Some Properties of Query Languages for Bags

In this paper we study the expressive power of query languages for nested bags. We deene 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 deenable in BQL. We consider augmenting BQL with powerbag and structural recursion to overcome this deeciency. In contrast to the rela-tional case, where powerset and structural recursion are equivalent, the latter is stronger than the former for bags. We discuss problems with using structural recursion and suggest a new bounded loop construct which works uniformly for bags, sets and lists. It has the power of structural recursion and does not require any preconditions to be veriied. We nd relational languages equivalent to BQL with powerbag and structural re-cursion/bounded loop. Finally, we discuss orderings on bags for rigorous treatment of partial information. 1 Summary Sets and bags are closely related structures. While sets have been studied intensively by the theoretical database community, bags have not received the same amount of attention. However, real implementations frequently use bags as the underlying data model. For example, the \select distinct" construct and the \select average of column" construct of SQL can be better explained if bags instead of sets are used. In an earlier paper 5], Breazu-Tannen, Buneman, and Wong deened a language based on monads 20, 29] and structural recursion 3] for querying sets. In section 2 of this report, the same syntax is given a bag-theoretic semantics. We use this language as our ambient bag language