Synthesizing Nested Relational Queries from Implicit Specifications

Abstract

Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset 𝑂 in terms of datasets ì 𝐼 ) is a logical specification involving the source data ì 𝐼 and the interface data 𝑂. It is a valid definition of 𝑂 in terms of ì 𝐼 , if any two models of the specification agreeing on ì 𝐼 agree on 𝑂. In contrast, an explicit definition is a query that produces 𝑂 from ì 𝐼 . Variants of Beth's theorem [7] state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous effective implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be effectively converted to explicit definitions in the nested relational calculus (NRC). As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views.