Extending the Applicability Condition in the Formal System \lambda\delta

The formal system \lambda\delta is a typed lambda calculus derived from \Lambda\infinity, aiming to support the foundations of Mathematics that require an underlying theory of expressions (for example the Minimal Type Theory). The system is developed in the context of the Hypertextual Electronic Library of Mathematics as a machine-checked digital specification, that is not the formal counterpart of previous informal material. The first version of the calculus appeared in 2006 and proved unsatisfactory for some reasons. In this article we present a revised version of the system and we prove three relevant desired properties: the confluence of reduction, the strong normalization of an extended form of reduction, known as the ``big tree'' theorem, and the preservation of validity by reduction. To our knowledge, we are presenting here the first fully machine-checked proof of the ``big tree'' theorem for a calculus that includes \Lambda\infinity

SMT Solving for Functional Programming over Infinite Structures

We develop a simple functional programming language aimed at manipulating infinite, but first-order definable structures, such as the countably infinite clique graph or the set of all intervals with rational endpoints. Internally, such sets are represented by logical formulas that define them, and an external satisfiability modulo theories (SMT) solver is regularly run by the interpreter to check their basic properties. The language is implemented as a Haskell module.Comment: In Proceedings MSFP 2016, arXiv:1604.0038

A survey of parallel execution strategies for transitive closure and logic programs

An important feature of database technology of the nineties is the use of parallelism for speeding up the execution of complex queries. This technology is being tested in several experimental database architectures and a few commercial systems for conventional select-project-join queries. In particular, hash-based fragmentation is used to distribute data to disks under the control of different processors in order to perform selections and joins in parallel. With the development of new query languages, and in particular with the definition of transitive closure queries and of more general logic programming queries, the new dimension of recursion has been added to query processing. Recursive queries are complex; at the same time, their regular structure is particularly suited for parallel execution, and parallelism may give a high efficiency gain. We survey the approaches to parallel execution of recursive queries that have been presented in the recent literature. We observe that research on parallel execution of recursive queries is separated into two distinct subareas, one focused on the transitive closure of Relational Algebra expressions, the other one focused on optimization of more general Datalog queries. Though the subareas seem radically different because of the approach and formalism used, they have many common features. This is not surprising, because most typical Datalog queries can be solved by means of the transitive closure of simple algebraic expressions. We first analyze the relationship between the transitive closure of expressions in Relational Algebra and Datalog programs. We then review sequential methods for evaluating transitive closure, distinguishing iterative and direct methods. We address the parallelization of these methods, by discussing various forms of parallelization. Data fragmentation plays an important role in obtaining parallel execution; we describe hash-based and semantic fragmentation. Finally, we consider Datalog queries, and present general methods for parallel rule execution; we recognize the similarities between these methods and the methods reviewed previously, when the former are applied to linear Datalog queries. We also provide a quantitative analysis that shows the impact of the initial data distribution on the performance of methods

Formal Derivation of Concurrent Garbage Collectors

Concurrent garbage collectors are notoriously difficult to implement correctly. Previous approaches to the issue of producing correct collectors have mainly been based on posit-and-prove verification or on the application of domain-specific templates and transformations. We show how to derive the upper reaches of a family of concurrent garbage collectors by refinement from a formal specification, emphasizing the application of domain-independent design theories and transformations. A key contribution is an extension to the classical lattice-theoretic fixpoint theorems to account for the dynamics of concurrent mutation and collection.Comment: 38 pages, 21 figures. The short version of this paper appeared in the Proceedings of MPC 201

Service oriented transitive closure solution

The goal of this project is a service based solution that utilizes parallel and distributed processing algorithms to solve the transitive closure problem for a large dataset. A dataset may be view conceptually as a table in a database, with a physical structure representing a file containing a sequence of records and fields. Two records are said to be transitively related if and only if they are directly related due to sharing of one or more specific fields, or a sequence may be made from one record to the other under the condition that all intermediate entries are related the immediate previous and subsequent entry. The transitive closure problem is to cluster the records in a dataset into groups such that all transitively related records are in one group. An approach to solve this problem is to divide the task into two separate problems. The first of these problems is the processing of the dataset, and thus generating a set of pairs. Each of these pairs would include two record identifiers, and these pairs would exist if and only if these two records were directly related. The second of these problems is to use the record pairs to cluster the records into transitive closures. The current software solution solves this second sub problem through the reading of record pairs, produced by a different software solution, and writes the completed results of the transitive closure problem to a file. This thesis studies how to enhance the current software solution in such a way that it becomes a service . The study includes designing, implementing, testing, and evaluating the enhanced solution. The service model identifies an aspect that would potentially benefit from restructuring or addition of functionality. A current issue is the lack of an ability to fetch transitive closure from within the solution upon the completion of a job, and is thus limited in itsdirect use with other processes or applications