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 ...

Strong normalization results are obtained for a general language for collection types. An induced normal form for sets and bags is then used to show that the class of functions whose input has height (that is, the maximal depth of nestings of sets/bags/lists in the complex object) at most i and output has height at most o definable in a nested relational query language without powerset operator is independent of the height of intermediate expressions used. Our proof holds regardless of whether the language is used for querying sets, bags, or lists, even in the presence of variant types. Moreover, the normal forms are useful in a general approach to query optimization. Paredaens and Van Gucht proved a similar result for the special case when i = o = 1. Their result is complemented by Hull and Su who demonstrated the failure of independence when powerset operator is present and i = o = 1. The theorem of Hull and Su was generalized to all i and o by Grumbach and Vianu. Our result genera...

We provide new techniques for the analysis of the expressive power of query languages for nested collections. These languages may use set or bag semantics and may be further complicated by the presence of aggregate functions. We exhibit certain classes of graphs and prove that the properties of these graphs that can be tested in such languages are either finite or cofinite. This result settles the conjectures of Grumbach, Milo, and Paredaens that parity test, transitive closure, and balanced binary tree test are not expressible in bag languages like the PTIME fragment of BALG of Grumbach and Milo and BQL of Libkin and Wong. Moreover, it implies that many recursive queries, including simple ones like the test for a chain, cannot be expressed in a nested relational language even when aggregate functions are available. In an attempt to generalize the finite-cofiniteness result, we study the bounded degree property which says that the number of distinct in- and out-degrees in the output of...

We address the problem of query containment and query equivalence for complex objects. We show that for a certain conjunctive query language for complex objects, query containment and weak query equivalence are decidable. Our results have two consequences. First, when the answers of the two queries are guaranteed not to contain empty sets, then weak equivalence coincides with equivalence, and our result answers partially an open problem about the equivalence of nest; unnest queries for complex objects [GPG90]. Second, we derive an NP-complete algorithm for checking the equivalence of certain conjunctive queries with grouping and aggregates. Our results rely on a translation of the containment and equivalence conditions for complex objects into novel conditions on conjunctive queries, which we call simulation and strong simulation. These conditions are more complex than containment of conjunctive queries, because they involve arbitrary numbers of quantifier alternations. We prove that c...

In this paper we study the expressive power of query languages for nested bags. We define the ambient bag language by generalizing the constructs of the relational language of Breazu-Tannen, Buneman and Wong, which is known to have precisely the power of the nested relational algebra. Relative strength of additional polynomial constructs is studied, and the ambient language endowed with the strongest combination of those constructs is chosen as a candidate for the basic bag language, which is called BQL (Bag Query Language). We prove that achieveing the power of BQL in the relational language amounts to adding simple arithmetic to the latter. We show that BQL has shortcomings of the relational algebra: it can not express recursive queries. In particular, parity test is not definable in BQL. We consider augmenting BQL with powerbag and structural recursion to overcome this deficiency. In contrast to the relational case, where powerset and structural recursion are equivalent...

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. 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...

We examine the power of incremental evaluation systems that use an SQL-like language for maintaining recursively-defined views. We show that recursive queries such as transitive closure, and "alternating paths" can be incrementally maintained in a nested relational language, when some auxiliary relations are allowed. In the presence of aggregate functions, even more queries can be maintained, for example, the "same generation" query. In contrast, it is still an open problem whether such queries are maintainable in relational calculus. We then restrict the language so that no nested relations are involved (but wekeep the aggregate functions). Such a language captures the capability of most practical relational database systems. We prove that this restriction does not reduce the incremental computational power; that is, any query that can be maintained in a nested language with aggregates, is still maintainable using only flat relations. We also show that one does not need auxiliar...

The syntax of comprehensions is very close to the syntax of a number of practical database query languages and is, we believe, a better starting point than first-order logic for the development of database languages. We give an informal account of a language based on comprehension syntax that deals uniformly with a variety of collection types; it also includes pattern matching, variant types and function definition. We show, again informally, how comprehension syntax is a natural fragment of structural recursion, a much more powerful programming paradigm for collection types. We also show that a very small "abstract syntax language" can serve as a basis for the implementation and optimization of comprehension syntax.

. We investigate and compare two forms of recursion on sets for querying nested collections. The first one is called sri and it corresponds to sequential processing of data. The second one is called sru and it corresponds to data-parallel processing. A uniform first-order translation from sru into sri was known from previous work. The converse translation is by necessity more difficult and we have obtained three main results concerning it. First, we exhibit a uniform translation of sri queries into sru queries over the nested relational algebra. We observe that this translation maps PTIME algorithms into exponential-space queries. The second result proves that any uniform translation of sri queries into sru queries over the nested relational algebra must map some PTIME queries into exponential-space ones. In fact, in the presence of certain external functions, we provide a PTIME sri query for which any equivalent sru query requires exponential space. Thus, as a mechanism for implemen...