OpenMath - AbsoluteAstronomy.com

OpenMath is the name of a markup language for specifying the meaning of mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

formulae. Among other things, it can be used to complement MathML

MathML

Mathematical Markup Language is an application of XML for describing mathematical notations and capturing both its structure and content. It aims at integrating mathematical formulae into World Wide Web pages and other documents...

, a standard which mainly focuses on the presentation of formulae, with information about their semantic meaning. OpenMath can be encoded in XML

Extensible Markup Language

Extensible Markup Language is a set of rules for encoding documents in machine-readable form. It is defined in the XML 1.0 Specification produced by the W3C, and several other related specifications, all gratis open standards....

or in a binary format.

Coverage

OpenMath consists of the definition of "OpenMath Objects", which is 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.

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, while OpenMath2 brought better XML integration, structure sharing and liberalized the notion of OpenMath Content dictionaries.

OpenMath Society

The OpenMath Effort is governed by the OpenMath Society, based in Helsinki

Helsinki

Helsinki is the capital and largest city in Finland. It is in the region of Uusimaa, located in southern Finland, on the shore of the Gulf of Finland, an arm of the Baltic Sea. The population of the city of Helsinki is , making it by far the most populous municipality in Finland. Helsinki is...

, Finland

Finland

Finland , officially the Republic of Finland, is a Nordic country situated in the Fennoscandian region of Northern Europe. It is bordered by Sweden in the west, Norway in the north and Russia in the east, while Estonia lies to its south across the Gulf of Finland.Around 5.4 million people reside...

. The Society brings together tool builders, software suppliers, publishers and authors. Membership is by invitation of the Societies Executive Committee, which welcomes self-nominations of individuals who have worked on OpenMath-related issues in research or application. As of 2007, Michael Kohlhase

Michael Kohlhase

Dr. Michael Kohlhase is a German computer scientist and professor at Jacobs University, Bremen, Germany, where he is head of the KWARC research group at the School of Engineering and Science.-Academic Positions:Dr...

is president of the OpenMath society. He succeeded Arjeh M. Cohen, who was the first president.

Example

The well-known quadratic formula:

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):

2

4

2

2

In the expression tree above symbols — i.e. elements like — 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), except for the plusminus, which comes from a non-standard place, hence the cdbase attribute here.

OpenMath Content Dictionaries

Content Dictionaries are structured XML documents that define mathematical 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 requirements, 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

OMDoc

OMDoc is a semantic markup format for mathematical documents. While MathML only covers mathematical formulae and the related OpenMath standard only supports formulae and “content dictionaries” containing definitions of the symbols used in formulae, OMDoc covers the whole range of written...

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.

External links

The OpenMath website — The definitive source of information on all things OpenMath.
The OpenMath Standard — The definition of OpenMath Objects.
OpenMath Content Dictionaries — The official repository of OpenMath Content Dictionaries.