Tony Tan

Tree automata over infinite alphabets

A number of models of computation on trees labeled with symbols from an infinite alphabet is considered. We study closure and decision properties of each of the models and compare their computation power.

Regular Expressions for Languages over Infinite Alphabets

In this paper we introduce a notion of a regular expression over infinite alphabets and show that a language is definable by an infinite alphabet regular expression if and only if it is accepted by finite-state unification based automaton - a model of computation that is tightly related to other models of automata over infinite alphabets.

On the complexity of query answering over incomplete XML documents

Previous studies of incomplete XML documents have identified three main sources of incompleteness -- in structural information, data values, and labeling -- and addressed data complexity of answering analogs of unions of conjunctive queries under the open world assumption. It is known that structural incompleteness leads to intractability, while incompleteness in data values and labeling still permits efficient computation of certain answers. The goal of this paper is to provide a complete picture of the complexity of query answering over incomplete XML documents. We look at more expressive languages, at other semantic assumptions, and at both data and combined complexity of query answering, to see whether some well-behaving tractable classes have been missed. To incorporate non-positive features into query languages, we look at gentle ways of introducing negation via inequalities and/or Boolean combinations of positive queries, as well as the analog of relational calculus. We also look at the closed world assumption which, due to the hierarchical structure of XML, has two variations. For all combinations of languages and semantics of incompleteness we determine data and combined complexity of computing certain answers. We show that structural incompleteness leads to intractability under all assumptions, while by dropping it we can recover efficient evaluation algorithms for some queries that go beyond those previously studied.

Feasible automata for two-variable logic with successor on data words

We introduce an automata model for data words, that is words that carry at each position a symbol from a finite alphabet and a value from an unbounded data domain. The model is (semantically) a restriction of data automata, introduced by Bojanczyk, et. al. in 2006, therefore it is called weak data automata. It is strictly less expressive than data automata and the expressive power is incomparable with register automata. The expressive power of weak data automata corresponds exactly to existential monadic second order logic with successor +1 and data value equality ˜, EMSO2(+1,˜). It follows from previous work, David, et. al. in 2010, that the nonemptiness problem for weak data automata can be decided in 2-NEXPTIME. Furthermore, we study weak Buchi automata on data ω-strings. They can be characterized by the extension of EMSO2(+1,˜) with existential quantifiers for infinite sets. Finally, the same complexity bound for its nonemptiness problem is established by a nondeterministic polynomial time reduction to the nonemptiness problem of weak data automata.

Graph Reachability and Pebble Automata over Infinite Alphabets

We study the graph reachability problem as a language over an infinite alphabet. Namely, we view a word of even lengtha0 b0 ... an b_n over an infinite alphabet as a directed graph with the symbols that appear in a0 b0 ... an bn as the vertices and (a0, b0),...,(an, bn) as the edges. We prove that for any positive integer k, k pebbles are sufficient for recognizing the existence of a path of length 2^k-1 from the vertex a0 to the vertex bn, but are not sufficient for recognizing the existence of a path of length 2^{k+1} - 2 from the vertex a0 to the vertex bn. Based on this result, we establish a number of relations among some classes of languages over infinite alphabets.

On the satisfiability of two-variable logic over data words

Data trees and data words have been studied extensively in connection with XML reasoning. These are trees or words that, in addition to labels from a finite alphabet, carry labels from an infinite alphabet (data). While in general logics such as MSO or FO are undecidable for such extensions, decidablity results for their fragments have been obtained recently, most notably for the two-variable fragments of FO and existential MSO. The proofs, however, are very long and non-trivial, and some of them come with no complexity guarantees. Here we give a much simplified proof of the decidability of two-variable logics for data words with the successor and data-equality predicates. In addition, the new proof provides several new fragments of lower complexity. The proof mixes database-inspired constraints with encodings in Presburger arithmetic.

On Pebble Automata for Data Languages with Decidable Emptiness Problem

In this paper we study a subclass of pebble automata (PA) for data languages for which the emptiness problem is decidable. Namely, we show that the emptiness problem for weak 2-pebble automata is decidable, while the same problem for weak 3-pebble automata is undecidable. We also introduce the so-called top view weak PA. Roughly speaking, top view weak PA are weak PA where the equality test is performed only between the data values seen by the two most recently placed pebbles. The emptiness problem for this model is still decidable.

Extending two-variable logic on data trees with order on data values and its automata

Data trees are trees in which each node, besides carrying a label from a finite alphabet, also carries a data value from an infinite domain. They have been used as an abstraction model for reasoning tasks on XML and verification. However, most existing approaches consider the case where only equality test can be performed on the data values. In this article we study data trees in which the data values come from a linearly ordered domain, and in addition to equality test, we can test whether the data value in a node is greater than the one in another node. We introduce an automata model for them which we call ordered-data tree automata (ODTA), provide its logical characterisation, and prove that its non-emptiness problem is decidable in 3-NExpTime. We also show that the two-variable logic on unranked data trees, studied by Bojanczyk et al. [2009], corresponds precisely to a special subclass of this automata model. Then we define a slightly weaker version of ODTA, which we call weak ODTA, and provide its logical characterisation. The complexity of the non-emptiness problem drops to NP. However, a number of existing formalisms and models studied in the literature can be captured already by weak ODTA. We also show that the definition of ODTA can be easily modified, to the case where the data values come from a tree-like partially ordered domain, such as strings.

A Note on Two-pebble Automata Over Infinite Alphabets

It is shown that the emptiness problemfor two-pebble automata languages is undecidable and that two-pebble automata are weaker than three-pebble automata.

Regular Graphs and the Spectra of Two-Variable Logic with Counting

The {\em spectrum} of a first-order logic sentence is the set of natural numbers that are cardinalities of its finite models. In this paper we show that when restricted to using only two variables, but allowing counting quantifiers, the spectra of first-order logic sentences are semilinear and hence, closed under complement. At the heart of our proof are semilinear characterisations for the existence of regular and biregular graphs, the class of graphs in which there are a priori bounds on the degrees of the vertices. Our proof also provides a simple characterisation of models of two-variable logic with counting -- that is, up to renaming and extending the relation names, they are simply a collection of regular and biregular graphs.