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On Quotients of Formal Power Series
Quotient is a basic operation of formal languages, which plays a key role in
the construction of minimal deterministic finite automata (DFA) and the
universal automata. In this paper, we extend this operation to formal power
series and systemically investigate its implications in the study of weighted
automata. In particular, we define two quotient operations for formal power
series that coincide when calculated by a word. We term the first operation as
(left or right) \emph{quotient}, and the second as (left or right)
\emph{residual}. To support the definitions of quotients and residuals, the
underlying semiring is restricted to complete semirings or complete
c-semirings. Algebraical properties that are similar to the classical case are
obtained in the formal power series case. Moreover, we show closure properties,
under quotients and residuals, of regular series and weighted context-free
series are similar as in formal languages. Using these operations, we define
for each formal power series two weighted automata and . Both weighted automata accepts , and is the minimal
deterministic weighted automaton of . The universality of is
justified and, in particular, we show that is a sub-automaton of
. Last but not least, an effective method to construct the
universal automaton is also presented in this paper.Comment: 48 pages, 3 figures, 30 conference