On transductions of formal power series over complete semirings

AbstractRational and pushdown transductions of formal languages are generalized to formal power series with coefficients in a complete semiring. A characterization similar to Nivat's Theorem is given. Commutativity requirements for the coefficients are especially studied

Algebraic Systems and Pushdown Automata

The theory of algebraic power series in noncommuting variables, as we un-derstand it today, was initiated in [2] and developed in its early stages by the French school. The main motivation was the interconnection with context-free grammars: the defining equations were made to correspond to context-fre

Survey : Weighted extended top-down tree transducers part I. : basics and expressive power

Weighted extended top-down tree transducers (transducteurs généralisés descendants [Arnold, Dauchet: Bi-transductions de forêts. ICALP'76. Edinburgh University Press, 1976]) received renewed interest in the field of Natural Language Processing, where they are used in syntax-based machine translation. This survey presents the foundations for a theoretical analysis of weighted extended top-down tree transducers. In particular, it discusses essentially complete semirings, which are a novel concept that can be used to lift incomparability results from the unweighted case to the weighted case even in the presence of infinite sums. In addition, several equivalent ways to define weighted extended top-down tree transducers are presented and the individual benefits of each presentation is shown on a small result

Statistical language models within the algebra of weighted rational languages

Statistical language models are an important tool in natural language processing. They represent prior knowledge about a certain language which is usually gained from a set of samples called a corpus. In this paper, we present a novel way of creating N-gram language models using weighted finite automata. The construction of these models is formalised within the algebra underlying weighted finite automata and expressed in terms of weighted rational languages and transductions. Besides the algebra we make use of five special constant weighted transductions which rely only on the alphabet and the model parameter N. In addition, we discuss efficient implementations of these transductions in terms of virtual constructions

Quantifiers on languages and codensity monads

This paper contributes to the techniques of topo-algebraic recognition for languages beyond the regular setting as they relate to logic on words. In particular, we provide a general construction on recognisers corresponding to adding one layer of various kinds of quantifiers and prove a corresponding Reutenauer-type theorem. Our main tools are codensity monads and duality theory. Our construction hinges on a measure-theoretic characterisation of the profinite monad of the free S-semimodule monad for finite and commutative semirings S, which generalises our earlier insight that the Vietoris monad on Boolean spaces is the codensity monad of the finite powerset functor.Comment: 30 pages. Presentation improved and details of several proofs added. The main results are unchange

Weighted Automata and Monadic Second Order Logic

Let S be a commutative semiring. M. Droste and P. Gastin have introduced in 2005 weighted monadic second order logic WMSOL with weights in S. They use a syntactic fragment RMSOL of WMSOL to characterize word functions (power series) recognizable by weighted automata, where the semantics of quantifiers is used both as arithmetical operations and, in the boolean case, as quantification. Already in 2001, B. Courcelle, J.Makowsky and U. Rotics have introduced a formalism for graph parameters definable in Monadic Second order Logic, here called MSOLEVAL with values in a ring R. Their framework can be easily adapted to semirings S. This formalism clearly separates the logical part from the arithmetical part and also applies to word functions. In this paper we give two proofs that RMSOL and MSOLEVAL with values in S have the same expressive power over words. One proof shows directly that MSOLEVAL captures the functions recognizable by weighted automata. The other proof shows how to translate the formalisms from one into the other.Comment: In Proceedings GandALF 2013, arXiv:1307.416

Weighted Operator Precedence Languages

In the last years renewed investigation of operator precedence languages (OPL) led to discover important properties thereof: OPL are closed with respect to all major operations, are characterized, besides the original grammar family, in terms of an automata family (OPA) and an MSO logic; furthermore they significantly generalize the well-known visibly pushdown languages (VPL). In another area of research, quantitative models of systems are also greatly in demand. In this paper, we lay the foundation to marry these two research fields. We introduce weighted operator precedence automata and show how they are both strict extensions of OPA and weighted visibly pushdown automata. We prove a Nivat-like result which shows that quantitative OPL can be described by unweighted OPA and very particular weighted OPA. In a Büchi-like theorem, we show that weighted OPA are expressively equivalent to a weighted MSO-logic for OPL