. 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...

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 study adding aggregate operators, such as summing up elements of a column of a relation, to logics with counting mechanisms. The primary motivation comes from database applications, where aggregate operators are present in all real life query languages. Unlike other features of query languages, aggregates are not adequately captured by the existing logical formalisms. Consequently, all previous approaches to analyzing the expressive power of aggregation were only capable of producing partial results, depending on the allowed class of aggregate and arithmetic operations. We consider a powerful counting logic, and extend it with the set of all aggregate operators. We show that the resulting logic satis es analogs of Hanf's and Gaifman's theorems, meaning that it can only express local properties. We consider a database query language that expresses all the standard aggregates found in commercial query languages, and show how it can be translated into the aggregate logic, thereby pro...

The expressive power of first-order logic over finite structures is limited in two ways: it lacks a recursion mechanism, and it cannot count. Overcoming the first limitation has been a subject of extensive study. A number of fixpoint logics have been introduced, and shown to be subsumed by an infinitary logic L ! 1! . This logic is easier to analyze than fixpoint logics, and it still lacks counting power, as it has a 0-1 law. On the counting side, there is no analog of L ! 1! . There are a number of logics with counting power, usually introduced via generalized quantifiers. Most known expressivity bounds are based on the fact that counting extensions of first-order logic preserve the locality properties. This paper has three main goals. First, we introduce a new logic L 1! (C) that plays the same role for counting as L ! 1! does for recursion -- it subsumes a number of extensions of first-order logic with counting, and has nice properties that make it easy to study. Second, we ...

A query is local if...

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This survey article introduces into the essential concepts and methods underlying rule-based query languages. It covers four complementary areas: declarative semantics based on adaptations of mathematical logic, operational semantics, complexity and expressive power, and optimisation of query evaluation. The treatment of these areas is foundation-oriented, the foundations having resulted from over four decades of research in the logic programming and database communities on combinations of query languages and rules. These results have later formed the basis for conceiving, improving, and implementing several Web and Semantic Web technologies, in particular query languages such as XQuery or SPARQL for querying relational, XML, and RDF data, and rule languages like the “Rule Interchange Framework (RIF) ” currently being developed in a working group of the W3C. Coverage of the article is deliberately limited to declarative languages in a classical setting: issues such as query answering in F-Logic or in description logics, or the relationship of query answering to reactive rules and events, are not addressed.

ing with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works, requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept, ACM Inc., 1515 Broadway, New York, NY 10036 USA, fax +1 (212) 869-0481, or permissions@acm.org. 2 \Delta 1. INTRODUCTION The expressive power of first-order logic (FO) on finite structures is rather limited. Two main limitations of first-order logic are its inability to count and the lack of a recursion mechanism. Since first-order logic over finite structures plays an important role in several areas of computer science (e.g., databases and complexity), various extensions have been proposed to deal with these shortcomings. On the recursion side, a beautiful theory has been developed over the past decade. Various fixpoint extensions of first-order logic have been introduced, including least, inflationary and partial fixpoint...

This paper studies expressivity bounds for extensions of first-order logic with counting and unary quantifiers in the presence of relations of large degree. There are several motivations for this work. First, it is known that first-order logic with counting quantifiers captures uniform TC 0 over ordered structures. Thus, proving expressivity bounds for first-order with counting can be seen as an attempt to show TC 0 $ DLOG using techniques of descriptive complexity. Second, the presence of auxiliary built-in relations (e.g., order, successor) is known to make a big impact on expressivity results in finite-model theory and database theory (where logics with counting and unary quantifiers have recently been used to model query languages with aggregation). For those logics, our goal is to extend techniques from "pure" setting to that of auxiliary relations. Until now, all known results on the limitations of expressive power of the counting and unary quantifier extensions of first-order...

We study the expressive power of counting logics in the presence of auxiliary relations such as orders and preorders. The simplest such logic, first-order with counting, captures the complexity class TC 0 over ordered structures. We also consider first-order logic with arbitrary unary quantifiers, and infinitary extensions. We start by giving a simple direct proof that first-order with counting, in the presence of preorders that are almost-everywhere linear orders, cannot express the transitive closure of a binary relation. The proof is based on locality of formulae. We then show that the technique cannot be extended to linear orders, and that the result does not say anything about the power of invariant queries in first-order with counting, in the presence of those preorders, vs. the class TC 0 . In the second part of the paper we then prove a separation result showing that for all the counting logics above, a linear order is more powerful than a preorder that is a linea...