The theory of two-dimensional languages, generalizing formal string languages, was motivated by problems arising from image processing and models of parallel computing. Weighted automata and series over pictures map pictures to some semiring and provide an extension to a quantitative setting. We establish a notion of a weighted MSO logics over pictures. The semantics of a weighted formula will be a picture series. We introduce weighted 2-dimensional online tessellation automata (W2OTA) extending the common automata-theoretic model for picture languages. We prove that the class of picture series defined by sentences of the weighted logics coincides with the family of picture series that are computable by W2OTA. Moreover, behaviours of W2OTA coincide precisely with the recognizable picture series characterized in [18].

Weighted picture automata and weighted logics

Review: C. C. Elgot, Decision Problems of Finite Automata Design and Related Arithmetics

Graphs have a variety of uses in natural language processing, particularly as representations of linguistic meaning. A deficit in this area of research is a formal framework for creating, combining ...

/pdf/weighted-dag-automata-for-semantic-graphs-589hcmxt1o.pdf

Weighted DAG Automata for Semantic Graphs

In this paper we analyze several models of 1-way quantum finite automata, in the light of formal power series theory. In this general context, we recall two well known constructions, by proving: 1. Languages generated with isolated cut-point by a class of bounded rational formal series are regular. 2. If a class of formal series is closed under f-complement, Hadamard product and convex linear combination, then the class of languages generated with isolated cut-point is closed under boolean operations. We introduce a general model of 1-way quantum automata and we compare their behaviors with those of measure-once, measure-many and reversible 1-way quantum automata.

Developments in Language Theory

Ce manuscrit regroupe differents travaux explorant les interactions entre les tenseurs et l'apprentissage automatique. Le premier chapitre est consacre a l'extension des modeles de series reconnaissables de chaines et d'arbres aux graphes. Nous y montrons que les modeles d'automates ponderes de chaines et d'arbres peuvent etre interpretes d'une maniere simple et unifiee a l'aide de reseaux de tenseurs, et que cette interpretation s'etend naturellement aux graphes ; nous etudions certaines proprietes de ce modele et presentons des resultats preliminaires sur leur apprentissage. Le second chapitre porte sur la minimisation approximee d'automates ponderes d'arbres et propose une approche theoriquement fondee a la problematique suivante : etant donne un automate pondere d'arbres a n etats, comment trouver un automate a m<n etats calculant une fonction proche de l'originale. Le troisieme chapitre traite de la regression de faible rang pour sorties a structure tensorielle. Nous y proposons un algorithme d'apprentissage rapide et efficace pour traiter un probleme de regression dans lequel les sorties des tenseurs. Nous montrons que l'algorithme propose est un algorithme d'approximation pour ce probleme NP-difficile et nous donnons une analyse theorique de ses proprietes statistiques et de generalisation. Enfin, le quatrieme chapitre introduit le modele de melanges algebriques de distributions. Ce modele considere des combinaisons affines de distributions (ou les coefficients somment a un mais ne sont pas necessairement positifs). Nous proposons une approche pour l'apprentissage de melanges algebriques qui etend la methode tensorielle des moments introduite recemment. .

A tensor perspective on weighted automata, low-rank regression and algebraic mixtures

Weighted automata model quantitative features of the behavior of systems and have been investigated for various structures like words, trees, traces, pictures, and nested words. In this paper, we introduce a general model of weighted automata acting on graphs, which form a quantitative version of Thomas’ unweighted model of graph acceptors. We derive a Nivat theorem for weighted graph automata which shows that their behaviors are precisely those obtainable from very particular weighted graph automata and unweighted graph acceptors with a few simple operations. We also show that a suitable weighted MSO logic is expressively equivalent to weighted graph automata. As a consequence, we obtain corresponding Buchi-type equivalence results known from the recent literature for weighted automata and weighted logics on words, trees, pictures, and nested words. Establishing such a general result has been an open problem for weighted logic for some time.

https://www.informatik.uni-leipzig.de/~dueck/DrosteDueckMFCS2015.pdf

Weighted Automata and Logics on Graphs

Nested words introduced by Alur and Madhusudan are used to capture structures with both linear and hierarchical order, e.g. XML documents, without losing valuable closure properties. Furthermore, Alur and Madhusudan introduced automata and equivalent logics for both finite and infinite nested words, thus extending B\"uchi's theorem to nested words. Recently, average and discounted computations of weights in quantitative systems found much interest. Here, we will introduce and investigate weighted automata models and weighted MSO logics for infinite nested words. As weight structures we consider valuation monoids which incorporate average and discounted computations of weights as well as the classical semirings. We show that under suitable assumptions, two resp. three fragments of our weighted logics can be transformed into each other. Moreover, we show that the logic fragments have the same expressive power as weighted nested word automata.

Weighted Automata and Logics for Infinite Nested Words

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 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 B\"uchi-like theorem, we show that weighted OPA are expressively equivalent to a weighted MSO-logic for OPL.

Weighted Operator Precedence Languages

Nested words introduced by Alur and Madhusudan are used to capture structures with both linear and hierarchical order, e.g. XML documents, without losing valuable closure properties. Furthermore, Alur and Madhusudan introduced automata and equivalent logics for both finite and infinite nested words, thus extending Buchi's theorem to nested words. Recently, average and discounted computations of weights in quantitative systems found much interest. Here, we will introduce and investigate weighted automata models and weighted MSO logics for infinite nested words. As weight structures we consider valuation monoids which incorporate average and discounted computations of weights as well as the classical semirings. We show that under suitable assumptions, two resp. three fragments of our weighted logics can be transformed into each other. Moreover, we show that the logic fragments have the same expressive power as weighted nested word automata.

Stefan Dück

Papers

Weighted Automata and Logics on Graphs

Weighted Automata and Logics for Infinite Nested Words

Weighted Operator Precedence Languages

Weighted Automata and Logics for Infinite Nested Words

Weighted automata and logics for infinite nested words