[Paper Review] Towards OpenMath Content Dictionaries as Linked Data

Christoph Lange|arXiv (Cornell University)|Jun 21, 2010

Semantic Web and Ontologies15 references3 citations

TL;DR

This paper proposes integrating OpenMath Content Dictionaries (CDs) into the Linked Data ecosystem by addressing conceptual and technical limitations in OpenMath 2, enabling machine-readable mathematical semantics in RDF-based datasets. It advocates for liberalized URI schemes, RDF-compatible metadata, and SPARQL extension with OpenMath entailment regimes to support scalable, semantically rich computations over linked statistical and scientific data.

ABSTRACT

"The term 'Linked Data' refers to a set of best practices for publishing and connecting structured data on the web". Linked Data make the Semantic Web work practically, which means that information can be retrieved without complicated lookup mechanisms, that a lightweight semantics enables scalable reasoning, and that the decentral nature of the Web is respected. OpenMath Content Dictionaries (CDs) have the same characteristics - in principle, but not yet in practice. The Linking Open Data movement has made a considerable practical impact: Governments, broadcasting stations, scientific publishers, and many more actors are already contributing to the "Web of Data". Queries can be answered in a distributed way, and services aggregating data from different sources are replacing hard-coded mashups. However, these services are currently entirely lacking mathematical functionality. I will discuss real-world scenarios, where today's RDF-based Linked Data do not quite get their job done, but where an integration of OpenMath would help - were it not for certain conceptual and practical restrictions. I will point out conceptual shortcomings in the OpenMath 2 specification and common bad practices in publishing CDs and then propose concrete steps to overcome them and to contribute OpenMath CDs to the Web of Data.

Motivation & Objective

• Address the lack of mathematical semantics in existing Linked Open Data, particularly in statistical and scientific datasets.
• Identify conceptual and practical shortcomings in OpenMath 2, such as rigid URI formats and limited linking capabilities.
• Propose a path to publish OpenMath CDs as true Linked Data by aligning them with Linked Data principles.
• Enable computation over linked data by integrating OpenMath-based symbolic reasoning into SPARQL query processing.
• Support the evolution of OpenMath 3 with technical and semantic guidelines for interoperability with the Semantic Web.

Proposed method

• Adopt Linked Data principles: use HTTP-resolvable URIs, publish machine-readable RDF, and include links to related data.
• Liberalize OpenMath symbol URIs beyond the cdbase/cd#name format to support slash-based URIs (e.g., http://cdba.se/cd/name) for efficient dereferencing.
• Introduce RDFa or FMP-based mechanisms to link OpenMath symbols to external resources (e.g., DBpedia), enabling cross-ontology linking.
• Propose an OpenMath entailment regime for SPARQL, allowing queries to infer derived values using symbolic rewriting from CDs.
• Leverage existing SPARQL extension functions and generalize them to support arbitrary functions defined in OpenMath CDs.
• Integrate OpenMath definitions into data models like SCOVO to make derived values (e.g., population density) semantically explicit and computable.

Experimental results

Research questions

• RQ1 How can OpenMath Content Dictionaries be published as true Linked Data to enhance machine-understandable mathematical semantics?
• RQ2 What technical and conceptual changes are needed to align OpenMath 2 with Linked Data best practices?
• RQ3 How can OpenMath symbols be semantically linked to external knowledge sources like DBpedia or Wikidata?
• RQ4 In what ways can SPARQL queries be extended to perform OpenMath-based symbolic computations over linked datasets?
• RQ5 What improvements to URI design and metadata encoding are required to support scalable, efficient discovery of mathematical symbols in Linked Data?

Key findings

• OpenMath 2’s rigid cdbase/cd#name URI scheme forces clients to download entire CDs even when only a few symbols are needed, reducing efficiency.
• The absence of standardized linking mechanisms prevents OpenMath symbols from being connected to external data, limiting their utility in the Web of Data.
• Current Linked Open Data lacks mathematical semantics, especially for derived values like population density, which are stored as raw numbers without explicit derivation rules.
• The proposal to use slash-based URIs (e.g., http://cdba.se/cd/name) enables efficient, granular access to individual symbols, improving performance over hash-based URIs.
• An OpenMath entailment regime for SPARQL could allow direct computation of derived values (e.g., density) within queries, reducing the need for multiple query and computation steps.
• Extending SPARQL with OpenMath-aware functions would enable scalable, semantically grounded reasoning over statistical and scientific datasets.