Martin Simons

V-nested parallelism in C

This paper describes the integration of nested data parallelism into imperative languages using the example of C. Unlike flat data parallelism, nested data parallelism directly provides means for handling irregular data structures and certain forms of control parallelism, such as divide-and-conquer algorithms, thus enabling the programmer to express such algorithms far more naturally. Existing work deals with nested data parallelism in a functional environment, which does help avoid a set of problems, but makes efficient implementations more complicated. Moreover functional languages are not readily accepted by programmers used to languages, such as Fortran and C, which are currently predominant in programming parallel machines. In this paper, we introduce the imperative data-parallel language V and give an overview of its implementation.

A Calculational Approach to Flattening Nested Data Parallelism in Functional Languages

The data-parallel programming model is currently the most successful model for programming massively parallel computers. Unfortunately, it is, in its present form, restricted to exploiting flat data parallelism, which is not suitable for some classes of algorithms, e.g. those operating on irregular structures. Recently, some effort has been made to implement nested data-parallel programs efficiently by compiling them into equivalent flat programs using a transformation called flattening. However, previous translations of nested into flat data-parallel programs have proved unwieldy when it comes to inventing and specifying optimizations and verifying the translation. This paper presents a new formalization of the flattening transformation in a calculational style. The formalization is easily verified and provides a good starting point for the development of new optimizations. Some optimizations invented on the basis of this new formalism are described. Furthermore, we present practical evidence obtained by experimenting with an implementation of the transformation.

Proof Presentation for Isabelle

We present an approach to the intelligible communication of formal proofs. Observing a close correspondence between the activities of formal-proof development and program development, and using this as a guideline, we apply well-known principles from program design to proof design and presentation, resulting in formal proofs presented in a literate style, that are hierarchically structured and emphasize calculation. We illustrate the practicability of this approach by describing its instantiation to the case of the interactive theorem prover Isabelle and by presenting a proof of the Church-Rosser theorem.

Enlarging the scope of vector-based computations: extending Fortran 90 by nested data parallelism

This paper describes the integration of nested data parallelism into Fortran 90. Unlike flat data parallelism, nested data parallelism directly provides means for handling irregular data structures and certain forms of control parallelism, such as divide-and-conquer algorithms thus enabling the programmer to express such algorithms far more naturally. Existing work deals with nested data parallelism in a functional environment, which does help avoid a set of problems, but makes efficient implementations more complicated. Moreover functional languages are not readily accepted by programmers used to languages such as Fortran and C, which are currently predominant in programming parallel machines. In this paper, we introduce the imperative data-parallel language Fortran 90V and give an overview of its implementation.

An approach to literate and structured formal developments

We present an approach to the literate and structured presentation of formal developments. We discuss the presentation of formal developments in a logical framework and distinguish three aspects: language-related aspects, structural aspects of proofs, and presentational aspects. We illustrate the approach by two examples: a simple mathematical proof of the Knaster-Tarski fixpoint theorem, and a formalization of the VDM development of a revision management system.

Algebraic Composition and Refinement of Proofs

We present an algebraic calculus for proof composition and refinement. Fundamentally, proofs are expressed at successive levels of abstraction, with the perhaps unconventional principle that a formula is considered to be its own most abstract proof, which may be refined into increasingly concrete proofs. Consequently, we suggest a new paradigm for expressing proofs, which views theorems and proofs as inhabiting the same semantic domain. This algebraic/model-theoretical view of proofs distinguishes our approach from conventional typetheoretical or sequent-based approaches in which theorems and proofs are different entities. All the logical concepts that make up a formal system — formulas, inference rules, and derivations — are expressible in terms of the calculus itself. Proofs are constructed and structured by means of a composition operator and a consequential rule-forming operator. Their interplay and their relation wrt. the refinement order are expressed as algebraic laws.

Darstellung formaler Beweise

In dieser Arbeit wird ein Ansatz zur verstandlichen Darstellung formaler Beweise vorgestellt. Die enge Ubereinstimmung zwischen dem Entwicklungsprozes formaler Beweise und dem Entwicklungsprozes eines Programms erlaubt, bekannte Prinzipien und Techniken des Programmentwurfs auf den Entwurf und die Darstellung formaler Beweise zu ubertragen. Dies fuhrt zu hierarchisch gegliederten formalen Beweisen, die in einem Stil prasentiert werden, der Knuth’s „literate programming” nachempfunden ist und der transformationelle Beweise betont. Die Beschreibung zweier Experimente illustriert die Praktikabilitat des Ansatzes. Hierbei werden, dem Prasentationsansatz folgend, formale Beweise dargestellt, die zum einen in einer statischen Beweissprache ausgedruckt und zum anderen fur einen interaktiven Theorembeweiser programmiert sind. Es wird jeweils beschrieben, wie der Prasentationsansatz fur den entsprechenden logischen Kalkul angepast wurde und welche Werkzeuge die Darstellung unterstutzen. Anschliesend werden mit diesen Werkzeugen formale Beweise von nichttrivialen mathematischen Satzen dargestellt. Die Beobachtung, das zwei wesentliche Elemente des Prasentationsansatzes — namlich die Komposition von formalen Herleitungen und deren hierarchische Verfeinerung — in herkommlichen logischen Beweiskalkulen fehlen, fuhrt zur Formulierung eines algebraischen Beweiskalkuls, in dem formale Herleitungen selbst Formeln sind, sowie miteinander komponiert und verfeinert werden konnen. Der Hauptbeitrag dieser Arbeit besteht demnach in dem praktischen Nachweis, das Komposition und Verfeinerung als primitive Konstruktionselemente formaler Beweiskalkule zur Verbesserung der verstandlichen Darstellung formaler Beweise beitragen, der Formulierung einer algebraischen Semantik fur einen solchen Beweiskalkul und der Entwicklung von Techniken und Werkzeugen zur Prasentation formaler Beweise in herkommlichen formalen Beweissystemen.

An experimental support system for formal mathematical reasoning

Requirements for tools which support the creation and the intelligible presentation of formal deductions are investigated. They are contrasted with requirements which emphasize the interactive construction of correct proofs. As an example, the design and the implementation of a set of support tools for Deva is described. Deva is a typed functional language and has been used in a number of case-studies on formal program development. The use of this toolset is illustrated by impressions of a working session.