Combining Algebra and Higher-Order Types

We study the higher-order rewrite/equational proof systems obtained by adding the simply typed lambda calculus to algebraic rewrite/equational proof systems. We show that if a many-sorted algebraic rewrite system has the Church-Rosser property, then the corresponding higher-order rewrite system which adds simply typed ß-reduction has the Church-Rosser property too. This result is relevant to parallel implementations of functional programming languages. We also show that provability in the higher-order equational proof system obtained by adding the simply typed ß and η axioms to some many-sorted algebraic proof system is effectively reducible to provability in that algebraic proof system. This effective reduction also establishes transformations between higher-order and algebraic equational proofs, transformations which can be useful in automated deduction

Faster Query Answering in Probabilistic Databases using Read-Once Functions

A boolean expression is in read-once form if each of its variables appears exactly once. When the variables denote independent events in a probability space, the probability of the event denoted by the whole expression in read-once form can be computed in polynomial time (whereas the general problem for arbitrary expressions is #P-complete). Known approaches to checking read-once property seem to require putting these expressions in disjunctive normal form. In this paper, we tell a better story for a large subclass of boolean event expressions: those that are generated by conjunctive queries without self-joins and on tuple-independent probabilistic databases. We first show that given a tuple-independent representation and the provenance graph of an SPJ query plan without self-joins, we can, without using the DNF of a result event expression, efficiently compute its co-occurrence graph. From this, the read-once form can already, if it exists, be computed efficiently using existing techniques. Our second and key contribution is a complete, efficient, and simple to implement algorithm for computing the read-once forms (whenever they exist) directly, using a new concept, that of co-table graph, which can be significantly smaller than the co-occurrence graph.Comment: Accepted in ICDT 201

On the Limitations of Provenance for Queries With Difference

The annotation of the results of database transformations was shown to be very effective for various applications. Until recently, most works in this context focused on positive query languages. The provenance semirings is a particular approach that was proven effective for these languages, and it was shown that when propagating provenance with semirings, the expected equivalence axioms of the corresponding query languages are satisfied. There have been several attempts to extend the framework to account for relational algebra queries with difference. We show here that these suggestions fail to satisfy some expected equivalence axioms (that in particular hold for queries on "standard" set and bag databases). Interestingly, we show that this is not a pitfall of these particular attempts, but rather every such attempt is bound to fail in satisfying these axioms, for some semirings. Finally, we show particular semirings for which an extension for supporting difference is (im)possible.Comment: TAPP 201

Provenance for Aggregate Queries

We study in this paper provenance information for queries with aggregation. Provenance information was studied in the context of various query languages that do not allow for aggregation, and recent work has suggested to capture provenance by annotating the different database tuples with elements of a commutative semiring and propagating the annotations through query evaluation. We show that aggregate queries pose novel challenges rendering this approach inapplicable. Consequently, we propose a new approach, where we annotate with provenance information not just tuples but also the individual values within tuples, using provenance to describe the values computation. We realize this approach in a concrete construction, first for "simple" queries where the aggregation operator is the last one applied, and then for arbitrary (positive) relational algebra queries with aggregation; the latter queries are shown to be more challenging in this context. Finally, we use aggregation to encode queries with difference, and study the semantics obtained for such queries on provenance annotated databases

Data Structures and Data Types in Object-Oriented Databases

The possibility of finding a static type system for object-oriented programming languages was initiated by Cardelli [Car88, CW85] who showed that it is possible to express the polymorphic nature of functions such a

Algorithms for Provisioning Queries and Analytics

Provisioning is a technique for avoiding repeated expensive computations in what-if analysis. Given a query, an analyst formulates $k$ hypotheticals, each retaining some of the tuples of a database instance, possibly overlapping, and she wishes to answer the query under scenarios, where a scenario is defined by a subset of the hypotheticals that are "turned on". We say that a query admits compact provisioning if given any database instance and any $k$ hypotheticals, one can create a poly-size (in $k$) sketch that can then be used to answer the query under any of the $2^{k}$ possible scenarios without accessing the original instance. In this paper, we focus on provisioning complex queries that combine relational algebra (the logical component), grouping, and statistics/analytics (the numerical component). We first show that queries that compute quantiles or linear regression (as well as simpler queries that compute count and sum/average of positive values) can be compactly provisioned to provide (multiplicative) approximate answers to an arbitrary precision. In contrast, exact provisioning for each of these statistics requires the sketch size to be exponential in $k$. We then establish that for any complex query whose logical component is a positive relational algebra query, as long as the numerical component can be compactly provisioned, the complex query itself can be compactly provisioned. On the other hand, introducing negation or recursion in the logical component again requires the sketch size to be exponential in $k$. While our positive results use algorithms that do not access the original instance after a scenario is known, we prove our lower bounds even for the case when, knowing the scenario, limited access to the instance is allowed

Lambda Calculus,Conservative Extension and Structural Induction

The issue of whether embedding algebraic theories in higher-order theories such as the simply typed and polymorphic lambda calculi is of interest in programming language design. The establishment of such a conservative extension result permits modularity in the verification of the correctness of datatype and function implementations. In earlier work [Breazu-Tannen & Meyer 1987a], [Breazu-Tannen & Meyer 1987b] and [Breazu-Tannen 1988], conservative extension results have been obtained for algebraic theories. However, in modelling inductive datatypes, the principle of structural induction needs to be admitted in the inference system, and the question of whether conservative extension holds in the presence of the principle of structural induction needs to be addressed. In this paper we look at the question of whether inductive algebraic theories are conservatively extended when embedded in the simply typed lambda calculus

A PC Chase

PC stands for path-conjunctive, the name of a class of queries and dependencies that we define over complex values with dictionaries. This class includes the relational conjunctive queries and embedded dependencies, as well as many interesting examples of complex value and oodb queries and integrity constraints. We show that some important classical results on containment, dependency implication, and chasing extend and generalize to this class