Two-Variable Logic with Two Order Relations

It is shown that the finite satisfiability problem for two-variable logic over structures with one total preorder relation, its induced successor relation, one linear order relation and some further unary relations is EXPSPACE-complete. Actually, EXPSPACE-completeness already holds for structures that do not include the induced successor relation. As a special case, the EXPSPACE upper bound applies to two-variable logic over structures with two linear orders. A further consequence is that satisfiability of two-variable logic over data words with a linear order on positions and a linear order and successor relation on the data is decidable in EXPSPACE. As a complementing result, it is shown that over structures with two total preorder relations as well as over structures with one total preorder and two linear order relations, the finite satisfiability problem for two-variable logic is undecidable

Structure Theorem and Strict Alternation Hierarchy for FO^2 on Words

It is well-known that every first-order property on words is expressible using at most three variables. The subclass of properties expressible with only two variables is also quite interesting and well-studied. We prove precise structure theorems that characterize the exact expressive power of first-order logic with two variables on words. Our results apply to both the case with and without a successor relation. For both languages, our structure theorems show exactly what is expressible using a given quantifier depth, n, and using m blocks of alternating quantifiers, for any m \leq n. Using these characterizations, we prove, among other results, that there is a strict hierarchy of alternating quantifiers for both languages. The question whether there was such a hierarchy had been completely open. As another consequence of our structural results, we show that satisfiability for first-order logic with two variables without successor, which is NEXP-complete in general, becomes NP-complete once we only consider alphabets of a bounded size

Distribution Constraints: The Chase for Distributed Data

This paper introduces a declarative framework to specify and reason about distributions of data over computing nodes in a distributed setting. More specifically, it proposes distribution constraints which are tuple and equality generating dependencies (tgds and egds) extended with node variables ranging over computing nodes. In particular, they can express co-partitioning constraints and constraints about range-based data distributions by using comparison atoms. The main technical contribution is the study of the implication problem of distribution constraints. While implication is undecidable in general, relevant fragments of so-called data-full constraints are exhibited for which the corresponding implication problems are complete for EXPTIME, PSPACE and NP. These results yield bounds on deciding parallel-correctness for conjunctive queries in the presence of distribution constraints

Dynamic Complexity of Formal Languages

The paper investigates the power of the dynamic complexity classes DynFO, DynQF and DynPROP over string languages. The latter two classes contain problems that can be maintained using quantifier-free first-order updates, with and without auxiliary functions, respectively. It is shown that the languages maintainable in DynPROP exactly are the regular languages, even when allowing arbitrary precomputation. This enables lower bounds for DynPROP and separates DynPROP from DynQF and DynFO. Further, it is shown that any context-free language can be maintained in DynFO and a number of specific context-free languages, for example all Dyck-languages, are maintainable in DynQF. Furthermore, the dynamic complexity of regular tree languages is investigated and some results concerning arbitrary structures are obtained: there exist first-order definable properties which are not maintainable in DynPROP. On the other hand any existential first-order property can be maintained in DynQF when allowing precomputation.Comment: Contains the material presenten at STACS 2009, extendes with proofs and examples which were omitted due lack of spac

Index of Authors

Author Index

A Little Bit Infinite? On Adding Data to Finitely Labelled Structures (Abstract)

Finite or infinite strings or trees with labels from a finite alphabet play an important role in computer science. They can be used to model many interesting objects including system runs in Automated Verification and XML documents in Database Theory. They allow the application of formal tools like logical formulas to specify properties and automata for their implementation. In this framework, many reasoning tasks that are undecidable for general computational models can be solved algorithmically, sometimes even efficiently. Nevertheless, the use of finitely labelled structures usually requires an early abstraction from the real data. For example, theoretical research on XML processing very often con- centrates on the document structure (including labels) but ignores attribute or text values. While this abstraction has led to many interesting results, some aspects like key or other integrity constraints can not be adequately handled. In Automated Verification of software systems or communication protocols, infinite domains occur even more naturally, e.g., induced by program data, recursion, time, com- munication or by unbounded numbers of concurrent processes. Usually one approximates infinite domains by finite ones in a very early abstraction step. An alternative approach that has been investigated in recent years is to extend strings and trees by (a limited amount of) data and to use logical languages with a restricted ex- pressive power concerning this data. As an example, in the most simple setting, formulas can only test equality of data values. The driving goal is to identify logical languages and corresponding automata models which are strong enough to describe interesting proper- ties of data-enhanced structures while keeping decidability or even feasibility of automatic reasoning. The talk gives a basic introduction into data-enhanced finitely labelled structures, presents examples of their use, and highlights recent decidability and complexity results