
a monoid morphism
$\phi : P_\Sigma\to \mbox{\rm Stoch}_{n\times n}$
compatible with $\phi$ and a vector
$\gamma\in\mbox{\rm Stoch}_n$, so that
$(S,t) = \psi(c)\phi(\tau)\gamma$ for $t = c\tau$
where $\mbox{\rm Stoch}_{n\times n}$ and
$\mbox{\rm Stoch}_n$ denote the monoid of stochastic
$n\times n$ matrices and stochastic $n$-vectors,
respectively.}
}