On Quotients of Formal Power Series

TL;DR

This paper extends the concept of quotient operations from formal languages to formal power series, exploring their algebraic properties, closure, and automaton constructions in weighted automata over complete semirings.

Contribution

It introduces two quotient operations for formal power series, establishes their algebraic properties, and constructs minimal and universal weighted automata based on these operations.

Findings

Defined quotient and residual operations for formal power series.

Proved closure properties of regular and weighted context-free series under these operations.

Presented an effective method to construct the universal automaton.

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 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 $A$ two weighted automata ${\cal M}_A$ and ${\cal U}_A$. Both weighted automata accepts $A$, and ${\cal M}_A$ is the minimal deterministic weighted automaton of $A$. The universality of ${\cal U}_A$ is justified and, in particular, we show that ${\cal M}_A$ is a sub-automaton of ${\cal U}_A$. Last but not least, an effective method to construct the universal automaton is also presented in this paper.