Abstract
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 which are called (left or right) quotients and (left or right) residuals respectively. Algebraic properties of the quotient operation and closure properties (w.r.t. regular and context-free series) under quotients and residuals are also investigated for formal power series. Using these operations, we define for each formal power series A two weighted automata that accept A. The first is the minimal deterministic weighted automaton of A, and the second is the universal weighted automaton of A. An effective algebraic method to construct the universal automaton is also presented.
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