Chase and Axioms for Pc Queries and Dependencies

1 Overview This report is the extended version of PT99] and while they are some overlapping parts (mainly the results) we don't repeat here some of the examples given in PT99], even though we frequently refer to them. The overview of this report is as follows. In section 2 we present some aspects of our internal framework, called CoDi 1. This is a language and equational theory that combines a treatment of dictionaries with our previous work BBW92, BNTW95, LT97] on collections and aggregates using the theory of monads. In addition we use dictionaries ((nite functions), which allow us to represent oodb schemas and queries. While here we focus on set-related queries, we show elsewhere 2 that the (full) CoDi collection, aggregation and dictionary primitives suuce for implementing the quasi-totality of ODMG/ODL/OQL Cat96]. Using boolean aggregates, CoDi can represent constraints as equalities between boolean-valued queries. We also give the basic equational laws of CoDi while the rest of the axiomatization can be found in section 11. In section 3 we give a simple form for the equational chase which connects dependencies to query equivalence (and therefore containment via intersection). We also show there how we can represent in CoDi relational conjunctive queries (with equality) CM77, ASU79] embedded dependencies, which are the multi-relation and un-typed versions of Fagin's embedded implicational dependencies Fag82], and that the relational chase of ABU79, MMS79, BV84b] is a particular case of our equational chase. In section 4 we ooer examples of dependencies and equivalences beyond the relational model on which we use the equational chase deened earlier. One example captures a many-many inverse relationship in an OO schema, while in PT99] we showed an example of an inverse between a class and a relation. In the same section we point out that view deenitions can be equivalently characterized by dependencies and, in the special case of path-conjunctive (PC) views, the dependencies are path-conjunctive as well (EPCDs). In section 5 we exhibit a class of queries and dependencies on complex values with dictionaries called path-conjunctive (PC queries and embedded PC dependencies (EPCDs)) for which the methods illustrated in earlier sections are complete, and in certain cases decidable. Section 6 gives the technical background that will be used throughout the rest of the paper. It describes well-deened path expressions, well-deened tableaux, and gives the construction of the canonical instance associated to each tableau. Theorem 7.9 in section 7 …