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<?xml version="1.0" encoding="utf-8"?><html xmlns="http://www.w3.org/1999/xhtml" xmlns:m="http://www.w3.org/1998/Math/MathML"><head><title>The OpenMath Standard</title><object id="mmlFactory" classid="clsid:32F66A20-7614-11D4-BD11-00104BD3F987"/><?import
namespace="m" implementation="#mmlFactory"
?><style>
p {text-align:justify;
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code {font-size: 125%;
font-family: monospace;
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.figure {
border-width:1px;
border-color: black;
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.footnote{
font-size: 75%;
font-style: italic;
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.literal {
font-size: 75%;
background-color: #cfcfcf;
border-color: black;
border-style: solid;
border-width: 1px;
padding: 1em;
}
.del {
color: red;
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.new {
color: green;
}
.changetoc {
border-style: solid;
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margin: 2em 2em 2em 2em;
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</style></head><body><h1>The <i>OpenMath</i> Standard</h1><div>
Version: 1.1b</div><div>The OpenMath Esprit Consortium</div><div>Editors<br/>O. Caprotti, D. P. Carlisle, A. M. Cohen</div><div>October 22 2002</div><div><h3>Abstract</h3><p>This document proposes <i>OpenMath</i> as a standard for the communication of
semantically rich mathematical objects. This draft of the <i>OpenMath</i>
standard comprises the following: a description of <i>OpenMath</i> objects, the
grammar of <span>xml</span> and of the binary encoding of objects, a
description of Content Dictionaries and an <span>xml</span> document type
definition for validating Content Dictionaries. The non-normative
<a href="#cha_his">Chapter 1</a> of this document briefly overviews the history
of <i>OpenMath</i>.</p></div><div class="changetoc"><h3>Change-marked edition notes</h3><p>
This edition contains colour coded change markings
relative to the OpenMath 1.0 document...</p><ul><li class="new">New text is marked with css class "new" (green).</li><li class="del">Deleted text is marked with css class "del" (red).</li></ul><p>Sections with new text</p><ul><a href="#sec_omin" class="new">3.2 Further Description of OpenMath Objects</a><br/><a href="#ssec_xml" class="new">4.1.1 A Grammar for the xml Encoding</a><br/></ul></div>
<h2>Contents</h2><a href="#cha_his">1 <i>OpenMath</i> Movement</a><br/>
<a href="#sec_hist">1.1 History</a><br/>
<a href="#sec_omsoc">1.2 <i>OpenMath</i> Society</a><br/><a href="#cha_int">2 Introduction to <i>OpenMath</i></a><br/>
<a href="#sec_om-arch">2.1 <i>OpenMath</i> Architecture</a><br/>
<a href="#sec_intro-obj">2.2 <i>OpenMath</i> Objects and Encodings</a><br/>
<a href="#sec_intro-cd">2.3 Content Dictionaries</a><br/>
<a href="#sec_addnfiles">2.4 Additional Files</a><br/>
<a href="#sec_phrasebooks">2.5 Phrasebooks</a><br/><a href="#cha_obj">3 <i>OpenMath</i> Objects</a><br/>
<a href="#sec_omabs">3.1 Formal Definition of <i>OpenMath</i> Objects</a><br/>
<a href="#sec_basic">3.1.1 Basic <i>OpenMath</i> objects</a><br/>
<a href="#sec_compound">3.1.2 Compound <i>OpenMath</i> Objects</a><br/>
<a href="#sec_omin">3.2 Further Description of <i>OpenMath</i> Objects</a><br/>
<a href="#sec_summary">3.3 Summary</a><br/><a href="#cha_enco">4 <i>OpenMath</i> Encodings</a><br/>
<a href="#sec_xml">4.1 The <span>xml</span> Encoding</a><br/>
<a href="#ssec_xml">4.1.1 A Grammar for the <span>xml</span> Encoding</a><br/>
<a href="#sec_xml-desc">4.1.2 Description of the Grammar</a><br/>
<a href="#xmldoc">4.1.3 Embedding OpenMath in XML Documents</a><br/>
<a href="#sec_binary">4.2 The Binary Encoding</a><br/>
<a href="#sec_binary_grammar">4.2.1 A Grammar for the Binary Encoding</a><br/>
<a href="#sec_bin-desc">4.2.2 Description of the Grammar</a><br/>
<a href="#sec_sharing">4.2.2.1 Sharing</a><br/>
<a href="#sec_impl_note">4.2.3 Implementation Note</a><br/>
<a href="#sec_bin_example">4.2.4 Example of Binary Encoding</a><br/>
<a href="#sec_enc_summary">4.3 Summary</a><br/><a href="#cha_cd">5 Content Dictionaries</a><br/>
<a href="#sec_cd_summary">5.1 Introduction</a><br/>
<a href="#sect_func">5.2 Content Dictionaries</a><br/>
<a href="#sec_xml_cd">5.3 The XML Encoding for Content Dictionaries</a><br/>
<a href="#sec_dtd_cd">5.3.1 The DTD Specification of Content Dictionaries</a><br/>
<a href="#sect_pcdata">5.3.2 Further Requirements of an <i>OpenMath</i> Content Dictionary</a><br/>
<a href="#addfiles">5.4 Additional Information</a><br/>
<a href="#sigfiles">5.4.1 Signature Files</a><br/>
<a href="#sec_dtd_sig">5.4.1.1 The DTD Specification of Signature Files</a><br/>
<a href="#sect_sigpcdata">5.4.1.2 Further Requirements of a Signature File</a><br/>
<a href="#sect_sigex">5.4.1.3 Examples</a><br/>
<a href="#ssec_cdgroups">5.4.2 CDGroups</a><br/>
<a href="#sec_dtd_cdg">5.4.2.1 The DTD Specification of CDGroups</a><br/>
<a href="#sect_cdgpcdata">5.4.2.2 Further Requirements of a CDGroup</a><br/>
<a href="#cdapprove">5.5 Content Dictionaries Reviewing Process</a><br/><a href="#cha_comp">6 <i>OpenMath</i> Compliance</a><br/>
<a href="#sec_compl_encoding">6.1 Encoding</a><br/>
<a href="#sec_compl_cd">6.2 Content Dictionaries</a><br/>
<a href="#sec_comp_lex">6.3 Lexical Errors</a><br/><a href="#cha_conc">7 Conclusion</a><br/><a href="#app_cdfiles">A CD Files</a><br/>
<a href="#app_cdcd">A.1 The <b>meta</b> Content Dictionary</a><br/>
<a href="#arith1.ocd">A.2 The <b>arith1</b> Content Dictionary File</a><br/>
<a href="#arith1.sts">A.3 The <b>arith1</b> STS Signature File</a><br/>
<a href="#mathml.cdg">A.4 The <b>MathML</b> CDGroup</a><br/>
<a href="#errorcd">A.5 The <b>error</b> Content Dictionary</a><br/><a href="#bibliography">B Bibliography</a><br/>
<h2>List of Figures</h2><a href="#fig_om">2.1 The <i>OpenMath</i> Architecture</a><br/><a href="#fig_obj">3.1 The <i>OpenMath</i> application and binding objects for
<m:math><m:mi>sin</m:mi> <m:mo>(</m:mo><m:mi>x</m:mi> <m:mo>)</m:mo></m:math> and
<m:math><m:mi>λ</m:mi> <m:mi>x</m:mi><m:mo>.</m:mo><m:mi>x</m:mi> <m:mo>+</m:mo>
<m:mn>2</m:mn></m:math> in tree-like notation.</a><br/><a href="#fig_objdtd">4.1 DTD for the <i>OpenMath</i> <span>xml</span> encoding of objects.</a><br/><a href="#fig_xml">4.2 Grammar for the <span>xml</span> encoding of <i>OpenMath</i> objects.</a><br/><a href="#fig_bin-enc">4.3 Grammar of the binary encoding of <i>OpenMath</i> objects.</a><br/><a href="#fig_cd-dtd">5.1 DTD Specification of Content Dictionaries</a><br/><a href="#fig_omcdsig.dtd">5.2 DTD Specification of Signature Files</a><br/><a href="#fig_cdgroup.dtd">5.3 DTD Specification of CDGroups</a><br/>
<h2 name="cha_his" id="cha_his">
Chapter 1<br/><i>OpenMath</i> Movement</h2>
<p>This chapter is a historical account of <i>OpenMath</i> and should be regarded
as non-normative.</p>
<p><i>OpenMath</i> is a standard for representing mathematical objects,
allowing them to be exchanged between computer programs, stored in
databases, or published on the worldwide web. While the original
designers were mainly developers of computer algebra systems, it is
now attracting interest from other areas of scientific computation and
from many publishers of electronic documents with a significant
mathematical content. There is a strong relationship to the MathML
recommendation <a href="#MathML_98">[5]</a> from the Worldwide Web
Consortium, and a large overlap between the two developer communities.
MathML deals principally with the <i>presentation</i> of
mathematical objects, while <i>OpenMath</i> is solely concerned with their
semantic meaning or <i>content</i>. While MathML does
have some limited facilities for dealing with content, it also allows
semantic information encoded in <i>OpenMath</i> to be embedded inside a MathML
structure. Thus the two technologies may be seen as highly
complementary.</p>
<div><h3 name="sec_hist" id="sec_hist">1.1 History</h3>
<p><i>OpenMath</i> was originally developed through a series of workshops held
in Zurich (1993 and 1996), Oxford (1994), Amsterdam (1995), Copenhagen
(1995), Bath (1996), Dublin (1996), Nice (1997), Yorktown Heights
(1997), Berlin (1998), and Tallahassee (1998). The participants in
these workshops formed a global <i>OpenMath</i> community which was coordinated
by a Steering Committee and operated through electronic mailing groups
and ad-hoc working parties. This loose arrangement has been
formalised through the establishment of an <i>OpenMath</i> Society. Up until the
end of 1996 much of the work of the community was funded through a
grant from the Human Capital and Mobility program of the European
Union, the contributions of several institutions and individuals. A
document outlining the objectives and basic design of <i>OpenMath</i> was
produced (later published as
<a href="#Abbott_Leeuwen_Strotmann_98">[3]</a>). By the end of 1996
a simplified specification had been agreed on and some prototype
implementations have come about
<a href="#Dalmas_Gaetano_Watt_97">[8]</a>.</p>
<p>In 1996 a group of European participants in <i>OpenMath</i> decided to bid
for funding under the European Union's Fourth Framework Programme for
strategic research in information technology. This bid was successful
and the project started in late 1997. The principal aims of the
project are to formalise <i>OpenMath</i> as a standard and to develop it
further through industrial applications; this document is a product of
that process and draws heavily on the previous work described earlier.
<i>OpenMath</i> participants from all over the world continue to meet
regularly and cooperate on areas of mutual interest, and
recent workshops in Tallahassee (November 1998) and Eindhoven (June
1999) endorsed drafts of this document as the current <i>OpenMath</i> standard.</p>
</div>
<div><h3 name="sec_omsoc" id="sec_omsoc">1.2 <i>OpenMath</i> Society</h3>
<p>In November 1998 the <i>OpenMath</i> Society has been established to coordinate
all <i>OpenMath</i> activities. The society is based in Helsinki, Finland and is
steered by the executive committee whose members are elected by the
society. The official web page of the society is
<a href="http://www.openmath.org">http://www.openmath.org</a>.</p>
</div>
<h2 name="cha_int" id="cha_int">
Chapter 2<br/>Introduction to <i>OpenMath</i></h2>
<p>This chapter briefly introduces <i>OpenMath</i> concepts and notions that are
referred to in the rest of this document.</p>
<div><h3 name="sec_om-arch" id="sec_om-arch">2.1 <i>OpenMath</i> Architecture</h3>
<div class="figure"><a name="fig_om" id="fig_om"/>
<img src="om-arch.png" alt="om-arch.png"/>
<div>
Figure 2.1 The <i>OpenMath</i> Architecture</div></div>
<p>The architecture of <i>OpenMath</i> is described in <a href="#fig_om">Figure 2.1</a> and
summarizes the interactions among the different <i>OpenMath</i> components.
There are three layers of representation of a mathematical object
<a href="#OM_98">[10]</a>. A private layer that is the internal representation used
by an application. An abstract layer that is the representation as an
<i>OpenMath</i> object. Third is a communication layer that translates the <i>OpenMath</i>
object representation to a stream of bytes. An application dependent
program manipulates the mathematical objects using its internal
representation, it can convert them to <i>OpenMath</i> objects and communicate
them by using the byte stream representation of <i>OpenMath</i> objects.</p>
</div>
<div><h3 name="sec_intro-obj" id="sec_intro-obj">2.2 <i>OpenMath</i> Objects and Encodings</h3>
<p><i>OpenMath</i> objects are representations of mathematical entities that
can be communicated among various software applications in a
meaningful way, that is, preserving their
<span>"semantics"</span>.</p>
<p><i>OpenMath</i> objects and encodings are described in detail in <a href="#cha_obj">Chapter 3</a> and <a href="#cha_enco">Chapter 4</a>.</p>
<p>The standard endorses encodings in <span>xml</span> and binary
format. These are the encodings supported by the official <i>OpenMath</i>
libraries. However they are not the only possible encodings of <i>OpenMath</i>
objects. Users that wish to define their own encoding using some other
specific language (e.g. Lisp) may do so provided there is an
effective translation of this encoding to an official one.</p>
</div>
<div><h3 name="sec_intro-cd" id="sec_intro-cd">2.3 Content Dictionaries</h3>
<p>Content Dictionaries (CDs) are used to assign informal and formal
semantics to all symbols used in the <i>OpenMath</i> objects. They define the
symbols used to represent concepts arising in a particular area of
mathematics.</p>
<p>The Content Dictionaries are public, they represent the actual
common knowledge among <i>OpenMath</i> applications. Content Dictionaries fix
the <span>"meaning"</span> 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.</p>
</div>
<div><h3 name="sec_addnfiles" id="sec_addnfiles">2.4 Additional Files</h3>
<p>Several
additional files are related to Content Dictionaries. Signature files
contain the signatures of symbols defined in some <i>OpenMath</i> Content
Dictionary and their format is endorsed by this standard.</p>
<p>Furthermore, the standard fixes how to define as a CDGroup a specific
set of Content Dictionaries.</p>
<p>Auxiliary files that define presentation and rendering or that
are used for manipulating and processing Content Dictionaries are not
discussed by the standard.</p>
</div>
<div><h3 name="sec_phrasebooks" id="sec_phrasebooks">2.5 Phrasebooks</h3>
<p>The conversion of an <i>OpenMath</i> object to/from the internal
representation in a software application is performed by an interface
program called <i>Phrasebook</i>. The translation is
governed by the Content Dictionaries and the specifics of the
application. It is envisioned that a software application dealing with
a specific area of mathematics declares which Content Dictionaries it
understands. As a consequence, it is expected that the Phrasebook of
the application is able to translate <i>OpenMath</i> objects built using symbols
from these Content Dictionaries to/from the internal mathematical
objects of the application.</p>
<p><i>OpenMath</i> objects do not
specify any compuational behaviour, they merely represent mathematical
expressions. Part of the <i>OpenMath</i> philosophy is to leave it to the
application to decide what it does with an object once it has received
it. <i>OpenMath</i> is not a query or programming language. Because of this,
<i>OpenMath</i> does not prescribe a way of forcing <span>"evaluation"</span> or
<span>"simplification"</span> of objects like
<m:math><m:mn>2</m:mn><m:mo>+</m:mo><m:mn>3</m:mn></m:math> or
<m:math><m:mi>sin</m:mi><m:mo>(</m:mo><m:mi>π</m:mi><m:mo>)</m:mo></m:math>. Thus,
the same object <m:math><m:mn>2</m:mn><m:mo>+</m:mo><m:mn>3</m:mn></m:math> could be
transformed to <m:math><m:mn>5</m:mn></m:math> by a computer algebra system,
or displayed as <m:math><m:mn>2</m:mn><m:mo>+</m:mo><m:mn>3</m:mn></m:math> by a
typesetting tool.</p>
</div>
<h2 name="cha_obj" id="cha_obj">
Chapter 3<br/><i>OpenMath</i> Objects</h2>
<p>In this chapter we provide a self-contained description of <i>OpenMath</i>
objects. We first do so at an informal level (<a href="#sec_omin">Section 3.2</a>) and next by means of an abstract grammar
description (<a href="#sec_omabs">Section 3.1</a>).</p>
<div><h3 name="sec_omabs" id="sec_omabs">3.1 Formal Definition of <i>OpenMath</i> Objects</h3>
<p><i>OpenMath</i> represents mathematical objects as terms or as labelled
trees that are called <i>OpenMath</i> objects or <i>OpenMath</i> expressions. The definition
of an abstract <i>OpenMath</i> object is then the following.</p>
<div><h4 name="sec_basic" id="sec_basic">3.1.1 Basic <i>OpenMath</i> objects</h4>
<p>The Basic <i>OpenMath</i> Objects form
the leaves of the <i>OpenMath</i> Object tree. A Basic <i>OpenMath</i> Object is of one of
the following.</p>
<ol class="lowerroman">
<li><p> Integer.</p><p>Integers in
the mathematical sense, with no predefined range. They are
<span>"infinite precision"</span> integers (also called
<span>"bignums"</span> in computer algebra).</p></li>
<li><p> IEEE floating point
number.</p><p>Double precision floating-point numbers
following the <span>ieee</span> 754-1985
standard <a href="#ieee754_85">[1]</a>.</p></li>
<li><p> Character string.</p><p>A Unicode Character string. This also corresponds to `characters' in
<span>xml</span>.</p></li>
<li><p> Bytearray.</p><p>A sequence of bytes.</p></li>
<li><p> Symbol.</p><p>A Symbol
encodes two fields of information, a <i>name</i> and a
<i>Content Dictionary</i>. Each is a sequence of
characters matching a regular expression, as described below.</p></li>
<li><p> Variable.</p><p>A Variable consists of a <i>name</i> which is a
sequence of characters matching a regular expression, as described
below.</p></li>
</ol>
</div>
<div><h4 name="sec_compound" id="sec_compound">3.1.2 Compound <i>OpenMath</i> Objects</h4>
<p><i>OpenMath</i> objects are built recursively as follows.
<ol class="lowerroman">
<li><p> Basic <i>OpenMath</i> objects are <i>OpenMath</i> objects.</p></li>
<li><p>
If
<m:math><m:msub><m:mi>A</m:mi><m:mn>1</m:mn></m:msub></m:math>, <span>…</span>,
<m:math><m:msub><m:mi>A</m:mi><m:mi>n</m:mi></m:msub></m:math>
<m:math><m:mo>(</m:mo><m:mi>n</m:mi><m:mo>></m:mo><m:mn>0</m:mn><m:mo>)</m:mo></m:math> are <i>OpenMath</i>
objects, then
<m:math display="block">
<m:mi mathvariant="bold">application</m:mi><m:mo>(</m:mo><m:msub><m:mi>A</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo> <m:mi>…</m:mi><m:mo>,</m:mo> <m:msub><m:mi>A</m:mi><m:mi>n</m:mi></m:msub><m:mo>)</m:mo>
</m:math>
is an <i>OpenMath</i> <i>application object</i>.</p></li> <li><p> If
<m:math><m:msub><m:mi>S</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo>
<m:mi>…</m:mi><m:mo>,</m:mo> <m:msub><m:mi>S</m:mi><m:mi>n</m:mi></m:msub></m:math>
are <i>OpenMath</i> symbols, and <m:math><m:mi>A</m:mi></m:math>,
<m:math><m:msub><m:mi>A</m:mi><m:mn>1</m:mn></m:msub></m:math>,
<span>…</span>, <m:math><m:msub><m:mi>A</m:mi><m:mi>n</m:mi></m:msub></m:math>, <m:math><m:mo>(</m:mo><m:mi>n</m:mi><m:mo>></m:mo><m:mn>0</m:mn><m:mo>)</m:mo></m:math> are <i>OpenMath</i> objects, then
<m:math display="block"><m:mi mathvariant="bold">attribution</m:mi>
<m:mo>(</m:mo><m:mi>A</m:mi><m:mo>,</m:mo> <m:msub><m:mi>S</m:mi><m:mn>1</m:mn></m:msub>
<m:mspace width=".3em"/>
<m:msub><m:mi>A</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo>
<m:mspace width=".3em"/> <m:mi>…</m:mi> <m:mspace width=".3em"/> <m:mo>,</m:mo> <m:msub><m:mi>S</m:mi><m:mi>n</m:mi></m:msub> <m:mspace width=".3em"/>
<m:msub><m:mi>A</m:mi><m:mi>n</m:mi></m:msub><m:mo>)</m:mo></m:math>
is an <i>OpenMath</i> <i>attribution object</i> and <m:math><m:mi>A</m:mi></m:math> is the object
<i>stripped of attributions</i>. The operation of recursively
applying stripping to the stripped object is called <i>flattening
of the attribution</i>. When the stripped object after flattening
is a variable, the attributed object is called <i>attributed
variable</i>.</p></li>
<li><p> If <m:math><m:mi>B</m:mi></m:math> and <m:math><m:mi>C</m:mi></m:math> are <i>OpenMath</i> objects, and <m:math><m:msub><m:mi>v</m:mi><m:mn>1</m:mn></m:msub></m:math>, <m:math><m:mi>…</m:mi></m:math>,
<m:math><m:msub><m:mi>v</m:mi><m:mi>n</m:mi></m:msub></m:math> <m:math><m:mo>(</m:mo><m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn><m:mo>)</m:mo></m:math> are <i>OpenMath</i> variables or attributed variables, then
<m:math display="block">
<m:mi mathvariant="bold">binding</m:mi> <m:mo>(</m:mo><m:mi>B</m:mi><m:mo>,</m:mo> <m:msub><m:mi>v</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo> <m:mi>…</m:mi><m:mo>,</m:mo> <m:msub><m:mi>v</m:mi><m:mi>n</m:mi></m:msub><m:mo>,</m:mo> <m:mi>C</m:mi><m:mo>)</m:mo>
</m:math>
is an <i>OpenMath</i> <i>binding object</i>.</p></li>
<li><p> If <m:math><m:mi>S</m:mi></m:math> is an
<i>OpenMath</i> symbol and <m:math><m:msub><m:mi>A</m:mi><m:mn>1</m:mn></m:msub></m:math>, <span>…</span>, <m:math><m:msub><m:mi>A</m:mi><m:mi>n</m:mi></m:msub></m:math> <m:math><m:mo>(</m:mo><m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn><m:mo>)</m:mo></m:math> are <i>OpenMath</i> objects, then
<m:math display="block"><m:mi mathvariant="bold">error</m:mi> <m:mo>(</m:mo><m:mi>S</m:mi><m:mo>,</m:mo> <m:msub><m:mi>A</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:mi>…</m:mi><m:mo>,</m:mo><m:msub><m:mi>A</m:mi><m:mi>n</m:mi></m:msub><m:mo>)</m:mo>
</m:math>
is an <i>OpenMath</i> <i>error object</i>.</p></li>
</ol>
</p>
</div>
</div>
<div><h3 name="sec_omin" id="sec_omin">3.2 Further Description of <i>OpenMath</i> Objects</h3>
<p>Informally, an <i>OpenMath</i> <span><i>object</i></span> can be
viewed as a tree and is also referred to as a term. The objects at
the leaves of <i>OpenMath</i> trees are called <span><i>basic
objects</i></span>. The basic objects supported by <i>OpenMath</i> are:
<dl>
<dt>Integer</dt><dd><p>Arbitrary Precision
integers.</p> </dd>
<dt>Float</dt><dd> <p><i>OpenMath</i> floats are
<span>ieee</span> 754 Double precision floating-point
numbers. Other types of floating point number may be encoded in <i>OpenMath</i>
by the use of suitable content dictionaries.</p>
</dd>
<dt>Character strings</dt><dd><p>are
sequences of characters. These characters come from the Unicode
standard <a href="#UNICODE">[13]</a>.</p>
</dd>
<dt>Bytearrays</dt><dd><p>are sequences of bytes. There is no <span>"byte"</span> in <i>OpenMath</i>
as an object of its own. However, a single byte can of course be
represented by a bytearray of length 1. The difference between
strings and bytearrays is the following: a character string is a
sequence of bytes with a fixed interpretation (as characters,
Unicode texts may require several bytes to code one character),
whereas a bytearray is an uninterpreted sequence of bytes with no
intrinsic meaning. Bytearrays could be used inside <i>OpenMath</i> errors to
provide information to, for example, a debugger; they could also
contain intermediate results of calculations, or `handles' into
computations or databases.</p>
</dd>
<dt>Symbols</dt><dd>
<p>
are uniquely defined by the Content Dictionary in which they occur
and by a name. In definition in <a href="#sec_omabs">Section 3.1</a> we have
left this information implicit. However, it should be kept in mind
that all symbols appearing in an <i>OpenMath</i> object are defined in a
Content Dictionary. The form of these definitions is explained in
<a href="#cha_cd">Chapter 5</a>. Each symbol has no more than one
definition in a Content Dictionary. Many Content Dictionaries may
define differently a symbol with the same name (e.g., the symbol
<code>union</code> is defined as
associative-commutativeset theoretic union in a Content Dictionary
<code>set1</code> but another Content Dictionary,
<code>multiset1</code> might define a symbol
<code>union</code> as the union of multi-sets. The name
of a symbol can only contain alphanumeric characters and
underscores. More precisely, a symbol name matches the following
regular expression: <blockquote><p>
[<code>A</code>-<code>Z</code><code>a</code>-<code>z</code>]
[<code>A</code>-<code>Z</code><code>a</code>-<code>z</code><code>0</code>-<code>9</code><code>_</code>]*
</p></blockquote></p>
<p>Notice that these symbol names are case sensitive. <i>OpenMath</i>
<i>recommends</i> that symbol names should be no longer than
100 characters.</p>
</dd>
<dt>Variables</dt><dd><p>are meant to
denote parameters, variables or indeterminates (such as bound
variables of function definitions, variables in summations and
integrals, independent variables of derivatives). Plain variable
names are restricted to use a subset of the printable ASCII
characters. Formally the names must match the regular expression:
<blockquote><p> [A-Za-z0-9=+(),-./:?!#$%*;=@[]^_`{|}]+
</p></blockquote></p>
</dd>
</dl> </p>
<p>The four following constructs can be used to make compound <i>OpenMath</i>
objects.</p>
<dl>
<dt>Application</dt><dd><p>constructs an
<i>OpenMath</i> object from a sequence of one or more <i>OpenMath</i> objects. The first
argument of application is referred to as <span>"head"</span> while
the remaining objects are called <span>"arguments"</span>. An <i>OpenMath</i>
application object can be used to convey the mathematical notion of
application of a function to a set of arguments. For instance,
suppose that the <i>OpenMath</i> symbol <code>sin</code> is defined
in a Content Dictionary for trigonometry, then <m:math><m:mi mathvariant="bold">application</m:mi><m:mo>(</m:mo><m:mi>sin</m:mi><m:mo>,</m:mo>
<m:mi>x</m:mi> <m:mo>)</m:mo></m:math> is the abstract <i>OpenMath</i> object
corresponding to <m:math><m:mi>sin</m:mi> <m:mo>(</m:mo><m:mi>x</m:mi>
<m:mo>)</m:mo></m:math>. More generally, an <i>OpenMath</i> application object can
be used as a constructor to convey a mathematical object built from
other objects such as a polynomial constructed from a set of
monomials. Constructors build inhabitants of some symbolic type,
for instance the type of rational numbers or the type of
polynomials. The rational number, usually denoted as
<m:math><m:mn>1</m:mn><m:mo>/</m:mo><m:mn>2</m:mn></m:math>, is represented by the
<i>OpenMath</i> application object <m:math><m:mi mathvariant="bold">application</m:mi><m:mo>(</m:mo><m:mi>Rational</m:mi><m:mo>,</m:mo>
<m:mn>1</m:mn><m:mo>,</m:mo> <m:mn>2</m:mn><m:mo>)</m:mo></m:math>. The symbol
<code>Rational</code> must be defined, by a Content
Dictionary, as a constructor symbol for the rational numbers.</p>
<div class="figure"><a name="fig_obj" id="fig_obj"/>
<img src="lambda.png" alt="lambda.png"/>
<div>
Figure 3.1 The <i>OpenMath</i> application and binding objects for
<m:math><m:mi>sin</m:mi> <m:mo>(</m:mo><m:mi>x</m:mi> <m:mo>)</m:mo></m:math> and
<m:math><m:mi>λ</m:mi> <m:mi>x</m:mi><m:mo>.</m:mo><m:mi>x</m:mi> <m:mo>+</m:mo>
<m:mn>2</m:mn></m:math> in tree-like notation.</div></div>
</dd>
<dt>Binding</dt><dd><p>objects are
constructed from an <i>OpenMath</i> object, and from a sequence of zero or more
variables followed by another <i>OpenMath</i> object. The first <i>OpenMath</i> object is
the <span>"binder"</span> object. Arguments 2 to
<m:math><m:mi>n</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:math> are always variables to
be bound in the <span>"body"</span> which is the
<m:math><m:msup><m:mi>n</m:mi><m:mi>th</m:mi></m:msup></m:math> argument object. It
is allowed to have no bound variables, but the binder object and the
body should be present. Binding can be used to express functions or
logical statements. The function <m:math><m:mi>λ</m:mi>
<m:mi>x</m:mi><m:mo>.</m:mo><m:mi>x</m:mi> <m:mo>+</m:mo><m:mn>2</m:mn></m:math>, in which
the variable <m:math><m:mi>x</m:mi></m:math> is bound by
<m:math><m:mi>λ</m:mi></m:math>, corresponds to a binding object having
as binder the <i>OpenMath</i> symbol <code>lambda</code>: <m:math display="block"><m:mi mathvariant="bold">binding</m:mi><m:mo>(</m:mo><m:mi>lambda</m:mi><m:mo>,</m:mo>
<m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="bold">application</m:mi><m:mo>(</m:mo><m:mi>plus</m:mi><m:mo>,</m:mo>
<m:mi>x</m:mi> <m:mo>,</m:mo>
<m:mn>2</m:mn><m:mo>)</m:mo><m:mo>)</m:mo><m:mtext>.</m:mtext></m:math></p>
<p class="del">Binding of several variables as in: <m:math display="block"><m:mi mathvariant="bold">binding</m:mi>
<m:mo>(</m:mo><m:mi>B</m:mi><m:mo>,</m:mo>
<m:msub><m:mi>v</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo>
<m:mi>…</m:mi><m:mo>,</m:mo>
<m:msub><m:mi>v</m:mi><m:mi>n</m:mi></m:msub><m:mo>,</m:mo> <m:mi>C</m:mi>
<m:mo>)</m:mo></m:math> is semantically equivalent to composition of
binding of a single variable, namely
<m:math display="block">
<m:mi mathvariant="bold">binding</m:mi>
<m:mo>(</m:mo>
<m:mi>B</m:mi>
<m:mo>,</m:mo>
<m:msub><m:mi>v</m:mi><m:mn>1</m:mn></m:msub>
<m:mo>,</m:mo>
<m:mo>(</m:mo>
<m:mi mathvariant="bold">binding</m:mi>
<m:mo>(</m:mo>
<m:mi>B</m:mi>
<m:mo>,</m:mo>
<m:msub><m:mi>v</m:mi><m:mn>2</m:mn></m:msub>
<m:mo>,</m:mo>
<m:mo>(</m:mo>
<m:mi>…</m:mi>
<m:mo>,</m:mo>
<m:mi mathvariant="bold">binding</m:mi>
<m:mo>(</m:mo>
<m:mi>B</m:mi>
<m:mo>,</m:mo>
<m:msub><m:mi>v</m:mi><m:mi>n</m:mi></m:msub>
<m:mo>,</m:mo>
<m:mi>C</m:mi>
<m:mo>)</m:mo>
<m:mi>…</m:mi>
<m:mo>)</m:mo>
<m:mtext>.</m:mtext>
</m:math>
</p>
<p class="del">Note that it follows from
this that repeated occurrences of the same variable in a binding
operator are allowed. For example the object
<m:math display="block">
<m:mi mathvariant="bold">binding</m:mi>
<m:mo>(</m:mo><m:mi>lambda</m:mi><m:mo>,</m:mo> <m:mi>v</m:mi> <m:mo>,</m:mo>
<m:mi>v</m:mi> <m:mo>,</m:mo><m:mi mathvariant="bold">application</m:mi>
<m:mo>(</m:mo><m:mi>times</m:mi><m:mo>,</m:mo><m:mi>v</m:mi>
<m:mo>,</m:mo><m:mi>v</m:mi><m:mo>)</m:mo> <m:mo>)</m:mo></m:math> is semantically
equivalent to: <m:math display="block"><m:mi mathvariant="bold">binding</m:mi><m:mo>(</m:mo><m:mi>lambda</m:mi><m:mo>,</m:mo>
<m:mi>v</m:mi><m:mo>,</m:mo> <m:mi mathvariant="bold">binding</m:mi>
<m:mo>(</m:mo><m:mi>lambda</m:mi><m:mo>,</m:mo> <m:mi>v</m:mi><m:mo>,</m:mo><m:mi mathvariant="bold">application</m:mi><m:mo>(</m:mo><m:mi>times</m:mi><m:mo>,</m:mo><m:mi>v</m:mi><m:mo>,</m:mo><m:mi>v</m:mi><m:mo>)</m:mo><m:mo>)</m:mo><m:mo>)</m:mo></m:math>
so that the outermost binding is actually a constant function
(<m:math><m:mi>v</m:mi></m:math> does not occur free in the body <m:math><m:mi mathvariant="bold">application</m:mi>
<m:mo>(</m:mo><m:mi>times</m:mi><m:mo>,</m:mo><m:mi>v</m:mi>
<m:mo>,</m:mo><m:mi>v</m:mi><m:mo>)</m:mo> <m:mo>)</m:mo></m:math>).</p>
<p>Phrasebooks are allowed to use <m:math><m:mi>α</m:mi></m:math>
conversion in order to avoid clashes of variable names. Suppose an
object <m:math><m:mi>Ω</m:mi></m:math> contains an occurrence of the
object <m:math><m:mi mathvariant="bold">binding</m:mi>
<m:mo>(</m:mo><m:mi>B</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> <m:mo>,</m:mo> <m:mi>C</m:mi>
<m:mo>)</m:mo></m:math>. This object <m:math><m:mi mathvariant="bold">binding</m:mi> <m:mo>(</m:mo><m:mi>B</m:mi> <m:mo>,</m:mo>
<m:mi>v</m:mi> <m:mo>,</m:mo> <m:mi>C</m:mi> <m:mo>)</m:mo></m:math> can be replaced
in <m:math><m:mi>Ω</m:mi></m:math> by <m:math><m:mi mathvariant="bold">binding</m:mi> <m:mo>(</m:mo><m:mi>B</m:mi> <m:mo>,</m:mo>
<m:mi>z</m:mi> <m:mo>,</m:mo> <m:mi>C'</m:mi><m:mo>)</m:mo></m:math> where
<m:math><m:mi>z</m:mi></m:math> is a variable not occurring free in
<m:math><m:mi>C</m:mi></m:math> and <m:math><m:mi>C'</m:mi></m:math> is obtained
from <m:math><m:mi>C</m:mi></m:math> by replacing each free (i.e., not bound
by any intermediate <b>binding</b> construct) occurrence
of <m:math><m:mi>v</m:mi></m:math> by <m:math><m:mi>z</m:mi></m:math>. This
operation preserves the semantics of the object
<m:math><m:mi>Ω</m:mi></m:math>. In the above example, a phrasebook is
thus allowed to transform the object to, e.g. <m:math class="del" display="block"><m:mi mathvariant="bold">binding</m:mi>
<m:mo>(</m:mo><m:mi>lambda</m:mi><m:mo>,</m:mo> <m:mi>v</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="bold">binding</m:mi> <m:mo>(</m:mo><m:mi>lambda</m:mi><m:mo>,</m:mo>
<m:mi>z</m:mi> <m:mo>,</m:mo><m:mi mathvariant="bold">application</m:mi>
<m:mo>(</m:mo><m:mi>times</m:mi><m:mo>,</m:mo><m:mi>z</m:mi>
<m:mo>,</m:mo><m:mi>z</m:mi><m:mo>)</m:mo><m:mo>)</m:mo><m:mo>)</m:mo><m:mtext>.</m:mtext></m:math>
<m:math class="new" display="block"><m:mi mathvariant="bold">binding</m:mi><m:mo>(</m:mo><m:mi>lambda</m:mi><m:mo>,</m:mo>
<m:mi>z</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="bold">application</m:mi><m:mo>(</m:mo><m:mi>plus</m:mi><m:mo>,</m:mo>
<m:mi>z</m:mi> <m:mo>,</m:mo>
<m:mn>2</m:mn><m:mo>)</m:mo><m:mo>)</m:mo><m:mtext>.</m:mtext></m:math>
</p>
<p class="new">Repeated occurrences of the same variable in a
binding operator are allowed. An OpenMath application should treat a
binding with multiple occurrences of the same variable as equivalent
to the binding in which all but the last occurrence of each variable
is replaced by a new variable which does not occur free in the body
of the binding. <m:math display="block"><m:mi mathvariant="bold">binding</m:mi> <m:mo>(</m:mo><m:mi>lambda</m:mi><m:mo>,</m:mo>
<m:mi>v</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> <m:mo>,</m:mo><m:mi mathvariant="bold">application</m:mi>
<m:mo>(</m:mo><m:mi>times</m:mi><m:mo>,</m:mo><m:mi>v</m:mi>
<m:mo>,</m:mo><m:mi>v</m:mi><m:mo>)</m:mo> <m:mo>)</m:mo></m:math> is semantically
equivalent to: <m:math display="block"><m:mi mathvariant="bold">binding</m:mi> <m:mo>(</m:mo><m:mi>lambda</m:mi> <m:mo>,</m:mo>
<m:msup><m:mi>v</m:mi><m:mo>'</m:mo></m:msup> <m:mo>,</m:mo> <m:mi>v</m:mi>
<m:mo>,</m:mo><m:mi mathvariant="bold">application</m:mi>
<m:mo>(</m:mo><m:mi>times</m:mi><m:mo>,</m:mo><m:mi>v</m:mi>
<m:mo>,</m:mo><m:mi>v</m:mi><m:mo>)</m:mo> <m:mo>)</m:mo></m:math> so that the
resulting function is actually a constant in its first argument
(<m:math><m:msup><m:mi>v</m:mi><m:mo>'</m:mo></m:msup></m:math> does not occur free
in the body <m:math><m:mi mathvariant="bold">application</m:mi>
<m:mo>(</m:mo><m:mi>times</m:mi><m:mo>,</m:mo><m:mi>v</m:mi>
<m:mo>,</m:mo><m:mi>v</m:mi><m:mo>)</m:mo> <m:mo>)</m:mo></m:math>).</p>
</dd>
<dt>Attribution</dt><dd><p>decorates an
object with a sequence of one or more pairs made up of an <i>OpenMath</i>
symbol, the <span>"attribute"</span>, and an associated <i>OpenMath</i> object,
the <span>"value of the attribute"</span>. The value of the
attribute can be an attribution object itself. As example of this,
consider the <i>OpenMath</i> objects representing groups, automorphism groups,
and group dimensions. It is then possible to attribute an <i>OpenMath</i>
object representing a group by its automorphism group, itself
attributed by its dimension.</p>
<p>Composition of attributions, as in
<m:math display="block">
<m:mi mathvariant="bold">attribution</m:mi><m:mo>(</m:mo><m:mi mathvariant="bold">attribution</m:mi><m:mo>(</m:mo><m:mi>A</m:mi><m:mo>,</m:mo>
<m:msub><m:mi>S</m:mi><m:mn>1</m:mn></m:msub> <m:mspace width=".3em"/>
<m:msub><m:mi>A</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:mi>…</m:mi><m:mo>,</m:mo><m:msub><m:mi>S</m:mi><m:mi>h</m:mi></m:msub>
<m:mspace width=".3em"/>
<m:msub><m:mi>A</m:mi><m:mi>h</m:mi></m:msub><m:mo>)</m:mo><m:mo>,</m:mo>
<m:msub><m:mi>S</m:mi><m:mrow><m:mi>h</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:msub>
<m:mspace width=".3em"/>
<m:msub><m:mi>A</m:mi><m:mrow><m:mi>h</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:msub><m:mo>,</m:mo>
<m:mi>…</m:mi><m:mo>,</m:mo> <m:msub><m:mi>S</m:mi><m:mi>n</m:mi></m:msub> <m:mspace width=".3em"/> <m:msub><m:mi>A</m:mi><m:mi>n</m:mi></m:msub><m:mo>)</m:mo></m:math> is
semantically equivalent to a single attribution, that is <m:math display="block"><m:mi mathvariant="bold">attribution</m:mi><m:mo>(</m:mo><m:mi>A</m:mi><m:mo>,</m:mo>
<m:msub><m:mi>S</m:mi><m:mn>1</m:mn></m:msub> <m:mspace width=".3em"/>
<m:msub><m:mi>A</m:mi><m:mn>1</m:mn></m:msub><m:mo>,</m:mo>
<m:mi>…</m:mi><m:mo>,</m:mo> <m:msub><m:mi>S</m:mi><m:mi>h</m:mi></m:msub> <m:mspace width=".3em"/> <m:msub><m:mi>A</m:mi><m:mi>h</m:mi></m:msub><m:mo>,</m:mo>
<m:msub><m:mi>S</m:mi><m:mrow><m:mi>h</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:msub>
<m:mspace width=".3em"/>
<m:msub><m:mi>A</m:mi><m:mrow><m:mi>h</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow></m:msub><m:mo>,</m:mo>
<m:mi>…</m:mi><m:mo>,</m:mo> <m:msub><m:mi>S</m:mi><m:mi>n</m:mi></m:msub> <m:mspace width=".3em"/>
<m:msub><m:mi>A</m:mi><m:mi>n</m:mi></m:msub><m:mo>)</m:mo><m:mtext>.</m:mtext></m:math>
The operation that produces an object with a single layer of
attribution is called <code>flattening</code>.</p>
<p>Multiple attributes with the same name are allowed. While the order
of the given attributes does not imply any notion of priority,
potentially it could be significant. For instance, consider the case
in which <m:math><m:msub><m:mi>S</m:mi><m:mi>h</m:mi></m:msub> <m:mo>=</m:mo>
<m:msub><m:mi>S</m:mi><m:mi>n</m:mi></m:msub></m:math>
(<m:math><m:mi>h</m:mi> <m:mo><</m:mo> <m:mi>n</m:mi></m:math>) in the example above. Then, the
object is to be interpreted as if the value <m:math><m:msub><m:mi>A</m:mi><m:mi>n</m:mi></m:msub></m:math> overwrites the
value <m:math><m:msub><m:mi>A</m:mi><m:mi>h</m:mi></m:msub></m:math>. (<i>OpenMath</i> however does not mandate that an application
preserves the attributes or their order.)</p>
<p>Objects can be decorated in a multitude of
ways. In <a href="#OM_D131b">[6]</a>, typing of <i>OpenMath</i> objects is
expressed by using an attribution. The object <m:math><m:mi mathvariant="bold">attribution</m:mi><m:mo>(</m:mo><m:mi>A</m:mi><m:mo>,</m:mo>
<m:mi>type</m:mi> <m:mspace width=".3em"/> <m:mi>t</m:mi> <m:mo>)</m:mo></m:math>
represents the judgment stating that object <m:math><m:mi>A</m:mi></m:math>
has type <m:math><m:mi>t</m:mi></m:math>. Note that both
<m:math><m:mi>A</m:mi></m:math> and <m:math><m:mi>t</m:mi></m:math> are <i>OpenMath</i>
objects.</p>
<p>Attribution can act as either annotation, in the sense of adornment,
or as modifier. In the former case, replacement of the adorned
object by the object itself is probably not harmful (preserves the
semantics). In the latter case however, it may very well be.
Therefore, attribution in general should by default be treated as a
construct rather than as adornment. Only when the CD definitions of
the attributes make it clear that they are adornments, can the
attributed object be viewed as semantically equivalent to the
stripped object.</p>
</dd>
<dt>Error</dt><dd><p>is made up of an <i>OpenMath</i>
symbol and a sequence of zero or more <i>OpenMath</i> objects. This object has
no direct mathematical meaning. Errors occur as the result of some
treatment on an <i>OpenMath</i> object and are thus of real interest only when
some sort of communication is taking place. Errors may occur inside
other objects and also inside other errors. Error objects might
consist only of a symbol as in the object: <m:math><m:mi mathvariant="bold">error</m:mi> <m:mo>(</m:mo><m:mi>S</m:mi>
<m:mo>)</m:mo></m:math>.</p>
</dd>
</dl>
</div>
<div><h3 name="sec_summary" id="sec_summary">3.3 Summary</h3>
<ul>
<li><p><i>OpenMath</i> supports basic objects like integers, symbols,
floating-point numbers, character strings, bytearrays, and
variables.</p></li>
<li><p><i>OpenMath</i> compound objects are of four kinds: applications, bindings,
errors, and attributions.</p></li>
<li><p><i>OpenMath</i> objects have the expressive power to cover all areas of
computational mathematics.</p></li>
</ul>
<p>Observe that an <i>OpenMath</i> application object is viewed as a <span>"tree"</span> by
software applications that do not understand Content Dictionaries,
whereas a Phrasebook that understands the semantics of the symbols, as
defined in the Content Dictionaries, should interpret the object as
functional application, constructor, or binding accordingly. Thus, for
example, for some applications, the <i>OpenMath</i> object corresponding
to <m:math><m:mn>2</m:mn><m:mo>+</m:mo><m:mn>5</m:mn></m:math> may result in a command that writes <m:math><m:mn>7</m:mn></m:math>.</p>
</div>
<h2 name="cha_enco" id="cha_enco">
Chapter 4<br/><i>OpenMath</i> Encodings</h2>
<p>In this chapter, two encodings are defined that map between <i>OpenMath</i>
objects and byte streams. These byte streams constitute a low level
representation that can be easily exchanged between processes (via
almost any communication method) or stored and retrieved from files.</p>
<p>The first encoding uses ISO 646:1983 characters <a href="#iso646_83">[2]</a>
(also known as <span>ascii</span> characters) and is an <span>xml</span>
application. Although the <span>xml</span> markup of the encoding uses only <span>ascii</span>
characters, OpenMath strings may use
arbitrary Unicode/ISO 10646:1988
characters <a href="#UNICODE">[13]</a>.
It can be used, for example, to send <i>OpenMath</i> objects via
e-mail, news, cut-and-paste, etc. The texts produced by this encoding
can be part of <span>xml</span> documents.</p>
<p>The second encoding is a binary encoding that is meant to be used when
the compactness of the encoding is important (interprocess communications
over a network is an example).</p>
<p>Note that these two encodings are sufficiently different for
autodetection to be effective: an application reading the bytes can
very easily determine which encoding is used.</p>
<div><h3 name="sec_xml" id="sec_xml">4.1 The <span>xml</span> Encoding</h3>
<p>This encoding has been designed with two main goals in mind:
<ol>
<li><p>to provide an encoding that uses the most common character set
(so that it can be easily included in most documents and transport
protocols) and that is both readable and writable by a human.</p></li>
<li><p>to provide an encoding that can be included (embedded) in
<span>xml</span> documents.</p></li>
</ol>
</p>
<div><h4 name="ssec_xml" id="ssec_xml">4.1.1 A Grammar for the <span>xml</span> Encoding</h4>
<p>The <span>xml</span> encoding of an OpenMath object is defined by the <span>dtd</span> given
in <a href="#fig_objdtd">Figure 4.1</a> below, with the following additional rules
not implied by the <span>xml</span> <span>dtd</span>.</p>
<ul>
<li><p>Comments are permitted only between
elements, not within element character data.</p></li>
<li><p>Processing Instructions are only allowed before the <span>OMOBJ</span>
element.</p></li>
<li><p>The content of an <span>OMB</span> element, is a valid base64-encoded text.</p></li>
<li><p>The character data forming element content and attribute values
matches the regular expressions of <a href="#fig_xml">Figure 4.2</a>.</p></li>
</ul>
<div class="figure"><a name="fig_objdtd" id="fig_objdtd"/>
<div class="literal"><pre>
<!-- DTD for OM Objects - sb 29.10.98 -->
<!-- sb 3.2.99 -->
<!--
general list of embeddable elements
: excludes OMATP as this is only embeddable in OMATTR
: excludes OMBVAR as this is only embeddable in OMBIND
-->
<!ENTITY % omel "OMS | OMV | OMI | OMB | OMSTR
| OMF | OMA | OMBIND | OME
| OMATTR ">
<!-- things which can be variables -->
<!ENTITY % omvar "OMV | OMATTR" >
<!-- symbol -->
<!ELEMENT OMS EMPTY>
<!ATTLIST OMS name CDATA #REQUIRED
cd CDATA #REQUIRED >
<!-- variable -->
<!ELEMENT OMV EMPTY>
<!ATTLIST OMV name CDATA #REQUIRED >
<!-- integer -->
<!ELEMENT OMI (#PCDATA) >
<!-- byte array -->
<!ELEMENT OMB (#PCDATA) >
<!-- string -->
<!ELEMENT OMSTR (#PCDATA) >
<!-- floating point -->
<!ELEMENT OMF EMPTY>
<!ATTLIST OMF dec CDATA #IMPLIED
hex CDATA #IMPLIED>
<!-- apply constructor -->
<!ELEMENT OMA (%omel;)+ >
<!-- binding constructor & variable -->
<!ELEMENT OMBIND ((%omel;), OMBVAR, (%omel;)) >
<!ELEMENT OMBVAR (%omvar;)+ >
<!-- error -->
<!ELEMENT OME (OMS, (%omel;)* ) >
<!-- attribution constructor & attribute pair constructor -->
<!ELEMENT OMATTR (OMATP, (%omel;)) >
<!ELEMENT OMATP (OMS, (%omel;))+ >
<!-- OM object constructor -->
<!ELEMENT OMOBJ (%omel;) >
</pre></div>
<div>
Figure 4.1 DTD for the <i>OpenMath</i> <span>xml</span> encoding of objects.</div></div>
<p>In addition, if the <span>xml</span> document encoding the <i>OpenMath</i> object is
linearised into the <span>xml</span> concrete syntax, the following further
constraints apply, which ensure thet the encoding may be read by <i>OpenMath</i>
applications that may not include a full <span>xml</span> parser.</p>
<ul>
<li><p>The document should use <span>utf-8</span> encoding.</p></li>
<li class="del"><p>Entity and character references should not be used.</p></li>
<li><p>A <code><!DOCTYPE</code> declaration should not be used.</p></li>
<li class="new"><p>Character references should not be used. As <code><!DOCTYPE</code>
is not used, the only entity references that are allowed are the five predefined entity
references:
<code>&apos;</code> ('),
<code>&quot;</code> ("),
<code>&lt;</code> (<),
<code>&gt;</code> (>),
<code>&amp;</code> (&).
</p></li>
<li><p>The <span>xml</span> empty element form <code><|<span>…</span>/></code> should always be
used to encode elements such as <span>omf</span> which are specified in the
<span>dtd</span> as
being <span>empty</span>. It should never be used for elements that may sometimes be
empty, such as <span>omstr</span>.</p></li>
</ul>
<p>Such a linearisation of an <span>xml</span> encoded <i>OpenMath</i> Object would match the
match the character based grammar given in <a href="#fig_xml">Figure 4.2</a>.</p>
<p>The notation used in this section and in <a href="#fig_xml">Figure 4.2</a> should
be quite straightforward (+ meaning <span>"one or more"</span>, ? meaning zero or
one, and <m:math><m:mi>|</m:mi></m:math> meaning <span>"or"</span>). The start symbol of the grammar is
<span>"start"</span>, <span>"space"</span> stands for the space character, <span>"cr"</span> for the
carriage return character, <span>"nl"</span> for the line feed character and
<span>"tab"</span> for the horizontal tabulation character.</p>
<div class="figure"><a name="fig_xml" id="fig_xml"/>
<table>
<tr>
<td>
S </td><td> <m:math><m:mo>→</m:mo></m:math> </td><td> (space<m:math><m:mi>|</m:mi></m:math>tab<m:math><m:mi>|</m:mi></m:math>cr<m:math><m:mi>|</m:mi></m:math>nl)+
</td>
</tr>
<tr>
<td>
integer </td><td> <m:math><m:mo>→</m:mo></m:math> </td><td>
(<code>-</code> S?)? [0-9]+ (S [0-9]+)* <m:math><m:mi>|</m:mi></m:math>
(<code>-</code> S?)? <code>x</code> S? [0-9A-F]+ (S [0-9A-F]+)*
</td>
</tr>
<tr>
<td>
cdname </td><td> <m:math><m:mo>→</m:mo></m:math> </td><td> [a-z][a-z0-9<code>_</code>]*
</td>
</tr>
<tr>
<td>
symbname </td><td> <m:math><m:mo>→</m:mo></m:math> </td><td> [A-Za-z][A-Za-z0-9<code>_</code>]*
</td>
</tr>
<tr>
<td>
fpdec </td><td> <m:math><m:mo>→</m:mo></m:math> </td><td>
(<code>-</code>?)([0-9]+)?(<code>.</code>[0-9]+)?(<code>e</code>([+-]?)[0-9]+)?
</td>
</tr>
<tr>
<td>
fphex </td><td> <m:math><m:mo>→</m:mo></m:math> </td><td> [0-9ABCDEF]+
</td>
</tr>
<tr>
<td>
varname </td><td> <m:math><m:mo>→</m:mo></m:math> </td><td> ([A-Za-z0-9+=(),-./:?!#$%*;@[]^_`{|}])+
</td>
</tr>
<tr>
<td>
base64 </td><td> <m:math><m:mo>→</m:mo></m:math> </td><td> ([A-Za-z0-9 +/=] <m:math><m:mi>|</m:mi></m:math> S)+
</td>
</tr>
<tr>
<td>
vv
char </td><td> <m:math><m:mo>→</m:mo></m:math> </td><td> <i>XML Character Data</i>
</td>
</tr>
</table>
<table>
<tr>
<td>
symbnameatt</td><td> <m:math><m:mo>→</m:mo></m:math></td><td>
<code>name</code> S? = S? (<code>"</code> symbname <code>"</code> <code>|</code> <code>'</code> symbname <code>'</code>)
</td>
</tr>
<tr>
<td>
cdnameatt</td><td> <m:math><m:mo>→</m:mo></m:math></td><td>
<code>cd</code> S? = S? (<code>"</code> cdname <code>"</code> <code>|</code> <code>'</code> cdname <code>'</code>)
</td>
</tr>
<tr>
<td>
varnameatt</td><td> <m:math><m:mo>→</m:mo></m:math></td><td>
<code>name</code> S? = S? (<code>"</code> varname <code>"</code> <code>|</code> <code>'</code> varname <code>'</code>)