On transformations of formal power series

Lecture Notes in Computer Science, 2001

Formal power series are an extension of formal languages. Recognizable formal power series can be captured by the so-called weighted finite automata, generalizing finite state machines. In this paper, motivated by codings of formal languages, we introduce and investigate two types of transformations for formal power series. We characterize when these transformations preserve rationality, generalizing the recent results of Zhang [16] to the formal power series setting. We show, for example, that the "square-root" operation, while preserving regularity for formal languages, preserves rationality for formal power series when the underlying semiring is commutative or locally finite, but not in general.

2012

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) quotient, and the second as (left or right) 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 M A and U A . Both weighted automata accepts A, and M A is the minimal deterministic weighted automaton of A. The universality of U A is justified and, in particular, we show that M A is a subautomaton of U A . Last but not least, an effective method to construct the universal automaton is also presented in this paper.

2009

Lecture Notes in Computer Science, 1997

We will describe the recognizable formal power series over arbitrary semirings and in partially commuting variables, i.e. over trace monoids. We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet. The converse is true if the underlying semi-ring is commutative. Moreover, if in addition the semiring is idempotent then the same result holds with a star restricted to series for which the elements in the support have connected (possibly di erent) alphabets. It is shown that these assumptions over the semiring are necessary. This provides a joint generalization of Kleene's, Sch utzenberger's and Ochma nski's theorems.

Information and Computation, 1999

Kleene's theorem on the coincidence of regular and rational languages in free monoids has been generalized by Sch tzenberger to a description of the recognizable formal power series in non-commuting variables over arbitrary semirings, and by Ochma ski to a characterization of the recognizable languages in trace monoids. We will describe the recognizable formal power series over arbitrary semirings and in partially commuting variables, i.e. over trace monoids. We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet. The converse is true if the underlying semi-ring is commutative. Moreover, if in addition the semiring is idempotent then the same result holds with a star restricted to series for which the elements in the support have connected (possibly di erent) alphabets. It is shown that these assumptions over the semiring are necessary. This provides a joint generalization of Kleene's, Sch tzenberger's and Ochma ski's theorems.