Mathematical Semantic Markup in a Wiki: The Roles of Symbols and Notations

We present semantic markup as a way to exploit the semantics of mathematics in a wiki. Semantic markup makes mathematical knowledge machine-processable and thus allows for a multitude of useful applications. But as it is hard to read and write for humans, an editor needs to understand its inherent semantics and allow for a humanreadable presentation. The semantic wiki SWiM offers this support for the OpenMath markup language. Using OpenMath as an example, we present a way of integrating a semantic markup language into a semantic wiki using a document ontology and extracting RDF triples from XML markup. As a benefit gained from making semantics explicit, we show how SWiM supports the collaborative editing of definitions of mathematical symbols and their visual appearance. 1 Making Mathematical Wikis More Semantic What does a wiki need in order to support mathematics in a semantic way? First, there needs to be a way to edit mathematical formulæ. Many wikis offer a LATEX-like syntax for that, and they have been used to build large mathematical knowledge collections, such as the mathematical sections of Wikipedia [30] or the mathematics-only encyclopædia PlanetMath [16]. But LATEX, which is mostly presentation-oriented, despite certain macros like \frac{num}{denom} or \binom{n}{k}, is not sufficient to capture the semantics of mathematics. One could write O(n2 + n), which could mean “O times n2 + n” (with redundant brackets), or “O (being a function) applied to n2 + n”, or the set of all integer functions not growing faster than n2 +n, and just by common notational convention we know that the latter is most likely to hold. For being able to express the semantics of O(n2+n), we need to make explicit that the Landau symbol O is a set construction operator and n is a variable. The meaning of O has to be defined in a vocabulary shared among mathematical applications such as our wiki. This is analogous to RDF, where a vocabulary— also called ontology—has to be defined before one can use it to create machineprocessable and exchangeable RDF statements. In a mathematical context, these vocabularies are called content dictionaries (CDs). As with ontology languages, one can usually do more than just listing symbols and their descriptions in a CD: defining symbols formally in terms of other symbols, declaring their types formally, and specifying their visual appearance. Thus, CDs themselves are special mathematical documents that could again be made available in a mathematical wiki. Then it would be possible to create an unambiguous link from any occurrence of O in a formula to its definition in the wiki, and knowledge from the wiki could be shared with any other mathematical application supporting this CD. As a practical solution, we present the OpenMath CD language in sect. 2 and its integration into the semantic wiki SWiM in sect. 4. 2 Semantic Markup for Mathematics with OpenMath Semantic markup languages for mathematics address the problems introduced in sect. 1 by offering an appropriate expressivity and semantics for defining symbols and other structures of mathematical knowledge. This is a common approach to knowledge representation not only in mathematics, but generally in science1. OpenMath [7] is a markup language for expressing the logical structure of mathematical formulæ. It provides its own sublanguage for defining CDs— collections of symbol definitions with formal and informal semantics. One symbol definition consists of a mandatory symbol name and a normative textual description of the symbol, as well as other metadata2. Formal mathematical properties (FMPs) of the symbol, such as the definition of the sine function, or the commutativity axiom that holds for the multiplication operator, can be added, written in OpenMath and possibly using other symbols (see fig. 1). Type signatures (such as sin : R→ R) and human-readable notations (see sect. 3) of symbols are defined separately from the CD in a similar fashion. As semantic markup makes mathematical formulæ machine-understandable, it has leveraged many applications. For OpenMath, it started with data exchange between computer algebra systems, then automated theorem provers, and more recently dynamic geometry systems. OpenMath is also used in multilingual publishing, adaptive learning applications, and web search [10]. OpenMath CDs foster exchange by their modularity. Usually, a CD contains a set of related symbols, e. g. basic operations on matrices (CD linalg1) or eigenvalues and related concepts (CD linalg4), and a CD group contains a set of related CDs, e. g. all standard CDs about linear algebra (CD group linalg). In this setting, agents exchanging mathematical knowledge need not agree upon one large, monolithic mathematical ontology, but can flexibly refer to a specific set of CDs or CD groups they understand3. 1 Consider e. g. the chemical markup language CML [23] 2 OpenMath 2 uses an idiosyncratic schema for metadata, but Dublin Core is likely to be adopted for OpenMath 3. 3 A communication protocol for such agreements is specified in [7, sect. 5.3].