An Interpolation-based Compiler and Optimizer for Relational Queries (System design Report)

Abstract

We outline the implementation of a query compiler for relational queries that generates query plans with respect to a database schema, that is, a set of arbitrary first-order constraints, and a distinguished subset of predicate symbols from the underlying signature that correspond to access paths. The compiler is based on a variant of the Craig interpolation theorem, with reasoning realized via a modified analytic tableau proof procedure. This procedure decouples the generation of candidate plans that are interpolants from the tableau proof procedure, and applies A*-based search with respect to an external cost model to arbitrate among the alternative candidate plans. The tableau procedure itself is implemented as a virtual machine that operates on a compiled and optimized byte-code that faithfully implements reasoning with respect to the database schema constraints and a user query.

How Can Reasoners Simplify Database Querying (And Why Haven't They Done It Yet)?

“…The more general enumeration approach via reduction to validity is discussed in [41,42,131]. Implementations of the approach have been developed by Toman and Weddell [98,132] as well as [36].…”

Craig Interpolation and Access Interpolation with Clausal First-Order Tableaux

“…In [59,30,60] query reformulation is indeed based on computing alternate interpolantseach considered as representing a query plan -and comparing them with a cost function. This approach has been refined in [30,60] with a condensed representation of a set of tableaux in a single structure.…”

“…In [59,30,60] query reformulation is indeed based on computing alternate interpolantseach considered as representing a query plan -and comparing them with a cost function. This approach has been refined in [30,60] with a condensed representation of a set of tableaux in a single structure. An advanced system that interleaves the generation of a pair of such condensed tableaux, one for each of the two interpolation input formulas, with detecting when their combination would be closed is described in [60].…”

“…By Craig's interpolation theorem [19], for two first-order formulas F and G such that F entails G there exists a third first-order formula H that is entailed by F, entails G and is such that all predicate and function symbols occurring in it occur in both F and G. Such a Craig interpolant H can be constructed from given formulas F and G, for example by a calculus that allows to extract H from a proof that F entails G, or, equivalently, that the implication F → G is valid. Automated construction of interpolants has many applications, in the area of computational logic most notably in symbolic model checking, initiated with [43], and in query reformulation [42,48,14,59,17,8,30,10,9,60]. The foundation for the E-mail: info@christophwerhard.com latter application field is the observation that a reformulated query can be viewed as a definiens of a given query where only symbols from a given set, the target language of the reformulation, occur in the definiens.…”

Craig Interpolation with Clausal First-Order Tableaux

“…that F → G is valid, or, again equivalently, a refutation of F ∧ ¬G. Automated construction of interpolants has many applications, in the area of computational logic most notably in symbolic model checking, initiated with [57], and in query reformulation [71,55,63,15,75,19,7,36,9,8,76]. The foundation for the latter application field is the observation that a reformulated query can be viewed as a definiens of a given query where only symbols from a given set, the target language of the reformulation, occur in the definiens.…”