Bags, i.e. sets with duplicates, are often used to implement relations in database systems. In this paper, we study the expressive power of algebras for manipulating bags. The algebra we present is a simple extension of the nested relation algebra. Our aim is to investigate how the use of bags in the language extends its expressive power, and increases its complexity. We consider two main issues, namely (i) the impact of the depth of bag nesting on the expressive power, and (ii) the complexity and the expressive power induced by the algebraic operations. We show that the bag algebra is more expressive than the nested relation algebra (at all levels of nesting), and that the difference may be subtle. We establish a hierarchy based on the structure of algebra expressions. This hierarchy is shown to be highly related to the properties of the powerset operator. Invited to a special issue of the Journal of Computer and System Sciences selected from ACM Princ. of Database Systems,...

Abstract. We compare the expressive power on finite models of two extensions of first order logic L with equality. L(Ct) is formed by adding an operator count{x: ϕ}, which builds a term of sort N that counts the number of elements of the finite model satisfying a formula ϕ. Our main result shows that the stronger operator count{t(x) : ϕ}, where t(x) is a term of sort N, cannot be expressed in L(Ct). That is, being able to count elements does not allow one to count terms. This paper also continues our interest in new proof techniques in database theory. The proof of the unary counter combines a number of modeltheoretic techniques that give powerful tools for expressivity bounds: in particular, we discuss here the use of indiscernibles, the Paris-Harrington form of Ramsey’s theorem, and nonstandard models of arithmetic. 1

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 ...

In this paper we present a new approach for studying aggregations in the context of database query languages. Starting from a broad definition of aggregate function, we address our investigation from two different perspectives. We first propose a declarative notion of uniform aggregate function that refers to a family of scalar functions uniformly constructed over a vocabulary of basic operators by a bounded Turing Machine. This notion yields an effective tool to study the effect of the embedding of a class of built-in aggregate functions in a query language. All the aggregate functions most used in practice are included in this classification. We then present an operational notion of aggregate function, by considering a high-order folding constructor, based on structural recursion, devoted to compute numeric aggregations over complex values. We show that numeric folding over a given vocabulary is sometimes not able to compute, by itself, the whole class of uniform aggre...

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. The main result of the paper is that all the counting logics above, in the presence of pre-orders that are almosteverywhere linear orders, exhibit a very tame behavior normally associated with first-order properties of unordered structures. This is in sharp contrast with the expressiveness of these logics in the presence of linear orders: such a tame behavior is not the case even for first-order logic with counting, and the most powerful logic we consider can express every property of ordered structures. The results attest to the difficulty of proving separation results for the ordered case, in particular, to proving the separation of TC 0 from NP. To prove th...

Most database theory focused on investigating databases containing sets of tuples. In practice databases often implement relations using bags, i.e. sets with duplicates. In this paper we study how database query languages are affected by the use of duplicates. We consider query languages that are simple extensions of the (nested) relational algebra, and investigate their resulting expressive power and complexity. 1 Introduction In the standard approach to database modeling, relations are assumed to be sets, and no duplicates are allowed. For real applications, many systems relax this restriction [Fis87, HM81] and support bags in their data model, often to save the cost of duplicate elimination. Efforts have been made for providing a theoretical framework for such systems. Algebras for manipulating bags were developed by extending the relational algebra [Alb91, Klu82, OOM87], and optimization techniques for these algebras were studied [BK90, Mum90, Alb91]. Computational aspects of...

Most database theory focused on investigating databases containing sets of tuples. In practice databases often implement relations using bags, i.e. sets with duplicates. In this paper we study how database query languages are affected by the use of duplicates. We consider query languages that are simple extensions of the (nested) relational algebra, and investigate their resulting expressive power and complexity. 1 Introduction In the standard approach to database modeling, relations are assumed to be sets, and no duplicates are allowed. For real applications, many systems relax this restriction [Fis87, HM81] and support bags in their data model, often to save the cost of duplicate elimination. Efforts have been made for providing a theoretical framework for such systems. Algebras for manipulating bags were developed by extending the relational algebra [Alb91, Klu82, OOM87], and optimization techniques for these algebras were studied [BK90, Mum90, Alb91]. Computational aspects of...