We study here the language Datalog(6=), which is the query language obtained from Datalog by allowing equalities and inequalities in the bodies of the rules. We view Datalog(6=) as a fragment of an innitary logic L ! and show that L ! can be characterized in terms of certain two-person pebble games. This characterization provides us with tools for investigating the expressive power of Datalog(6=). As a case study, we classify the expressibility of fixed subgraph homeomorphism queries on directed graphs. Fortune et al. [FHW80] classied the computational complexity of these queries by establishing two dichotomies, which are proper only if P 6= NP. Without using any complexity-theoretic assumptions, we show here that the two dichotomies are indeed proper in terms of expressibility in Datalog(6=).

. Expressiveness of database query languages remains the major motivation for research in finite model theory. However, most techniques in finite model theory are based on Ehrenfeucht-Fraisse games, whose application often involves a rather intricate argument. Furthermore, most tools apply to first-order logic and some of its extensions, but not to languages that resemble real query languages, like SQL. In this paper we use locality to analyze expressiveness of query languages. A query is local if, to determine if a tuple belongs to the output, one only has to look at a certain predetermined portion of the input. We study local properties of queries in a context that goes beyond the pure first-order case, and then apply the resulting tools to analyze expressive power of SQL-like languages. We first prove a general result describing outputs of local queries, that leads to many easy inexpressibility proofs. We then consider a closely related bounded degree property, which d...

A combinatorial argument for two finite structures to agree on all sentences with bounded quantifier rank in first-order logic with any set of unary generalized quantifiers, is given. It is known that connectivity of finite structures is neither in monadic \Sigma 1 1 nor in L !! (Q u ), where Q u is the set of all unary generalized quantifiers. Using this combinatorial argument and a combination of second-order EhrenfeuchtFra iss'e games developed by Ajtai and Fagin, we prove that connectivity of finite structures is not in monadic \Sigma 1 1 with any set of unary quantifiers, even if sentences are allowed to contain built-in relations of moderate degree. The combinatorial argument is also used to show that no class (if it is not in some sense trivial) of finite graphs defined by forbidden minors, is in L !! (Q u ). Especially, the class of planar graphs is not in L !! (Q u ). 1. Introduction The expressive power of first-order logic L !! is rather limited. This is beca...

In this paper we survey the major developments in understanding the complexity of the graph connectivity problem in several computational models, and highlight some challenging open problems. 1

this article is a proof that each FO problem can be solved in linear time if only relational structures of bounded degree are considered. The basic idea of the proof is a localization technique based on a method that was originally developed by Hanf (Hanf 1965) to show that the elementary theories of two structures are equal under certain conditions, i.e., that two structures agree on all first-order sentences. Fagin, Stockmeyer and Vardi (Fagin et al. 1993) developed a variant of this technique, which is applicable in descriptive complexity theory to classes of finite relational structures of uniformly bounded degree. Variants of this result can also be found in Gaifman (1982) (see also Thomas (1991)). The essential content of this result, which is also called the Hanf-Sphere Lemma, is that two relational structures of bounded degree satisfy the same first-order sentences of a certain quantifier-rank if both contain, up to a certain number m, the same number of isomorphism types of substructures of a bounded radius r. In addition, a technique of model interpretability from Rabin (1965) (see also Arnborg et al. (1991), Seese (1992), Compton and Henson (1987) and Baudisch et al. (1982)) is adapted to descriptive complexity classes, and proved to be useful for reducing the case of an arbitrary class of relational structures to a class of structures consisting only of the domain and one binary irreflexive and symmetric relation, i.e., the class of simple graphs. It is shown that the class of simple graphs is lintime-universal with respect to first-order logic, which shows that many problems on descriptive complexity classes, described in languages extending first-order logic for arbitrary structures, can be reduced to problems on simple graphs. This paper is organized as f...

Many known tools for proving expressibility bounds for first-order logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is one of the easiest tools for proving expressibility bounds. These results apply beyond the first-order case. We use them to derive expressibility bounds for first-order logic with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree. 1 Introduction It is well known that first-order logic has limited expressive power. Typically, inexpressibility proofs are based on either a compactness argument, or Ehrenfeucht-Fraiss'e games. In ...

We define matchings, and show that they capture the essence of context-freeness. More precisely, we show that the class of context-free languages coincides with the class of those sets of strings which can be defined by sentences of the form 9 b', where ' is first order, b is a binary predicate symbol, and the range of the second order quantifier is restricted to the class of matchings. Several variations and extensions are discussed.

Inexpressibility results in Finite Model Theory are often proved by showing that Duplicator, one of the two players of an Ehrenfeucht game, has a winning strategy on certain structures. In this article a new method is introduced that allows, under certain conditions, the extension of a winning strategy of Duplicator on some small parts of two finite structures to a global winning strategy. As applications of this technique it is shown that (*) Graph Connectivity is not expressible in existential monadic second-order logic (MonNP), even in the presence of a built-in linear order, (*) Graph Connectivity is not expressible in MonNP even in the presence of arbitrary built-in relations of degree n^o(1), and (*) the presence of a built-in linear order gives MonNP more expressive power than the presence of a built-in successor relation.

It is a well-known result of Fagin that the complexity class NP coincides with the class of

Ehrenfeucht games are a useful tool in proving that certain properties of finite structures are not expressible by formulas of a certain type. In this paper a new method is introduced that allows the extension of a local winning strategy for Duplicator, one of the two players in Ehrenfeucht games, to a global winning strategy. As an application it is shown that Graph Connectivity cannot be expressed by existential second-order formulas, where the second-order quantification is restricted to unary relations (Monadic NP), even in the presence of a built-in linear order. This settles an open problem from [1] and [11]. As a second application it is stated, that, on the other hand, the presence of a linear order increases the power of Monadic NP more than the presence of a successor relation. 1 Introduction Fagin [8] showed that the complexity class NP coincides with the class of all sets of finite structures that can be characterized by existential second-order formulas (\Sigma 1 1 ). Th...