Heiko Vogler

Handbook of Weighted Automata

Weighted finite automata are classical nondeterministic finite automata in which the transitions carry weights. These weights may model, for example, the cost involved when executing a transition, the resources or time needed for this, or the probability or reliability of its successful execution. Weights can also be added to classical automata with infinite state sets like pushdown automata, and this extension constitutes the general concept of weighted automata. Since their introduction in the 1960s they have stimulated research in related areas of theoretical computer science, including formal language theory, algebra, logic, and discrete structures. Moreover, weighted automata and weighted context-free grammars have found application in natural-language processing, speech recognition, and digital image compression. This book covers all the main aspects of weighted automata and formal power series methods, ranging from theory to applications. The contributors are the leading experts in their respective areas, and each chapter presents a detailed survey of the state of the art and pointers to future research. The chapters in Part I cover the foundations of the theory of weighted automata, specifically addressing semirings, power series, and fixed point theory. Part II investigates different concepts of weighted recognizability. Part III examines alternative types of weighted automata and various discrete structures other than words. Finally, Part IV deals with applications of weighted automata, including digital image compression, fuzzy languages, model checking, and natural-language processing. Computer scientists and mathematicians will find this book an excellent survey and reference volume, and it will also be a valuable resource for students exploring this exciting research area.

Macro tree transducers

Macro tree transducers are a combination of top-down tree transducers and macro grammars. They serve as a model for syntax-directed semantics in which context information can be handled. In this paper the formal model of macro tree transducers is studied by investigating typical automata theoretical topics like composition, decomposition, domains, and ranges of the induced translation classes. The extension with regular look-ahead is considered.

Attributed Tree Transducers

In this chapter we introduce the third formal model of syntax-directed semantics: the attributed tree transducer. It is created by abstraction from attribute grammars. It can also handle context information like a macro tree transducer, and in this sense it too is more powerful than a top-down tree transducer. However, attributed tree transducers treat context information in an explicit way rather than implicitly as macro tree transducers do.

Weighted Tree Automata and Tree Transducers

We survey some important results for weighted tree automata and weighted tree transducers over finite ranked trees and semirings as weight structure. In particular, we address closure properties of the class of recognizable tree series, results on the support of such tree series, the determinization of weighted tree automata, pumping lemmata and decidability results, and finite algebraic characterizations of recognizable tree series. We discuss the equivalence between recognizable tree series and equational, rational, and MSO-definable tree series, and we present a comparison of several other models of recognizability. For weighted tree transducers we show composition and decomposition results, an inclusion diagram of some fundamental classes of tree series transformations, and hierarchies obtained by composing weighted tree transducers. We also discuss other models of weighted tree transducers.

Weighted tree automata and weighted logics

We define a weighted monadic second order logic for trees where the weights are taken from a commutative semiring. We prove that a restricted version of this logic characterizes the class of formal tree series which are accepted by weighted bottom-up finite state tree automata. The restriction on the logic can be dropped if additionally the semiring is locally finite. This generalizes corresponding classical results of Thatcher, Wright, and Doner for tree languages and it extends recent results of Droste and Gastin [Weighted automata and weighted logics, in: Automata, Languages and Programrning--32nd International Colloquium, ICALP 2005, Lisbon, Portugal, 2005, Proceedings, Lecture Notes in Computer Science, Vol. 3580, Springer, Berlin, 2005, pp. 513-525, full version in Theoretical Computer Science, to appear.] from formal power series on words to formal tree series.

Pushdown machines for the macro tree transducer

The macro tree transducer can be considered as a system of recursive function procedures with parameters, where the recursion is on a tree (e.g., the syntax tree of a program). We investigate characterizations of the class of tree (tree-to-string) translations which is induced by macro tree transducers (macro tree-to-string transducers, respectively). For this purpose we define several pushdown machines of which the control is recursive without parameters, or even iterative, and which work on a generalized pushdown as storage. Because of the relevance for semantics of programming languages, we stress (besides the nondeterministic case) the study of machines for the total deterministic macro tree(-to-string) transducer, which translates every input tree into exactly one output tree (string, respectively). Finally, we characterize the n-fold composition of total deterministic macro tree transducers by recursive pushdown machines with an iterated pushdown as storage, which is a pushdown of pushdowns of … of pushdowns.

A Kleene Theorem for Weighted Tree Automata

Abstract In this paper we prove Kleene’s result for formal tree series over a commutative semiring A (which is not necessarily complete or continuous or idempotent), i.e., the class of formal tree series over A which are accepted by weighted tree automata, and the class of rational tree series over A are equal. We show the result by direct automata-theoretic constructions and prove their correctness.

Weighted finite automata over strong bimonoids

We investigate weighted finite automata over strings and strong bimonoids. Such algebraic structures satisfy the same laws as semirings except that no distributivity laws need to hold. We define two different behaviors and prove precise characterizations for them if the underlying strong bimonoid satisfies local finiteness conditions. Moreover, we show that in this case the given weighted automata can be determinized.

High level tree transducers and iterated pushdown tree transducers

Summaryn-level tree transducers (n≧0) combine the features ofn-level tree grammars and of top-down tree transducers in the sense that the derivations of the tree grammars are syntax-directed by input trees. For runningn, the sequence ofn-level tree transducers starts with top-down tree transducers (n=0) and macro tree transducers (n=1). In this paper the class of tree-to-tree translations computed byn-level tree transducers is characterized byn-iterated pushdown tree transducers. Such a transducer can be considered as a regular tree grammar of which the derivations are syntax-directed byn-iterated pushdowns of trees; ann-iterated pushdown of trees is a pushdown of pushdowns of ... of pushdowns (n times) of trees. In particular, we investigate the total deterministic case, which is relevant for syntax-directed semantics of programming languages.

Fundamental study: modular tree transducers

Abstract A new tree transducer, called a modular tree transducer, is introduced. This device specifies operations on trees and can be considered as a formalization of the concept of nested simultaneous primitive recursion on trees. Roughly speaking, a modular tree transducer is a special left-linear and non-overlapping term rewriting system of which the set of rules is partitioned into modules, each module being equipped with a non-negative integer: the number of the module. Then, a module with number i may call modules with numbers not less than i . Three properties are proved: (1) modular tree transducers compute exactly the (inductively defined) class of primitive recursive functions on trees; (2) the number of modules in modular tree transducers induces a strict hierarchy on the class of all modular tree transductions; and (3) by appropriately restricting the calling structure between modules, modular tree transducers characterize the compositions of macro tree transducers where the number of modules and the number of compositions coincide.