Generating collection transformations from proofs

Abstract

Nested relations, built up from atomic types via product and set types, form a rich data model. Over the last decades the nested relational calculus, NRC, has emerged as a standard language for defining transformations on nested collections. NRC is a strongly-typed functional language which allows building up transformations using tupling and projections, a singleton-former, and a map operation that lifts transformations on tuples to transformations on sets. In this work we describe an alternative declarative method of describing transformations in logic. A formula with distinguished inputs and outputs gives an implicit definition if one can prove that for each input there is only one output that satisfies it. Our main result shows that one can synthesize transformations from proofs that a formula provides an implicit definition, where the proof is in an intuitionistic calculus that captures a natural style of reasoning about nested collections. Our polynomial time synthesis procedure is based on an analog of Craig's interpolation lemma, starting with a provable containment between terms representing nested collections and generating an NRC expression that interpolates between them. We further show that NRC expressions that implement an implicit definition can be found when there is a classical proof of functionality, not just when there is an intuitionistic one. That is, whenever a formula implicitly defines a transformation, there is an NRC expression that implements it.

Synthesizing Nested Relational Queries from Implicit Specifications

“…Our effective Beth-like theorem for nested relations extends two results in [5]. One is an ineffective result, which makes use of an idea in [15], but without any effectivity.…”

“…Our result implies that if we have NRC views ì 𝑉 that determine an NRC query 𝑄, then we can generate -from a suitable proof -an NRC rewriting of 𝑄 in terms of ì 𝑉 . The fact that such an NRC rewriting exists whenever there is a functional relationship was proven in [5]. But the argument for existence given there is model-theoretic.…”

“…But the argument for existence given there is model-theoretic. Indeed, [5] listed coming up with an algorithm for constructing such a rewriting for a classically-complete proof system as a major open issue.…”

Query Lifting

“…Van den Bussche's work on simulating queries on nested sets using flat ones [54] has also inspired subsequent work on query shredding, flattening and (in this paper) lifting, though the simulation technique itself does not appear practical (as discussed in the extended version of Cheney et al [9]). More recently, Benedikt and Pradic [1] presented results on representing queries on nested collections using a bounded number of interpretations (first-order logic formulas corresponding to definable flat query expressions) in the context of their work on synthesizing N RC queries from proofs. This approach considers set-valued N RC only, and its relationship to our approach should be investigated further.…”

Query Lifting: Language-integrated query for heterogeneous nested collections

“…Van den Bussche's work on simulating queries on nested sets using flat ones [53] has also inspired subsequent work on query shredding, flattening and (in this paper) lifting, though the simulation technique itself does not appear practical (as discussed in the extended version of Cheney et al [9]). More recently, Benedikt and Pradic [1] presented results on representing queries on nested collections using a bounded number of interpretations (first-order logic formulas corresponding to definable flat query expressions) in the context of their work on synthesizing N RC queries from proofs. This approach considers set-valued N RC only, and its relationship to our approach should be investigated further.…”