3rd SCIEnce Workshop

SCIEnce -- Symbolic Computation Infrastructure for Europe [Slides]

Plenary talk introducing the EU Framework 6 project SCIEnce that brings together developers of computer algebra systems, experts in computational algebra, OpenMath, and parallel computations.

A New Lingua Franca for Symbolic Computation: Easy Composition of Symbolic Computation Software [Slides]

The SCIEnce project aims to provide key infrastructure for symbolic computation research. A primary outcome of the project is that we have developed a new way of combining computer algebra systems using the Symbolic Computation Software Composability Protocol (SCSCP), in which both protocol messages and data are encoded in the OpenMath format. We describe SCSCP middleware and APIs, outline some implementations for various Computer Algebra Systems (CAS), and show how SCSCP-compliant components may be combined to solve scientific problems that can not be solved within a single CAS, or may be organised into a system for distributed parallel computations.

Computing braid orbits with SCSCP [Slides]

Braid orbits on generating tuples of a finite group G classify Hurwitz loci of curves admitting G as a group of automorphisms. They can be computed using the GAP package BRAID. As applications require larger and larger orbits, in this talk we will discuss several ideas how parallel computation can be utilized in order to handle greater orbit lengths. We also present a particular realization, based on SCSCP, which we used to compute an orbit well beyond what was possible before.

SCSCP support in the computer algebra system TRIP [Slides]

As users use general computer algebra systems but need optimized functions available in the computer algebra system TRIP, dedicated to celestial mechanics and perturbation series, the SCSCP C Library had been developed to implement the SCSCP protocol and allows TRIP to run as a client or server. We present the recent improvements of the library and the status of SCSCP support in TRIP. The performance to compute the expansions of the basic functions of the two-body problem over SCSCP will be discussed.

Type-Theoretic Interface for Connecting Coq Theorem Prover to a Computer Algebra System [Slides]

Communication between theorem provers or proof assistants on one side and computer algebra systems (CAS) on the other has been dealt with many times before. We are trying to harness a particular instance of this manifold problem. The recent protocol SCSCP provides an abstraction over particular choice of CAS and allows to work with a relatively uniform representation of mathematical objects in the form of OpenMath terms. The proof assistant Coq is based on constructive type theory and is an extremely expressive pure functional language. From Coq, we are beginning to exploit computational versatility of GAP through SCSCP. Problems we are studying range from the most general ones such as the absence of a formal semantics for OpenMath, which makes it impossible to solve the question in theory, to more specific ones such as recovery of type theoretic structure from barebone OpenMath terms. More precisely, the structure should be that of packed classes of SSReflect (influenced by type classes of Haskell) created by the Mathematical Components team of INRIA-MSR.

Ever-Decreasing Circles: a Skeleton for Parallel Orbit Calculations in Eden [Slides]

The Orbit algorithm possesses a central place in computational algebra. We consider how the Orbit algorithm can be encoded in the functional programming language Haskell , and thence how a parallel skeleton can be developed to cover a variety of Orbit calculations, using the Eden parallel dialect of Haskell. Our results with a synthetic benchmark on a modern multicore machine clearly demonstrate the potential of this skeleton to allow simple but effective parallel implementation of the Orbit algorithm, giving relative speedups of up to 8.243 on eight cores.

Presentations of software products developed in the SCIEnce project:

• SCSCP package for GAP (Alexander Konovalov)
• Computations with TRIP and interactions with other SCSCP engines (Mickaël Gastineau)
• A new framework for dynamic composition of symbolic grid services (Georgiana Macariu)

A practical method to create mathematical e-contents for MathDox using LaTeX

In using e-learning material for mathematics to let the student practice behind the computer, it is useful that exercises are reusable and provide feedback. The MathDox software developed at the TU Eindhoven allows for exercises in multiple steps, with random values, and custom feedback. Creating such exercises, however involves a lot of work and programming. One cannot expect authors to be familiar with the MathDox system, so for them tools are needed which are adapted to their usual word processing routines.

To generate content for the MathDox system, we have constructed a format based on LaTeX (Lamport, 1994), which is well-known to university mathematicians. LaTeX enables the author to create (static) theory pages and (possibly) elaborate exercises. Our format can then be used to generate theory pages and exercises with the desired interactivity from LaTeX input. However, to enable automatic grading of the exercises, some of the mathematics is encoded in Popcorn. This is a language, developed within the SCIEnce project to semantically encode mathematics according to OpenMath standard. This encoding makes not only authoring of exercises easier, it also makes it possible to use various Openmath aware computer algebra systems to be used for automatic grading of the exercises.

This set-up is currently being used in the projects Experience Mathness and TELMME. These two projects are aimed at large groups of students entering either a bachelor or master program in the sciences.

Summarizing what we have presented today, we would like to invite for cooperation both developers and users of computer algebra systems and other scientific software. We hope that more SCSCP-compliant software tools will appear in the future, and existing/coming APIs may facilitate their appearance (see e.g. the screencast about the Java API). It's important to see that SCSCP is a standard, which may be implemented in various systems following different conventions used in them, and that it's not necessary to implement full range of OpenMath content dictionaries (CDs) to support SCSCP. Instead, CDs may be classified to required, recommended and optional, and to be SCSCP-compliant a software package need only to understand CDs in which SCSCP messages and their elements are defined. We would be very glad to offer advise and hear about interesting applications and further feedback from users who will use SCSCP to combine different systems.