Logics for Unranked Trees: An Overview

Labeled unranked trees are used as a model of XML documents, and logical languages for them have been studied actively over the past several years. Such logics have different purposes: some are better suited for extracting data, some for expressing navigational properties, and some make it easy to relate complex properties of trees to the existence of tree automata for those properties. Furthermore, logics differ significantly in their model-checking properties, their automata models, and their behavior on ordered and unordered trees. In this paper we present a survey of logics for unranked trees

Temporalized logics and automata for time granularity

Suitable extensions of the monadic second-order theory of k successors have been proposed in the literature to capture the notion of time granularity. In this paper, we provide the monadic second-order theories of downward unbounded layered structures, which are infinitely refinable structures consisting of a coarsest domain and an infinite number of finer and finer domains, and of upward unbounded layered structures, which consist of a finest domain and an infinite number of coarser and coarser domains, with expressively complete and elementarily decidable temporal logic counterparts. We obtain such a result in two steps. First, we define a new class of combined automata, called temporalized automata, which can be proved to be the automata-theoretic counterpart of temporalized logics, and show that relevant properties, such as closure under Boolean operations, decidability, and expressive equivalence with respect to temporal logics, transfer from component automata to temporalized ones. Then, we exploit the correspondence between temporalized logics and automata to reduce the task of finding the temporal logic counterparts of the given theories of time granularity to the easier one of finding temporalized automata counterparts of them.Comment: Journal: Theory and Practice of Logic Programming Journal Acronym: TPLP Category: Paper for Special Issue (Verification and Computational Logic) Submitted: 18 March 2002, revised: 14 Januari 2003, accepted: 5 September 200

A first-order axiomatization of the theory of finite trees

We provide first-order axioms for the theories of finite trees with bounded branching and finite trees with arbitrary (finite) branching. The signature is chosen to express, in a natural way, those properties of trees most relevant to linguistic theories. These axioms provide a foundation for results in linguistics that are based on reasoning formally about such properties. We include some observations on the expressive power of these theories relative to traditional language complexity classes

Decidability Results for the Boundedness Problem

We prove decidability of the boundedness problem for monadic least fixed-point recursion based on positive monadic second-order (MSO) formulae over trees. Given an MSO-formula phi(X,x) that is positive in X, it is decidable whether the fixed-point recursion based on phi is spurious over the class of all trees in the sense that there is some uniform finite bound for the number of iterations phi takes to reach its least fixed point, uniformly across all trees. We also identify the exact complexity of this problem. The proof uses automata-theoretic techniques. This key result extends, by means of model-theoretic interpretations, to show decidability of the boundedness problem for MSO and guarded second-order logic (GSO) over the classes of structures of fixed finite tree-width. Further model-theoretic transfer arguments allow us to derive major known decidability results for boundedness for fragments of first-order logic as well as new ones

What Does a Grammar Formalism Say About a Language?

Over the last ten or fifteen years there has been a shift in generative linguistics away from formalisms based on a procedural interpretation of grammars towards constraint-based formalisms—formalisms that define languages by specifying a set of constraints that characterize the set of well-formed structures analyzing the strings in the language. A natural extension of this trend is to define this set of structures model-theoretically—to define it as the set of mathematical structures that satisfy some set of logical axioms. This approach raises a number of questions about the nature of linguistic theories and the role of grammar formalisms in expressing them. We argue here that the crux of what theories of syntax have to say about language lies in the abstract properties of the sets of structures they license. This is the level that is most directly connected to the empirical basis of these theories and it is the level at which it is possible to make meaningful comparisons between the approaches. From this point of view, grammar formalisms, or (formal frameworks) are primarily means of presenting these properties. Many of the apparent distinctions between formalisms, then, may well be artifacts of their presentation rather than substantive distinctions between the properties of the structures they license. The model-theoretic approach offers a way in which to abstract away from the idiosyncrasies of these presentations. Having said that, we must distinguish between the class of sets of structures licensed by a linguistic theory and the set of structures licensed by a specific instance of the theory—by a grammar expressing that theory. Theories of syntax are not simply accounts of the structure of individual languages in isolation, but rather include assertions about the organization of the structure of human languages in general. These universal aspects of the theories present two challenges for the model-theoretic approach. First, they frequently are not properties of individual structures, but are rather properties of sets of structures. Thus, in capturing these model-theoretically one is not defining sets of structures but is rather defining classes of sets of structures; these are not first order properties. Secondly, the universal aspects of linguistic theories are frequently not explicit, but are consequences of the nature of the formalism that embodies the theory. In capturing these one must develop an explicit axiomatic treatment of the formalism. This is both a challenge and a powerful beneft of the approach. Such re-interpretations tend to raise a variety of issues that are often overlooked in the original formalization. In this report we examine these issues within the context of a model-theoretic reinterpretation of Generalized Phrase-Structure Grammar. While there is little current active research on GPSG, it provides an ideal laboratory for exploring these issues. First, the formalism of GPSG is expressly intended to embody a great deal of the accompanying linguistic theory. Thus it provides a variety of opportunities for examining principles expressed as restrictions on the formalism from a model-theoretic point of view. At the same time, the fact that these restrictions embody universal grammar principles provides us with a variety of opportunities to explore the way in which the linguistic theory expressed by a grammar can transcend the mathematical theory of the structures it licenses. Finally, GPSG, although defined declaratively, is a formalism with restricted generative capacity, a characteristic more typical of the earlier procedural formalisms. As such, one component of the theory it embodies is a claim about the language-theoretic complexity of natural languages. Such claims are difficult to establish for any of the constraint-based approaches to grammar. We can show, however, that the class of sets of trees that are definable within the logical language we employ in reformalizing GPSG is nearly exactly the class of sets of trees definable within the basic GPSG formalism. Thus we are able to capture the language-theoretic consequences of GPSGs restricted formalism by employing a restricted logical language