Hierarchical index sets in algebraic modelling languages

Multi-dimensional algebraic modelling languages make extensive use of simple and compound index sets. In this paper the multi-dimensional modelling paradigm is extended with the concept of a hierarchical index set to support the use of hierarchical data structures. The appropriate reference and indexing mechanisms are introduced, together with mechanisms to support various set operations. Special attention is paid to the Cartesian product of two hierarchical index sets. The modelling of multi-stage programming models is supported through the introduction of a hierarchical indexing mechanism. The extensions proposed in this paper are compared to existing facilities designed to support the modelling of hierarchical structures

Towards MKM in the Large: Modular Representation and Scalable Software Architecture

MKM has been defined as the quest for technologies to manage mathematical knowledge. MKM "in the small" is well-studied, so the real problem is to scale up to large, highly interconnected corpora: "MKM in the large". We contend that advances in two areas are needed to reach this goal. We need representation languages that support incremental processing of all primitive MKM operations, and we need software architectures and implementations that implement these operations scalably on large knowledge bases. We present instances of both in this paper: the MMT framework for modular theory-graphs that integrates meta-logical foundations, which forms the base of the next OMDoc version; and TNTBase, a versioned storage system for XML-based document formats. TNTBase becomes an MMT database by instantiating it with special MKM operations for MMT.Comment: To appear in The 9th International Conference on Mathematical Knowledge Management: MKM 201

Reify Your Collection Queries for Modularity and Speed!

Modularity and efficiency are often contradicting requirements, such that programers have to trade one for the other. We analyze this dilemma in the context of programs operating on collections. Performance-critical code using collections need often to be hand-optimized, leading to non-modular, brittle, and redundant code. In principle, this dilemma could be avoided by automatic collection-specific optimizations, such as fusion of collection traversals, usage of indexing, or reordering of filters. Unfortunately, it is not obvious how to encode such optimizations in terms of ordinary collection APIs, because the program operating on the collections is not reified and hence cannot be analyzed. We propose SQuOpt, the Scala Query Optimizer--a deep embedding of the Scala collections API that allows such analyses and optimizations to be defined and executed within Scala, without relying on external tools or compiler extensions. SQuOpt provides the same "look and feel" (syntax and static typing guarantees) as the standard collections API. We evaluate SQuOpt by re-implementing several code analyses of the Findbugs tool using SQuOpt, show average speedups of 12x with a maximum of 12800x and hence demonstrate that SQuOpt can reconcile modularity and efficiency in real-world applications.Comment: 20 page

Feat: Functional Enumeration of Algebraic Types

In mathematics, an enumeration of a set S is a bijective function from (an initial segment of) the natural numbers to S. We define "functional enumerations" as efficiently computable such bijections. This paper describes a theory of functional enumeration and provides an algebra of enumerations closed under sums, products, guarded recursion and bijections. We partition each enumerated set into numbered, finite subsets. We provide a generic enumeration such that the number of each part corresponds to the size of its values (measured in the number of constructors). We implement our ideas in a Haskell library called testing-feat, and make the source code freely available. Feat provides efficient "random access" to enumerated values. The primary application is property-based testing, where it is used to define both random sampling (for example QuickCheck generators) and exhaustive enumeration (in the style of SmallCheck). We claim that functional enumeration is the best option for automatically generating test cases from large groups of mutually recursive syntax tree types. As a case study we use Feat to test the pretty-printer of the Template Haskell library (uncovering several bugs)

Refactoring pattern matching

Defining functions by pattern matching over the arguments is advantageous for understanding and reasoning, but it tends to expose the implementation of a datatype. Significant effort has been invested in tackling this loss of modularity; however, decoupling patterns from concrete representations while maintaining soundness of reasoning has been a challenge. Inspired by the development of invertible programming, we propose an approach to program refactoring based on a right-invertible language rinv—every function has a right (or pre-) inverse. We show how this new design is able to permit a smooth incremental transition from programs with algebraic datatypes and pattern matching, to ones with proper encapsulation, while maintaining simple and sound reasoning