OpenMath

The OpenMath Standard is the official reference for the OpenMath language and has been approved by the OpenMath Society. It is not intended as an introductory document or a user's guide, for the latest available material of this nature, and the latest version of the standard, please consult the OpenMath web-site at . This chapter briefly introduces OpenMath concepts and notions that are referred to in the rest of this document. This standard does not describe the mechanisms by which software systems may offer, or make use of, computational services. The currently-suggested mechanism is the Symbolic Computation Software Composability Protocol (SCSCP) [[8]]( . OpenMath object s are representations of mathematical entities that can be communicated among various software applications in a meaningful way, that is, preserving their “semantics”. The standard endorses two encodings in XML (an innate one described here, and one in Strict Content MathML), a binary format and a JSON encoding. At the time of writing, these are the encodings supported by most existing OpenMath tools and applications, however they are not the only possible encodings of OpenMath objects. Users who wish to define their own encoding, are free to do so provided that there is a well-defined correspondence between the new encoding and the abstract model defined in Chapter 2 . The Content Dictionaries are public, they represent the actual common knowledge among OpenMath applications. Content Dictionaries fix the “meaning” of objects independently of the application. The application receiving the object may then recognize whether or not, according to the semantics of the symbols defined in the Content Dictionaries, the object can be transformed to the corresponding internal representation used by the application. Several additional files are related to Content Dictionaries. Signature Dictionaries contain the signatures of symbols defined in some OpenMath Content Dictionary and their format is endorsed by this standard. Auxiliary files that define presentation and rendering or that are used for manipulating and processing Content Dictionaries are not discussed by the standard. OpenMath represents mathematical objects as terms or as labelled trees that are called OpenMath objects or OpenMath expressions. The definition of an abstract OpenMath object is then the following. The Basic OpenMath Objects form the leaves of the OpenMath Object tree. A Basic OpenMath Object is of one of the following. Integers in the mathematical sense, with no predefined range. They are “infinite precision” integers (also called “bignums” in computer algebra). (i) Basic OpenMath objects are OpenMath object s. (Note that derived OpenMath object s are not OpenMath object s, but are used to construct OpenMath objects as described below.) OpenMath object s that are constructed via rules (ii) to (v) are jointly called compound OpenMath objects. OpenMath floats are IEEE 754 Double precision floating-point numbers. Other types of floating point number may be encoded in OpenMath by the use of suitable content dictionaries. are sequences of characters. These characters come from the Unicode standard [[16]]( . are meant to denote parameters, variables or indeterminates (such as bound variables of function definitions, variables in summations and integrals, independent variables of derivatives). Although foreign objects can come with a standarized encoding field, their interpretation is an issue beyond the OpenMath standard. In particular, a foreign object is primarily data that has been encoded in some format, and there is no promise that foreign objects encountered within one encoding of OpenMath can be faithfully represented in another. Derived OpenMath objects are constructed from non- OpenMath data. They differ from bytearrays in that they can have any structure. Currently there is only one way of making a derived OpenMath object . The four following constructs can be used to ma