OpenMath

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OpenMath is the name of a document markup language for mathematical formulae. Among other things, it can be used to complement MathML, a standard which mainly focuses on the presentation of formulae, with information about their semantic meaning. OpenMath can be encoded in XML or in a binary format.

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[edit] Coverage

OpenMath consists of the definition of "OpenMath Objects", an abstract datatype for describing the logical structure of a mathematical formula, and the definition of "OpenMath Content Dictionaries", or collections of names for mathematical concepts. The names available from the latter type of collections are specifically intended for use in extending MathML, and conversely, a basic set of such "Content Dictionaries" has been designed to be compatible with the small set of mathematical concepts defined in Content MathML, the non-presentational subset of MathML.

[edit] History

OpenMath has been developed in a long series of workshops and (mostly European) research projects that began in 1993 and continues through today. The OpenMath 1.0 Standard was released in February 2000, and revised as OpenMath 1.1 in October 2002. Two years later, the OpenMath 2.0 Standard was released in June 2004. OpenMath 1 fixed the basic language architecture, whild OpenMath2 brought better XML integration, structure sharing and liberalized the notion of OpenMath Content dictionaries.

[edit] Example

The well-known quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

would be marked up like this in OpenMath (the representation is an expression tree made up from functional elements like OMA for function application or OMV for variables):

<OMOBJ  xmlns="http://www.openmath.org/OpenMath">
  <OMA cdbase="http://www.openmath.org/cd">
    <OMS cd="relation1" name="eq"/>
    <OMV name="x"/>
    <OMA>
      <OMS cd="arith1" name="divide">
      <OMA>
        <OMS cdgroup="http://www.example.com/mathops" cd="multiops" name="plusminus"/>
        <OMA>
          <OMS cd="arith1" name="unary_minus"/>
          <OMV name="b"/>
        </OMA>
        <OMA>
          <OMS cd="arith1" name="root"/>
          <OMI>2</OMI>
          <OMA>
            <OMS cd="arith1" name="minus"/>
            <OMA>
              <OMS cd="arith1 name="power"/>
              <OMV name="b"/>
              <OMI>2<OMI>
            </OMA>
            <OMA>
              <OMS cd="arith1" name="minus"/>
              <OMI>4</OMI>
              <OMV name="a"/>
              <OMV name="c"/>
            </OMA>
          </OMA>
        </OMA>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="times"/>
        <OMI>2</OMI>
        <OMV name="a"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

In the expression tree above symbols — i.e. elements like <OMS cd="arith1" name="times"/> — stand for mathematical functions that are applied to sibling expressions in an OMA which are interpreted as arguments. The OMS element is a generic extension element that means whatever is specified in the content dictionary referred to in the cd attribute (this document can be found at the URI specified in the innermost cdbase attribute dominating the respective OMS element. In the example above, all symbols come from the content dictionary for arithmetics (arith1, see below).

[edit] OpenMath Content Dictionaries

Content Dictionaries are structured XML documents that define mathemtical symbols that can be referred to by OMS elements in OpenMath Objects. The OpenMath 2 standard does not prescribe a canonical encoding for content dictionaries, but only requires an infrastructure sufficient for unique referencing in OMS elements. OpenMath provides a very basic XML encoding that meets these requirement, and a set of specific content dictionaries for some areas of mathematics, in particular covering the K-14 fragment covered by content MathML.

For more richly structured content dictionaries (and generally for arbitrary mathematical documents) the OMDoc format extends OpenMath by a “statement level” (including structures like definitions, theorems, proofs and examples, as well as means for interrelating them) and a “theory level”, where a theory is a collection of several contextually related statements. OMDoc's theories are designed to be compatible to OpenMath content dictionaries, but they can also be set into inheritance and import relations.

[edit] See also

[edit] External links

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