Closure properties of linear context-free tree languages with an application to optimality theory

Traditionally, data that has both linear and hierarchical structure, such as annotated linguistic data, is modeled using ordered trees and queried using tree automata. In this paper, we argue that nested words and automata over nested words offer a better way to capture and process the dual structure. Nested words generalize both words and ordered trees, and allow both word and tree operations. We study various classes of automata over nested words, and show that while they enjoy expressiveness and succinctness benefits over word and tree automata, their analysis complexity and closure properties are analogous to the corresponding word and tree special cases. In particular, we show that finite-state nested word automata can be exponentially more succinct than tree automata, and pushdown nested word automata include the two incomparable classes of context-free word languages and context-free tree languages.

An operation of concatenation is defined for graphs. This allows strings to be viewed as expressions denoting graphs, and string languages to be interpreted as graph languages. For a class \(K\) of string languages, \({\rm Int}(K)\) is the class of all graph languages that are interpretations of languages from \(K\). For the classes REG and LIN of regular and linear context-free languages, respectively, \({\rm Int}({\rm REG}) = {\rm Int}({\rm LIN})\). \({\rm Int}({\rm REG})\) is the smallest class of graph languages containing all singletons and closed under union, concatenation and star (of graph languages). \({\rm Int}({\rm REG})\) equals the class of graph languages generated by linear HR (= Hyperedge Replacement) grammars, and \({\rm Int}(K)\) is generated by the corresponding \(K\)-controlled grammars. Two characterizations are given of the largest class \(K'\) such that \({\rm Int}(K') = {\rm Int}(K)\). For the class CF of context-free languages, \({\rm Int}({\rm CF})\) lies properly inbetween \({\rm Int}({\rm REG})\) and the class of graph languages generated by HR grammars. The concatenation operation on graphs combines nicely with the sum operation on graphs. The class of context-free (or equational) graph languages, with respect to these two operations, is the class of graph languages generated by HR grammars.

It is shown that the simple deterministic pushdown transductions of {0,1}∗ equal the class of context free languages (by means of an elementary proof). Several new results are presented concerning the class of source languages which can be compiled into Dyck languages by a DPDA transducer, i.e., the class DPDT-1 (D 2). It is shown that this language family does not 2 have the Parikh property and is incomparable with the unambiguous context free languages, the bounded context free languages, the linear context free languages, the one-way deterministic stack automaton languages and the one-way non-deterministic stack automaton languages.

We study the problem of learning regular tree languages from text. We show that the framework of function distinguishability as introduced in our ALT 2000 paper is generalizable from the case of string languages towards tree languages, hence providing a large source of identifiable classes of regular tree languages. Each of these classes can be characterized in various ways. Moreover, we present a generic inference algorithm with polynomial update time and prove its correctness. In this way, we generalize previous works of Angluin, Sakakibara and ourselves. Moreover, we show that this way all regular tree languages can be identified approximately.

The generative capacity of combinatory categorial grammars as acceptors of tree languages is investigated. It is demonstrated that the such obtained tree languages can also be generated by simple monadic context-free tree grammars. However, the subclass of pure combinatory categorial grammars cannot even accept all regular tree languages. Additionally, the tree languages accepted by combinatory categorial grammars with limited rule degrees are characterized: If only application rules are allowed, then they can accept only a proper subset of the regular tree languages, whereas they can accept exactly the regular tree languages once first degree composition rules are permitted.

Regular tree expressions are a formalism for describing regular tree languages, which can be accepted by a finite tree automaton as a standard model of computation. It was proved that the class of regular tree languages is a proper subclass of tree languages whose linear notations can be accepted by deterministic string pushdown automata. In this paper, we present a new algorithm for transforming regular tree expressions to equivalent real-time height-deterministic pushdown automata that accept the trees in their postfix notation.

We present effective criteria for first-order definability of regular tree languages. It is known that over words the absence of modulo counting (the “noncounting property”) characterizes the expressive power of first-order logic (McNaughton, Schutzenberger), whereas non-counting regular tree languages exist which are not first-order definable. We present new conditions on regular tree languages (more precisely, on tree automata) which imply nondefinability in first-order logic. One method is based on tree homomorphisms which allow to deduce nondefinability of one tree language from nondefinability of another tree language. Additionly we introduce a structural property of tree automata (the socalled Λ-∨-patterns) which also causes tree languages to be undefinable in first-order logic. Finally, it is shown that this notion does not yet give a complete characterization of first-order logic over trees. The proofs rely on the method of Ehrenfeucht-Fraisse games.

We investigate regular tree languages exact learning from positive examples and membership queries. Input data are trees of the language to infer. The learner computes new trees from the inputs and asks to the oracle whether or not they belong to the language. From the answers, the learner may ask further membership queries until he finds the correct grammar that generates the target language. This paradigm was introduced by Angluin in the seminal work [1] for the case of regular word language. Neither negative examples, equivalence queries nor counter examples are allowed in this paradigm. We describe an efficient algorithm which is polynomial in the size of the examples for learning the whole class of regular tree languages. The convergence is insured when the set of examples contains a representative sample of the language to guess. A finite subset e of a regular tree language £ is representative for L if every transition of the minimal tree automaton for L is used at least once for the derivation of an element of the set e.

Indexed grammars are a generalization of context-free grammars and recognize a proper subset of context-sensitive languages. The class of languages recognized by indexed grammars are called indexed languages and they correspond to the languages recognized by nested stack automata. For example indexed grammars can recognize the language which is not context-free, but they cannot recognize which is context-sensitive. Indexed grammars identify a set of languages that are more expressive than context-free languages, while having decidability results that lie in between the ones of context-free and context-sensitive languages. In this work we study indexed grammars in order to formalize the relation between indexed languages and the other classes of languages in the Chomsky hierarchy. To this end, we provide a fixpoint characterization of the languages recognized by an indexed grammar and we study possible ways to abstract, in the abstract interpretation sense, these languages and their grammars into context-free and regular languages.

Recently, the question has been raised whether the derivation tree languages of Minimalist grammars (MGs; [14, 16]) are closed under intersection with regular tree languages [4, 5]. Using a variation of a proof technique devised by Thatcher [17], I show that even though closure under intersection does not obtain, it holds for every MG and regular tree language that their intersection is identical to the derivation tree language of some MG modulo category labels. It immediately follows that the same closure property holds with respect to union, relative complement, and certain kinds of linear transductions. Moreover, enriching MGs with the ability to put regular constraints on the shape of their derivation trees does not increase the formalism’s weak generative capacity. This makes it straightforward to implement numerous linguistically motivated constraints on the Move operation.KeywordsMinimalist GrammarsDerivation Tree LanguagesClosure PropertiesRegular Tree LanguagesDerivational Constraints

The idea for this special issue came up during the preparations of the International Workshop on Finite-State Methods in Natural Language Processing, that was held at Bilkent University in Ankara, Turkey in the summer of 1998. The number of the submissions had exceeded our initial expectations and we were able to select quite a good set of papers from those submitted. Further, the workshop and the preceding tutorial by Kenneth Beesley, on finite-state methods, was attended by quite a large number of participants. This led us to believe that interest in the theory and applications of finitestate machinery was alive and well, and that some of the papers from this workshop along with further additional submissions could make a very good special issue for this journal. The five papers in this issue are the result of this process. The last decade has seen a quite a substantial surge in the use of finite-state methods in all aspects of natural language applications. Fueled by the theoretical contributions of Kaplan and Kay (1994), Mohri’s recent contributions on the use of finite-state techniques in various NLP problems (Mohri 1996, 1997), the success of finite-state approaches especially in computational morphology, for example, Koskenniemi (1983), Karttunen (1983), and Karttunen, Kaplan, and Zaenen (1992), and, finally, the availability of state-of-the-art tools for building and manipulating large-scale finite-state systems (Karttunen 1993; Karttunen and Beesley 1992; Karttunen et al. 1996; Mohri, Pereira, and Riley 1998; van Noord 1999), recent years have seen many successful applications of finite-state approaches in tagging, spell checking, information extraction, parsing, speech recognition, and text-to-speech applications. This is a remarkable comeback considering that in the dawn of modern linguistics (Chomsky 1957), finitestate grammars were dismissed as fundamentally inadequate. As a result, most of the work in computational linguistics in the past few decades has been focused on far more powerful formalisms. Recent publications on finite-state technology include two collections of papers (Roche and Schabes 1997; Kornai 1999) with contributions covering a wide range of these topics. This special issue, we hope, will add to these contributions. The five papers in this collection cover many aspects of finite-state theory and applications. The papers Treatment of Epsilon Moves in Subset Construction by van Noord and Incremental Construction of Minimal Acyclic Finite-State Automata and Transducers by Daciuk, Watson, Watson, and Mihov, address two fundamental aspects in the construction of finite-state recognizers. Van Noord presents results for various methods for producing a deterministic automaton with no epsilon transitions from a nondeterministic automaton with a large number of epsilon transitions, especially those resulting from finite-state approximations of context-free and more powerful formalisms. Daciuk et al. present a new method for constructing minimal, deterministic, acyclic

We show that the class of string languages generated by linear context-free rewriting systems is equal to the class of output languages of deterministic tree-walking transducers. From equivalences that have previously been established we know that this class of languages is also equal to the string languages generated by context-free hypergraph grammars, multicomponent tree-adjoining grammars, and multiple context-free grammars and to the class of yields of images of the regular tree languages under finite-copying top-down tree transducers.

Bounded nesting in macro grammars

We consider the framework of regular tree model checking where sets of configurations of a system are represented by regular tree languages and its dynamics is modeled by a term rewriting system (or a regular tree transducer). We focus on the computation of the reachability set R*(L) where R is a regular tree transducer and L is a regular tree language. The construction of this set is not possible in general. Therefore, we present a general acceleration technique, called regular tree widening which allows to speed up the convergence of iterative fixpoint computations in regular tree model checking. This technique can be applied uniformly to various kinds of transformations. We show the application of our framework to different analysis contexts: verification of parameterized tree networks and data-flow analysis of multithreaded programs. Parametrized networks are modeled by relabeling tree transducers, and multithreaded programs are modeled by term rewriting rules encoding transformations on control structures. We prove that our widening technique can emulate many existing algorithms for special classes of transformations and we show that it can deal with transformations beyond the scope of these algorithms.

Degree-languages: A new concept of acceptance