On Descriptive Complexity, Language Complexity, and GB

We introduce $L^2_{K,P}$, a monadic second-order language for reasoning about trees which characterizes the strongly Context-Free Languages in the sense that a set of finite trees is definable in $L^2_{K,P}$ iff it is (modulo a projection) a Local Set---the set of derivation trees generated by a CFG. This provides a flexible approach to establishing language-theoretic complexity results for formalisms that are based on systems of well-formedness constraints on trees. We demonstrate this technique by sketching two such results for Government and Binding Theory. First, we show that {\em free-indexation\/}, the mechanism assumed to mediate a variety of agreement and binding relationships in GB, is not definable in $L^2_{K,P}$ and therefore not enforcible by CFGs. Second, we show how, in spite of this limitation, a reasonably complete GB account of English can be defined in $L^2_{K,P}$. Consequently, the language licensed by that account is strongly context-free. We illustrate some of the issues involved in establishing this result by looking at the definition, in $L^2_{K,P}$, of chains. The limitations of this definition provide some insight into the types of natural linguistic principles that correspond to higher levels of language complexity. We close with some speculation on the possible significance of these results for generative linguistics.Comment: To appear in Specifying Syntactic Structures, papers from the Logic, Structures, and Syntax workshop, Amsterdam, Sept. 1994. LaTeX source with nine included postscript figure

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

Eliminating Recursion from Monadic Datalog Programs on Trees

We study the problem of eliminating recursion from monadic datalog programs on trees with an infinite set of labels. We show that the boundedness problem, i.e., determining whether a datalog program is equivalent to some nonrecursive one is undecidable but the decidability is regained if the descendant relation is disallowed. Under similar restrictions we obtain decidability of the problem of equivalence to a given nonrecursive program. We investigate the connection between these two problems in more detail

Query Containment for Highly Expressive Datalog Fragments

The containment problem of Datalog queries is well known to be undecidable. There are, however, several Datalog fragments for which containment is known to be decidable, most notably monadic Datalog and several "regular" query languages on graphs. Monadically Defined Queries (MQs) have been introduced recently as a joint generalization of these query languages. In this paper, we study a wide range of Datalog fragments with decidable query containment and determine exact complexity results for this problem. We generalize MQs to (Frontier-)Guarded Queries (GQs), and show that the containment problem is 3ExpTime-complete in either case, even if we allow arbitrary Datalog in the sub-query. If we focus on graph query languages, i.e., fragments of linear Datalog, then this complexity is reduced to 2ExpSpace. We also consider nested queries, which gain further expressivity by using predicates that are defined by inner queries. We show that nesting leads to an exponentially increasing hierarchy for the complexity of query containment, both in the linear and in the general case. Our results settle open problems for (nested) MQs, and they paint a comprehensive picture of the state of the art in Datalog query containment.Comment: 20 page

Quantified CTL: Expressiveness and Complexity

While it was defined long ago, the extension of CTL with quantification over atomic propositions has never been studied extensively. Considering two different semantics (depending whether propositional quantification refers to the Kripke structure or to its unwinding tree), we study its expressiveness (showing in particular that QCTL coincides with Monadic Second-Order Logic for both semantics) and characterise the complexity of its model-checking and satisfiability problems, depending on the number of nested propositional quantifiers (showing that the structure semantics populates the polynomial hierarchy while the tree semantics populates the exponential hierarchy)

The VC-Dimension of Graphs with Respect to k-Connected Subgraphs

We study the VC-dimension of the set system on the vertex set of some graph which is induced by the family of its $k$-connected subgraphs. In particular, we give tight upper and lower bounds for the VC-dimension. Moreover, we show that computing the VC-dimension is $\mathsf{NP}$-complete and that it remains $\mathsf{NP}$-complete for split graphs and for some subclasses of planar bipartite graphs in the cases $k = 1$ and $k = 2$. On the positive side, we observe it can be decided in linear time for graphs of bounded clique-width

Challenges for Efficient Query Evaluation on Structured Probabilistic Data

Query answering over probabilistic data is an important task but is generally intractable. However, a new approach for this problem has recently been proposed, based on structural decompositions of input databases, following, e.g., tree decompositions. This paper presents a vision for a database management system for probabilistic data built following this structural approach. We review our existing and ongoing work on this topic and highlight many theoretical and practical challenges that remain to be addressed.Comment: 9 pages, 1 figure, 23 references. Accepted for publication at SUM 201