We present a new principle for the development of database query languages that the primitive operations should be organized around types. Viewing a relational database as consisting of sets of records, this principle dictates that we should investigate separately operations for records and sets. There are two immediate advantages of this approach, which is partly inspired by basic ideas from category theory. First, it provides a language for structures in which record and set types may be freely combined: nested relations or complex objects. Second, the fundamental operations for sets are closely related to those for other "collection types" such as bags or lists, and this suggests how database languages may be uniformly extended to these new types. The most general operation on sets, that of structural recursion, is one in which not all programs are welldefined. In looking for limited forms of this operation that always give rise to well-defined operations, we find a number of close ...

Abstract. Various models and languages for describing and manipulating hierar-chically structured data have been proposed. Algebraic, calculus-based, and logic-programming oriented languages have all been considered. This article presents a general model for complex values (i.e., values with hierarchical structures), and languages for it based on the three paradigms. The algebraic language generalizes those presented in the literature; it is shown to be related to the functional style of programming advocated by Backus (1978). The notion of domain independence (from relational databases) is defined, and syntactic restrictions (referred to as safety conditions) on calculus queries are formulated to guarantee domain inde-pendence. The main results are: The domain-independent calculus, the safe cal-culus, the algebra, and the logic-programming oriented language have equivalent expressive power. In particular, recursive queries, such as the transitive closure, can be expressed in each of the languages. For this result, the algebra needs the powerset operation. A more restricted version of safety is presented, such that the restricted safe calculus is equivalent to the algebra without the powerset. The results are extended to the case where arbitrary functions and predicates are used in the languages. Key Words. Database, query language, complex value, complex object, database model.

This article studies the complexity of evaluating functional query languages for complex values such as monad algebra and the recursion-free fragment of XQuery. We show that monad algebra with equality restricted to atomic values is complete for the class TA[2O(n) , O(n)] of problems solvable in linear exponential time with a linear number of alternations. The monotone fragment of monad algebra with atomic value equality but without negation is complete for nondeterministic exponential time. For monad algebra with deep equality, we establish TA[2O(n) , O(n)] lower and exponential-space upper bounds. We also study a fragment of XQuery, Core XQuery, that seems to incorporate all the features of a query language on complex values that are traditionally deemed essential. A close connection between monad algebra on lists and Core XQuery (with “child ” as the only axis) is exhibited, and it is shown that these languages are expressively equivalent up to representation issues. We show that Core XQuery is just as hard as monad algebra w.r.t. query and combined complexity, and that it is in TC0 if the query is assumed fixed. As Core XQuery is NEXPTIME-hard, it is commonly believed that any algorithm for evaluating Core XQuery has to require exponential amounts of working memory and doubly exponential time in the worst case. We present a property of queries – the lack of a certain form of composition – that virtually all real-world XQueries have and that allows for query evaluation in singly exponential time and polynomial space. Still, we are able to show for an important special case – Core XQuery with equality testing restricted to atomic values – that the composition-free language is just as expressive as the language with composition. Thus, under widely-held complexitytheoretic assumptions, the composition-free language is an exponentially less succinct version of the language with composition.

Databases are often incomplete because of the presence of disjunctive information, due to conflicts, partial knowledge and other reasons. Queries against such databases often ask questions about various possibilities encoded by the stored data, rather than the stored data itself. Normalization, which is a mechanism for asking such queries, was presented in [LW93a]; however, it had exponential space complexity. The main goal of this paper is to develop a general theory of answering queries against incomplete databases with disjunctive information, and use it to design practical algorithms for query evaluation. We define the semantics of such databases and prove normalization theorems for set- and bag-based complex objects. These theorems provide us with programming primitives that one needs in order to obtain the list of all possibilities encoded by a complex object with disjunctions. We study two ways of making query evaluation faster and more space efficient. Partial normalization a...

We present a functional framework for descriptive computational complexity, in which the Regular, Firstorder, Ptime, Pspace, k-Exptime, k-Expspace (k 1), and Elementary sets have syntactic characterizations. In this framework, typed lambda terms represent inputs and outputs as well as programs. The lambda calculi describing the above computational complexity classes are simply or let-polymorphically typed with functionalities of fixed order. They consist of: order 0 atomic constants, order 1 equality among these constants, variables, application, and abstraction. Increasing functionality order by one for these languages corresponds to increasing the computational complexity by one alternation. This exact correspondence is established using a semantic evaluation of languages for each fixed order, which is the primary technical contribution of this paper.

The relational data model has simple and clear foundations on which significant theoretical and systems research has flourished. By contrast, most research on data mining has focused on algorithmic issues. A major open question is “what’s an appropriate foundation for data mining, which can accommodate disparate mining tasks. ” We address this problem by presenting a database model and an algebra for data mining. The database model is based on the 3W-model introduced by Johnson et al. [2000]. This model relied on black box mining operators. A main contribution of this paper is to open up these black boxes, by using generic operators in a data mining algebra. Two key operators in this algebra are regionize, which creates regions (or models) from data tuples, and a restricted form of looping called mining loop. Then, the resulting data mining algebra MA is studied and properties concerning expressive power and complexity are established. We present results in three directions: (1) expressiveness of the mining algebra; (2) relations with alternative frameworks, and (3) interactions between regionize and mining loop.

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...

Theoretical foundations for querying databases based on bags are studied in this paper. We fully determine the strength of many polynomial-time bag operators relative to an ambient query language. Then we obtain BQL, a query language for bags, by picking the strongest combination of these operators. The relationship between the nested relational algebra and various fragments of BQL is investigated. The precise amount of extra power that BQL possesses over the nested relational algebra is determined. It is shown that the additional expressiveness of BQL amounts to adding aggregate functions to a relational language. The expressive power of BQL and related languages is investigated in depth. We prove that these languages possess the conservative extension property. That is, the expressibility of queries in these languages is independent of the nesting height of intermediate data. Using this result, we show that recursive queries, such as transitive closure, are not definable in...

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...