Mathematical expressions in RDF

Question

Are there any established ways of representing mathematical formulas or expressions in RDF?

A web search turned up a paper by Marchiori (2003) which provides an overview of relevant Semantic Web standards, and even contains an example of how a MathML fragment like this:

<apply>
  <csymbol encoding="text"
           definitionURL="http://www.mathsw.org/scalarplus">
           p
           </csymbol>
  <cn> 2 </cn>
  <cn> 6 </cn>
</apply>

could be represented in RDF like this:

:_1 <http://www.w3.org/TR/MathML2#apply> :_2
:_1 <http://www.w3.org/TR/MathML2#csymbol> "p"
:_1 <http://www.w3.org/TR/MathML2#definitionURL> 
                                 <http://www.mathsw.org/scalarplus>
:_1 <http://www.w3.org/TR/MathML2#encoding> "text"
:_2 <http://www.w3.org/1999/02/22-rdf-syntax-ns#:_1> "2"
:_2 <http://www.w3.org/1999/02/22-rdf-syntax-ns#:_2> "6"

However, that was just an example, and I couldn't find anything which would describe a complete framework for something like that.

Edit for clarification. I am looking for something which would allow to represent mathematical expressions which refer to other resources described in RDF as sub-expressions. Even perhaps something that would allow to construct mathematical expressions by some sort of rule language (like SPARQL CONSTRUCT queries). Therefore just using MathML (or OpenMath) markup in a XML Literal does not seem like a good solution.

Accepted Answer

In terms of the direct question I don't know of an established way of doing this.

What you want to achieve?

If you want to convey some mathematical markup over RDF but that maths is not "executable" or doesn't need to interact with the RDF itself then the obvious approach would be to use MathML to encode the content and embed the MathML fragments as rdf:XMLLiterals.

OTOH if you are trying to express some mathematics that will interact with the RDF data - for example integrity constraint checking or formulae for deriving computed values - then you might want to consider an RDF rule language. RIF Core can express some interesting maths and the semantics of RIF + RDF combinations is well defined, though there is not yet a standardized way of embedding RIF rules directly in RDF. SWRL, is another alternative, has been around for longer and has reasonable tool support.

Answer 2

As Dave says, there is no established way. In section 3.4.1 and 3.4.2 of this survey – http://www.semantic-web-journal.net/content/new-submission-ontologies-and-languages-representing-mathematical-knowledge-semantic-web, page 16 –, I provide a review of all ways known to date:

complete representation of formulae in RDF: Marchiori's way was a mere proposal that has never been worked out, but there is a more elaborate alternative by Robbins (not applied in pratice either, AFAIK); see http://straymindcough.blogspot.com/2009/06/semantic-mathml.html. Using the N3 extensions over RDF makes the whole thing more elegant but less compatible with RDF tools. RDF-based approaches hardly give you access to the semantic essence of a mathematical formula.
MathML or OpenMath fragments as rdf:XMLLiterals: basically what Dave says above, and that has actually been worked out at least once, in combination with an RDF representation of "higher-level" math structures, such as symbols and definitions, by an ontology. But for the formulae it means that they are mostly opaque to RDF tools (with a few exceptions).
RDFa: There is the theoretical possibility of annotating MathML with RDFa, e.g. using Robbins's ontology mentioned unter (1). However that doesn't conform with the specification of MathML, and combined XML/RDF queries are not yet well supported.
Standoff markup: represent the math as XML, but have RDF "pointers" from externally, which point to those subterms/symbols in your formula that are relevant for your application. This has been practiced in several projects and implementations. Again there is no single standard solution or ontology for these RDF pointers into math XML, but from my experience I'd say that this approach works best.

Answer 3

I've presented an approach for the integration of mathematical expressions into RDF datasets for the purpose of mathematical reasoning on the OpenMath workshop 2012.

The paper Mathematical Computations for Linked Data Applications with OpenMath covers an OpenMath content dictionary for RDF that allows to reference RDF resources and their properties from mathematical expressions. This is complemented by an OWL ontology for OpenMath objects (available at http://numerateweb.org/vocab/math) to enable the encoding of mathematical expressions in RDF for cross-referencing between mathematical expressions and RDF data.

The simple example taken from our paper (in an extended Popcorn notation)

[foaf:Person] : 
@e:bmi = @e:mass / @e:height ^ 2

[foaf:Group] :
@e:aBMI = sum(@@foaf:member, $x -> @e:bmi($x)) / set1.size(@@foaf:member)

is based on the FOAF vocabulary and defines the properties e:mass and e:height for the class foaf:Person and uses them to compute the e:bmi properties of individual persons and their average e:aBMI over a group.

An interesting point is that we can use an RDF graph for storing intermediate results of computations. For example, e:bmi is stored as a property of an foaf:Person and later reused for computing e:aBMI.

Mathematical expressions in RDF

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