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The traditional formulation of the program synthesis problem is to find a program that meets a logical correctness specification. When synthesis is successful, there is a guarantee that the implementation satisfies the specification. Unfortunately, synthesis engines are typically monolithic algorithms, and obscure the correspondence between the specification, implementation and user intent. In contrast, humans often include comments in their code to guide future developers towards the purpose and design of different parts of the codebase. In this paper, we introducesubspecificationsas a mechanism to augment the synthesized implementation with explanatory notes of this form. In this model, the user may ask for explanations of different parts of the implementation; the subspecification generated in response is a logical formula that describes the constraints induced on that subexpression by the global specification and surrounding implementation. We develop algorithms to construct and verify subspecifications and investigate their theoretical properties. We perform an experimental evaluation of the subspecification generation procedure, and measure its effectiveness and running time. Finally, we conduct a user study to determine whether subspecifications are useful: we find that subspecifications greatly aid in understanding the global specification, in identifying alternative implementations, and in debugging faulty implementations. 

Synthesizing relational queries from data is challenging in the presence of recursion and invented predicates. We propose a fully automated approach to synthesize such queries. Our approach comprises of two steps: it first synthesizes a non-recursive query consistent with the given data, and then identifies recursion schemes in it and thereby generalizes to arbitrary data. This generalization is achieved by an iterative predicate unification procedure which exploits the notion of data provenance to accelerate convergence. In each iteration of the procedure, a constraint solver proposes a candidate query, and a query evaluator checks if the proposed program is consistent with the given data. The data provenance for a failed query allows us to construct additional constraints for the constraint solver and refine the search. We have implemented our approach in a tool named Mobius. On a suite of 21 challenging recursive query synthesis tasks, Mobius outperforms three state-of-the-art baselines Gensynth, ILASP, and Popper, both in terms of runtime and accuracy. We also demonstrate that the synthesized queries generalize well to unseen data.