Brokering of Distributed Mathematical Services

We propose a software framework for brokering mathematical services that are distributed among networked servers. These services can be provided by systems from various mathematical areas, such as numerical computing, computer algebra, constrained solving, or automated theorem proving. The clients of such services may not only be human users but other programs which thus need to understand the meaning of the offered services. Therefore clients and servers require a common language that allows them to understand each others offers respectively demands with respect to the functionality of the services, their interaction protocols, and data formats. The foundation of the proposed framework is a an XML-based language for describing the mathematical problems solved by the services, as well as the characteristic features of the algorithms applied, the constraints of the actual implementations, and the low-level details of the service accesses. A client may then submit task descriptions to a "semantical broker" which locates from the registered service descriptions suitable servers for invocation by the client. The task descriptions may be based on references to registered problems or on formal definitions; in the later case, the broker needs to deduce (in cooperation with external deduction systems) those services that satisfy the corresponding demands. This brokering mechanisms hides low-level implementation issues and can thus serve as the technical component of future mathematical problem solving environments. As a first step towards such an environment, we propose a runtime system that uses the descriptions of compound tasks to coordinate the invocation of the services offered by the broker. Embedded into XML-documents and interpreted by browser applets, such descriptions may yield interactive hypermedia interfaces for distributed mathematical applications.

Nowadays the Internet/World Wide Web is mainly considered as a platform for communication and information. However, more and more it evolves into an infrastructure for providing computerized services such as online banking, book shopping, or travel booking. New software frameworks and standards are developed such that these Web services can be found and appropriately used, not only by humans but also by other services. Our project has aimed at opening the world of Web services to a new application area, that of computer mathematics. The background of this work is the following: on the one hand, researchers have developed a lot of software for a great variety of difficult mathematical problems, from solving algebraic equations to proving logical theorems. On the other hand, many applications in science and engineering are in need of such mathematical problem solving capabilities, e.g. physical or economical simulations. One approach to close this gap between supply and demand is to turn mathematical software into mathematical Web services that application software can access via the Internet. But then the major challenge for an application is to determine the right service to solve its particular problem, to invoke this service appropriately, and to finally interpret its results. This challenge has been addressed by the three major outcomes of our project: 1. We have employed open source software developed by the Web service community for building mathematical services. This has for the first time demonstrated how the technology developed for another purpose can be also successfully used in the area of computer mathematics. 2. We have developed a structured language for describing the form and the meaning of mathematical services, i.e., the rules for their invocation and for the interpretation of their behavior. A major European research consortium has adopted this idea using a simplified version of our description language. 3. We have implemented a Web-based repository for registering our structured service descriptions and querying it according to different criteria. Our next steps will focus on elaborating a brokering mechanism that with the help of this basic technology will provide a semi-intelligent "matching service" between mathematical service providers and client applications.