Olga Caprotti

Mathematics on the (Semantic) NET

Although web service technology is becoming more prevalent the mechanisms for advertising and discovering web services are still at a rudimentary stage. WSDL provides information about service name and parameters for the purpose of invocation. UDDI provides a set of WSDL documents matching keywords in a query. The aim of the Mathematics On the NET (MONET) project is to deliver a proof-of-concept demonstration of a framework for mathematical web services which uses semantic web technologies to broker between user requirements and deployed services. This requires mechanisms for describing mathematical objects and properties so that a piece of software can evaluate the applicability of a particular service to a given problem. Thus we describe our Mathematical Service Description Language (MSDL), with its ontological grounding in OpenMath and outline its role in service brokerage and service composition within MONET. We believe similar issues arise in many other (scientific) domains, and the leverage obtained here, through the formal background of mathematics, suggests a road-map for the development of similar domain-specific service description languages.

Connecting Proof Checkers and Computer Algebra Using OpenMath

Interactive mathematics over the Internet is particularly appealing in that it can take advantage of the wealth of resources available online. In particular, a problem-solving framework integrating computing and proving tools is of great interest. Several modes of advantageous cooperation can be envisioned. Proving is undoubtedly useful to computation for checking assertions and side conditions, whereas computation is useful for shortening the lengths of proofs. Both modes are definitely needed in interactive mathematics. In the interaction we do not want to waste resources by sending or receiving meaningless requests. There Strong OpenMath comes in, as it is possible to type-check the well-typedeness, hence the meaningfullness, of the mathematical objects it represents.

JAVA Phrasebooks for computer algebra and automated deduction

We discuss the developments within the OpenMath framework regarding programs that make it possible for a software package to interact with other packages or agents, namely Phrasebooks. Recently, several implementations of Phrasebooks have come about; we shall describe some of them. Most of the software is freely available, so, by downloading it and inspecting implemented examples, builders of software packages can pick up the examples and provide the mathematical community with further computational servers that can easily be interfaced.

On the Role of OpenMath in Interactive Mathematical Documents

The standard OpenMath is an enabling technology for creating an integrated computer environment in which software packages for computer algebra and for proof checking can be combined. Here we demonstrate how OpenMath can be employed for generating interactive mathematical documents containing primality proofs. Our case study takes place within a browser; once a prime number is specified, a document appears summarizing the proof in a number of assertions. By clicking an assertion regarding the truth of an arithmetic equality, a computer algebra calculation is invoked verifying the equality. By clicking an assertion regarding a specific mathematical lemma called Pocklington?s Criterion, a verification of the corresponding formal proof is carried out by a proof checker. Moreover, the whole document is structured in such a way that it can be easily translated to a formal proof object. OpenMath supports the interaction between the document as it appears in the browser and the mathematical software packages. This paper begins with an introduction to OpenMath and a brief comparison with MathML.

Integrating Computational and Deduction Systems Using OpenMath

The standard OpenMath is a crucial ingredient for creating an integrated environment combining systems for computer algebra with proof checkers. OpenMath consists of a formal grammar of OpenMath objects, their encodings, Content Dictionaries, Phrasebooks and other tools. The OpenMath standard allows integration of computational systems of different kind. Here we demonstrate how OpenMath works by setting up an environment in which Maple expressions are type-checked by the proof checkers Lego and Coq.

OpenMath Technology for Interactive Mathematical Documents

New technologies such as XML, XSL and both MathML and OpenMath make it possible to bring mathematics to the Internet. Indeed, OpenMath, a markup language for mathematical content, and OmDoc, its extension to mathematical documents, open a way of communicating mathematics between computers, between software applications and over the Internet without losing information. In this paper we describe the latest applications of OpenMath related technologies for Interactive Mathematical Documents. As an example we describe the way we incorporate these new technologies in a new version of Algebra Interactive, an interactive course on first and second year university algebra.

On Communicating Proofs in Interactive Mathematical Documents

There is a wealth of interactive mathematics available on the web. Examples range from animated geometry to computing the nth digit in the expansion of π. However, proofs seem to remain static and at most they provide interaction in the form of links to definitions and other proofs. In this paper, we want to show how interactivity can be included in proofs themselves by making them executable, human-readable, and yet formal. The basic ingredients are formal proof-objects, OpenMath-related languages, and the latest eXtensible Markup Language (XML) technology. We exhibit, by an example taken from a formal development in number theory, the final product of which we believe to be a truly interactive mathematical document.