The rational skimming theorem (original) (raw)

On Quotients of Formal Power Series

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.

Super-state automata and rational trees

Lecture Notes in Computer Science, 1998

We introduce the notion of super-state automaton constructed from another automaton. This construction is used to solve an open question about enumerative sequences of leaves of rational trees. We prove that any N-rational sequence s = (s n) n 0 of nonnegative numbers satisfying the Kraft inequality P n 0 s n k n 1 is the enumerative sequence of leaves by height of a k-ary rational tree. This result had been conjectured and was known only in the case of strict inequality. We also give new proofs, based on the notion of super-state automata, to the following known result about enumerative sequences of nodes in trees: any N-rational series t that has a primitive linear representation, such that t 0 = 1, 8n 1; t n kt n 1 , and whose convergence radius is strictly greater than 1=k, is the enumerative sequence of nodes by height in a k-ary rational tree.

On the Equivalence of mathbbZ{\mathbb Z}mathbbZ -Automata

Automata, Languages and Programming, 2005

We prove that two automata with multiplicity in Z are equivalent, i.e. define the same rational series, if and only if there is a sequence of Z-coverings, co-Z-coverings, and circulations of −1, which transforms one automaton into the other. Moreover, the construction of these transformations is effective. This is obtained by combining two results: the first one relates coverings to conjugacy of automata, and is modeled after a theorem from symbolic dynamics; the second one is an adaptation of Schützenberger's reduction algorithm of representations in a field to representations in an Euclidean domain (and thus in Z).

The Kleene–Schützenberger Theorem for Formal Power Series in Partially Commuting Variables

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.

Rational 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.

A construction on finite automata that has remained hidden

Theoretical Computer Science, 1998

We show how a construction on matrix representations of two tape automata proposed by Schiitzenberger to prove that rational functions are unambiguous can be given a central rble in the theory of relations and functions realized by finite automata, in such a way that the other basic results such as the "Cross-Section Theorem", its dual the theorem of rational uniformisation, or the decomposition theorem of rational functions into sequential functions, appear as direct and formal consequences of it.

On recognizable and rational formal power series in partially commuting variables

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.