Chapter 4
Content Dictionaries

4.1 Introduction

Content Dictionaries (CDs) are central to the OpenMath philosophy of transmitting mathematical information. It is the OpenMath Content Dictionaries which actually hold the meanings of the objects being transmitted.

For example if application A is talking to application B, and sends, say, an equation involving multiplication of matrices, then A and B must agree on what a matrix is, and on what matrix multiplication is, and even on what constitutes an equation. All this information is held within some Content Dictionaries which both applications agree upon.

A Content Dictionary holds the meanings of (various) mathematical "words". These words are OpenMath basic objects referred to as symbols in Section 2.1.

With a set of symbol definitions (perhaps from several Content Dictionaries), A and B can now talk in a common "language".

It is important to stress that it is not Content Dictionaries themselves which are being transmitted, but some "mathematics" whose definitions are held within the Content Dictionaries. This means that the applications must have already agreed on a set of Content Dictionaries which they "understand" (i.e., can cope with to some degree).

In many cases, the Content Dictionaries that an application understands will be constant, and be intrinsic to the application's mathematical use. However the above approach can also be used for applications which can handle every Content Dictionary (such as an OpenMath parser, or perhaps a typesetting system), or alternatively for applications which understand a changeable number of Content Dictionaries (perhaps after being sent Content Dictionaries in some way).

The primary use of Content Dictionaries is thought to be for designers of Phrasebooks, the programs which translate between the OpenMath mathematical object and the corresponding (often internal) structure of the particular application in question. For such a use the Content Dictionaries have themselves been designed to be as readable and precise as possible.

Another possible use for OpenMath Content Dictionaries could rely on their automatic comprehension by a machine (e.g., when given definitions of objects defined in terms of previously understood ones), in which case Content Dictionaries may have to be passed as data. Towards this end, a Content Dictionary has been written which contains a set of symbols sufficient to represent any other Content Dictionary. This means that Content Dictionaries may be passed in the same way as other (OpenMath) mathematical data.

Finally, the syntax of the reference encoding for Content Dictionaries has been designed to be relatively easy to learn and to write, and also free from the need for any specialist software. This is because it is acknowledged that there is an enormous amount of mathematical information to represent, and so most Content Dictionaries are written by "ordinary" mathematicians, encoding their particular fields of expertise. A further reason is that the mathematics conveyed by a specific Content Dictionary should be understandable independently of any application.

The key points from this section are:

• Content Dictionaries should be readable and precise to help Phrasebook designers,

• Content Dictionaries should be readily write-able to encourage widespread use,

• It ought to be possible for a machine to understand a Content Dictionary to some degree.

4.2 Abstract Content Dictionaries

In this section we define the abstract structure of Content Dictionaries.

A Content Dictionary consists of the following mandatory pieces of information:

1. A name corresponding to the rules described in Section 2.3.

2. A description of the Content Dictionary.

3. A revision date, the date of the last change to the Content Dictionary. Dates should be stored in the ISO-compliant format YYYY-MM-DD, e.g. 1966-02-03.

4. A review date, a date until which the content dictionary is guaranteed to remain unchanged.

5. A version number which consists of a major and minor part (see Section 4.2.2).

6. A status, as described in Section 4.2.1.

7. A CD base which, when combined with the CD name, forms a unique identifier for the Content Dictionary. It may or may not refer to an actual location from which it can be retrieved.

8. A series of one or more symbol definitions as described below.

A symbol definition consists of the following pieces of information:

1. A mandatory name corresponding to the rules described in Section 2.3.

2. A mandatory description of the symbol, which can be as formal or informal as the author likes.

3. An optional role as described in Section 2.1.4.

4. Zero or more commented mathematical properties which are mathematical properties of the symbol expressed in a mechanism other than OpenMath.

5. Zero or more formal mathematical properties which are mathematical properties of the symbol expressed in OpenMath. Note that it is common for commented and formal mathematical properties to be introduced in pairs, with the former describing the latter.

   A Formal Mathematical Property may be given an optional kind attribute. An author of a Content Dictionary may use this to indicate whether, for example, the property provides an algorithm for evaluation of the concept it is associated with. At present no fixed scheme is mandated for how this information should be encoded or used by an application.

6. Zero or more examples which are intended to demonstrate the use of the symbol within an OpenMath object.

Some pieces of information which might logically be thought to be part of a Content Dictionary, such as the types or signatures of symbols, are better represented externally. This allows for new variants to be associated with Content Dictionaries without the Dictionaries themselves being changed. A model for signatures is given in Section 4.4.1.

Content Dictionaries may be grouped into CD Groups. These groups allow applications to easily refer to collections of Content Dictionaries. One particular CDGroup of interest is the "MathML CDGroup". This group consists of the collection of core Content Dictionaries that is designed to have the same semantic scope as the content elements of MathML 2 [17]. OpenMath objects built from symbols that come from Content Dictionaries in this CDGroup may be expected to be easily transformed between OpenMath and MathML encodings. The detailed structure of a CDGroup is described in Section 4.4.2 below.

4.3 The Reference Encoding for Content Dictionaries

The reference encoding of Content Dictionaries are as XML documents. A valid Content Dictionary document should conform to the Relax NG Schema for Content Dictionaries given in Section 4.3.1.

An example of a complete Content Dictionary is given in Appendix Appendix A.1, which is the Meta Content Dictionary for describing Content Dictionaries themselves. A more typical Content Dictionary is given in Appendix A.2, the arith1 Content Dictionary for basic arithmetic functions.

	# *********************************************
# 
# Relax NG Schema for OpenMath CD
# 
# *********************************************

default namespace = "http://www.openmath.org/OpenMathCD"

include "openmath2.rnc" {start = CD}

CDComment = element CDComment { text }
CDName = element CDName { xsd:NCName }
CDUses = element CDUses { CDName* }
CDURL = element CDURL { xsd:anyURI }
CDBase = element CDBase { xsd:anyURI }
text-or-om = (text | OMOBJ)*
CDReviewDate = element CDReviewDate { xsd:date }
CDDate = element CDDate { xsd:date }
CDVersion = element CDVersion { xsd:nonNegativeInteger }
CDRevision = element CDRevision { xsd:nonNegativeInteger }
CDStatus = element CDStatus {
   "official" |
   "experimental" |
   "private" |
   "obsolete"}
Description = element Description { text }
Name = element Name { xsd:NCName }
Role = element Role {
   "binder" |
   "attribution" |
   "semantic-attribution" |
   "error" |
   "application" |
   "constant" }
CMP = element CMP { text }
FMP = element FMP { 
  attribute kind {xsd:string}?,
  OMOBJ
  }
# allow embedded OM
Example = element Example { text-or-om }
CDDefinition =
  element CDDefinition {
    CDComment*,
    (Name & Role? & Description),
   (CDComment | Example | FMP | CMP)*
  }
CD =
  element CD {
    (CDComment* & Description? &
     CDName & CDURL? & CDBase? &
     CDReviewDate? & CDDate & CDStatus & 
     CDUses? & 
     CDVersion & CDRevision),
    ( CDDefinition,CDComment* )+
  }


      

4.4 Additional Information

Content Dictionaries contain just one part of the information that can be associated to a symbol in order to define its meaning and its functionality. OpenMath Signature dictionaries, CDGroups, and possibly collections of extra mathematical properties, are used to convey the different aspects that as a whole make up a mathematical definition.

# *********************************************
# 
# Relax NG Schema for OpenMath CD Signatures
# 
# *********************************************

default namespace = "http://www.openmath.org/OpenMathCDS"

include "openmath2.rnc" { start = CDSignatures }

CDSComment = element CDSComment { text }
CDSReviewDate = element CDSReviewDate { text }
CDSStatus = element CDSStatus {
   "official" |
   "experimental" |
   "private" |
   "obsolete"}
CDSignatures =
  element CDSignatures {
    attlist.CDSignatures,
    (CDSComment)*,
    (CDSReviewDate? & CDSStatus),
    (CDSComment | Signature)*
  }
attlist.CDSignatures =
  attribute cd { xsd:NCName },
  attribute type { xsd:NCName }
Signature = element Signature { attlist.Signature, OMOBJ? }
attlist.Signature = attribute name { text }

<Signature name="plus">
<OMOBJ version="2.0">
 <OMA>
  <OMS name="mapsto" cd="sts"/>
  <OMA>
   <OMS name="nassoc" cd="sts"/> 
   <OMV name="AbelianSemiGroup"/>
  </OMA>
  <OMV name="AbelianSemiGroup"/>
 </OMA>
</OMOBJ>
</Signature>

# Schema for OpenMath CD groups

# info on the CD group itself

default namespace = "http://www.openmath.org/OpenMathCDG"

CDGroupName = element CDGroupName { xsd:NCName }
CDGroupVersion = element CDGroupVersion { xsd:nonNegativeInteger }
CDGroupRevision = element CDGroupRevision { xsd:nonNegativeInteger }
CDGroupURL = element CDGroupURL { text }
CDGroupDescription = element CDGroupDescription { text }
# info on the CDs in the group
CDComment = element CDComment { text }
CDGroupMember =
  element CDGroupMember {
    CDComment?, CDName, CDVersion?, CDURL?
  }
CDName = element CDName { xsd:NCName }
CDVersion = element CDVersion { xsd:nonNegativeInteger }
CDURL = element CDURL { text }
# structure of the group
CDGroup =
  element CDGroup {
    CDGroupName,
    CDGroupVersion,
    CDGroupRevision?,
    CDGroupURL,
    CDGroupDescription,
    (CDGroupMember | CDComment)*
  }
start = CDGroup

4.5 Content Dictionaries Reviewing Process

The OpenMath Society is responsible for implementing a review and referee process to assess the accuracy of the mathematical content of Content Dictionaries. The status (see CDStatus) and/or the version number (see CDVersion ) of a Content Dictionary may change as a result of this review process.