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Derived datasets can be defined implicitly or explicitly. An implicitdefinition (of dataset O in terms of datasets I) is a logical specificationinvolving the source data I and the interface data O. It is a valid definitionof O in terms of I, if any two models of the specification agreeing on I agreeon O. In contrast, an explicit definition is a query that produces O from I.Variants of Beth's theorem state that one can convert implicit definitions toexplicit ones. Further, this conversion can be done effectively given a proofwitnessing implicit definability in a suitable proof system.  We prove the analogous implicit-to-explicit result for nested relations:implicit definitions, given in the natural logic for nested relations, can beconverted to explicit definitions in the nested relational calculus (NRC) Wefirst provide a model-theoretic argument for this result, which makes someadditional connections that may be of independent interest. between NRCqueries, interpretations, a standard mechanisms for definingstructure-to-structure translation in logic, and between interpretations andimplicit to definability "up to unique isomorphism". The latter connectionmakes use of a variation of a result of Gaifman concerning "relativelycategorical" theories.  We also provide a proof-theoretic result that provides an effective argument:from a proof witnessing implicit definability, we can efficiently produce anNRC definition. This will involve introducing the appropriate proof system forreasoning with nested sets, along with some auxiliary Beth-type results forthis system. As a consequence, we can effectively extract rewritings of NRCqueries in terms of NRC views, given a proof witnessing that the query isdetermined by the views.

Synthesizing nested relational queries from implicit specifications: via model theory and via proof theory