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<?xml version="1.0" encoding="iso-8859-1"?>
<?xml-stylesheet type="text/xsl" href="pmathml.xsl"?>
<html xmlns="http://www.w3.org/1999/xhtml" xmlns:m="http://www.w3.org/1998/Math/MathML" xml:space="preserve">
<head>
<title>OpenMath Objects</title>
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<body>
<div class="minitoc"><a href="omstd20html-0.xml">OpenMath 2</a> <a href="omstd20html-0.xml#toc">Table of Contents</a><br/><br/><a href="omstd20html-1.xml">Previous: 1 Introduction to <i>OpenMath</i></a><br/><a href="#cha_obj">This: 2 <i>OpenMath</i> Objects</a><br/>
<a href="omstd20html-2.xml#sec_omabs">2.1 Formal Definition of <i>OpenMath</i> Objects</a><br/>
<a href="omstd20html-2.xml#sec_basic">2.1.1 Basic <i>OpenMath</i> objects</a><br/>
<a href="omstd20html-2.xml#sec_derived">2.1.2 Derived <i>OpenMath</i> Objects</a><br/>
<a href="omstd20html-2.xml#sec_compound">2.1.3 <i>OpenMath</i> Objects</a><br/>
<a href="omstd20html-2.xml#sec_roles">2.1.4 <i>OpenMath</i> Symbol Roles</a><br/>
<a href="omstd20html-2.xml#sec_omin">2.2 Further Description of <i>OpenMath</i> Objects</a><br/>
<a href="omstd20html-2.xml#sec_names">2.3 Names</a><br/>
<a href="omstd20html-2.xml#sec_summary">2.4 Summary</a><br/><a href="omstd20html-3.xml">Next: 3 <i>OpenMath</i> Encodings</a><br/></div>
<div><h1 name="cha_obj" id="cha_obj">
Chapter 2<br/><i>OpenMath</i> Objects</h1>
<p>In this chapter we provide a self-contained description of <i>OpenMath</i>
objects. We first do so by means of an abstract grammar
description (<a href="omstd20html-2.xml#sec_omabs">Section 2.1</a>)
and then give a more informal description (<a href="omstd20html-2.xml#sec_omin">Section 2.2</a>).</p>
<div><h2 name="sec_omabs" id="sec_omabs">2.1 Formal Definition of <i>OpenMath</i> Objects</h2>
<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><h3 name="sec_basic" id="sec_basic">2.1.1 Basic <i>OpenMath</i> objects</h3>
<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>
<ul>
<li><p><span>(i)</span> 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><span>(ii)</span> <acronym>IEEE</acronym> floating point
number.</p><p>Double precision floating-point numbers
following the <acronym>IEEE</acronym> 754-1985
standard <a href="omstd20html-g.xml#ieee754_85">[6]</a>.</p></li>
<li><p><span>(iii)</span> Character string.</p><p>A Unicode Character string. This also corresponds to <span>"characters"</span> in
<acronym>XML</acronym>.</p></li>
<li><p><span>(iv)</span> Bytearray.</p><p>A sequence of bytes.</p></li>
<li><p><span>(v)</span> Symbol.</p><p>A Symbol encodes three fields of
information, a <i>symbol name</i>, a <i>Content
Dictionary name</i>, and (optionally) a <i>Content
Dictionary base URI</i>, The name of a symbol is a sequence
of characters matching the regular expression described in <a href="omstd20html-2.xml#sec_names">Section 2.3</a>. The Content Dictionary is the location of
the definition of the symbol, consisting of a name (a sequence of
characters matching the regular expression described in <a href="omstd20html-2.xml#sec_names">Section 2.3</a>) and, optionally, a unique prefix called a
<i>cdbase</i> which is used to disambiguate multiple
Content Dictionaries of the same name. There are other properties
of the symbol that are not explicit in these fields but whose
values may be obtained by inspecting the Content Dictionary
specified. These include the symbol definition, formal properties
and examples and, optionally, a <i>Role</i> which is
a restriction on where the symbol may appear in an <i>OpenMath</i> object. The
possible roles are described in <a href="omstd20html-2.xml#sec_roles">Section 2.1.4</a>.
</p></li>
<li><p><span>(vi)</span> Variable.</p><p>A Variable
<span>must have</span> a
<i>name</i> which is a sequence of characters matching a
regular expression, as described in <a href="omstd20html-2.xml#sec_names">Section 2.3</a>.
</p></li>
</ul>
</div>
<div><h3 name="sec_derived" id="sec_derived">2.1.2 Derived <i>OpenMath</i> Objects</h3>
<p>Derived <i>OpenMath</i> objects are currently used as a way by which non-<i>OpenMath</i>
data is embedded inside an <i>OpenMath</i> object.
A derived <i>OpenMath</i> object is built as follows:
<ul>
<li><p><span>(i)</span> If <m:math><m:mi>A</m:mi></m:math> is
<i>not</i> an <i>OpenMath</i> object, then <m:math><m:mi mathvariant="bold">foreign</m:mi><m:mfenced><m:mi>A</m:mi></m:mfenced></m:math> is an <i>OpenMath</i>
<i>foreign object</i>. An <i>OpenMath</i> foreign object may optionally
have an <i>encoding</i> field which describes how its contents
should be interpreted.</p></li>
</ul>
</p>
</div>
<div><h3 name="sec_compound" id="sec_compound">2.1.3 <i>OpenMath</i> Objects</h3>
<p><i>OpenMath</i> objects are built recursively as follows.
<ul>
<li><p><span>(i)</span> Basic <i>OpenMath</i> objects are <i>OpenMath</i> objects.
(<span>Note that derived <i>OpenMath</i> objects are
<i>not</i> <i>OpenMath</i> objects, but are used to construct <i>OpenMath</i>
objects as described below.)</span></p></li>
<li><p>
<span>(ii)</span> 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><span>(iii)</span> 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
<span>
<m:math><m:mi>A</m:mi></m:math> is an <i>OpenMath</i> object, 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 or <i>OpenMath</i> derived objects, then
</span>
<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>.
</p><p>
<m:math><m:mi>A</m:mi></m:math> is the object <i>stripped of attributions</i>. <span>
<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 referred to as <i>keys</i> 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> as their associated
<i>values</i></span>.
<span>If, after recursively
applying stripping to remove attributions, the resulting
un-attributed object is a
variable, the original attributed object is called an <i>attributed
variable</i>.</span>
</p></li>
<li><p><span>(iv)</span> 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><span>(v)</span> 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 <span>or
<i>OpenMath</i> derived objects</span>, 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>
</ul>
<span><i>OpenMath</i> objects that are contstructed via rules (ii)
to (v) are jointly called <span><i>compound <i>OpenMath</i> objects</i></span></span>
</p>
</div>
<div><h3 name="sec_roles" id="sec_roles">2.1.4 <i>OpenMath</i> Symbol Roles</h3>
<p>
We say that an <i>OpenMath</i> symbol is used to <i>construct</i>
an <i>OpenMath</i> object if it is the first child of an <i>OpenMath</i> application,
binding or error object, or an even-indexed child of an <i>OpenMath</i>
attribution object (i.e. the <i>key</i> in a
<i>(key, value)</i> pair).
The <i>role</i> of an <i>OpenMath</i> symbol is a restriction
on how it may be used to construct a compound <i>OpenMath</i> object and, in the
case of the key in an attribution object, a clarification of how that
attribution should be interpreted. <span class="finaledit">The </span>possible roles are:
<ol class="lowerroman">
<li><p><i>binder</i> The symbol may
appear as the first child of an <i>OpenMath</i> binding object.
</p></li>
<li><p> <i>attribution</i> The symbol may
be used as key in an <i>OpenMath</i> attribution object, i.e. as the first
element of a (key, value) pair, or in an equivalent context (for example
to refer to the value of an attribution). This form of attribution
may be ignored by an application, so should be used for information
which does not change the meaning of the attributed <i>OpenMath</i> object.
</p></li>
<li><p> <i>semantic-attribution</i> This is the
same as <i>attribution</i> except that it modifies the
meaning of the attributed <i>OpenMath</i> object and thus cannot be ignored by an
application<span class="finaledit">, without changing the meaning</span>. </p></li>
<li><p> <i>error</i> The symbol <span class="finaledit">may</span> appear
as the first child of an <i>OpenMath</i> error object. </p></li>
<li><p> <i>application</i> The symbol <span class="finaledit">may</span> appear
as the first child of an <i>OpenMath</i> application object. </p></li>
<li><p> <i>constant</i> The symbol cannot be
used to construct an <i>OpenMath</i> compound object.
</p></li>
</ol>
A symbol cannot have more than one role and
cannot be used to construct a compound <i>OpenMath</i> object in a way
which requires a different role (using the definition of construct given
earlier in this section).
This means that one cannot use a symbol which binds some variables to
construct, say, an application object. However it does not prevent
the use of that symbol as an <i>argument</i> in an
application object (where by argument we mean a child with index
greater than 1).
</p>
<p>
If no role is indicated then the symbol can be used anywhere. Note
that this is not the same as saying that the symbol's role is
<i>constant</i>.
</p>
</div>
</div>
<div><h2 name="sec_omin" id="sec_omin">2.2 Further Description of <i>OpenMath</i> Objects</h2>
<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
<acronym>IEEE</acronym> 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="omstd20html-g.xml#UNICODE">[12]</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
<span>"handles"</span> 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.
The form of these definitions is explained in
<a href="omstd20html-4.xml#cha_cd">Chapter 4</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
<small><code>union</code></small> is defined as
associative-commutative set theoretic union in a Content Dictionary
<small><code>set1</code></small> but another Content Dictionary,
<small><code>multiset1</code></small> might define a symbol
<small><code>union</code></small> as the union of multi-sets).
</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).
</p>
</dd>
</dl> </p>
<p>Derived <i>OpenMath</i> objects are constructed from
non-<i>OpenMath</i> data. They differ from bytearrays in that they can have any
structure. Currently there is only one way of making a derived <i>OpenMath</i>
object.</p>
<dl>
<dt>Foreign</dt><dd><p>is used to import a
non-<i>OpenMath</i> object into an <i>OpenMath</i> attribution. Examples of its use could
be to annotate a formula with a visual or aural rendering, an
animation, etc. They may also appear in <i>OpenMath</i> error objects, for
example to allow an application to report an error in processing such
an object.
</p>
</dd>
</dl>
<p>The four following constructs can be used to make compound
<i>OpenMath</i> objects <span> out of basic or derived <i>OpenMath</i>
objects</span>.</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
<span class="finaledit">child</span>
of an application is referred to as its <span>"head"</span> while
the remaining objects are called its <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 <m:math><m:mi>sin</m:mi></m:math> is defined in
a <span>suitable</span> Content Dictionary,
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
<m:math><m:mi>Rational</m:mi></m:math> 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 class="caption">
Figure 2.1 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 <m:math><m:mi>lambda</m:mi></m:math>: <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>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 revisionflag="deleted" 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 revisionflag="added" 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>Repeated occurrences of the same variable in a binding operator
are allowed. An <i>OpenMath</i> 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 object, the <span>"value of the
attribute"</span>. The value of the attribute can be an <span><i>OpenMath</i></span> attribution object itself. As an
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>
<i>OpenMath</i> objects can be attributed with <i>OpenMath</i> foreign objects, which are
containers for non-<i>OpenMath</i> structures. For example a mathematical
expression could be attributed with its spoken or visual rendering.
</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 <i>flattening</i>.</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>Attribution acts as either adornment
annotation or as semantical annotation. When the key has role
<i>attribution</i>, then replacement of the
attributed object by the object itself is not harmful and preserves
the semantics. When the key has role
<i>semantic-attribution</i> then the attributed
object is modified by the attribution and cannot be viewed as
semantically equivalent to the stripped object. If the attribute
lacks the role specification then attribution is acting as adornment
annotation.
</p>
<p>Objects can be decorated in a multitude of ways.
<span>An example of the use of an adornment attribution
would be to indicate the colour in which an <i>OpenMath</i> object should be
displayed, for example <m:math><m:mi mathvariant="bold">attribution</m:mi><m:mo>(</m:mo><m:mi>A</m:mi><m:mo>,</m:mo>
<m:mi>colour</m:mi> <m:mspace width=".3em"/> <m:mi>red</m:mi> <m:mo>)</m:mo></m:math>.
Note that both <m:math><m:mi>A</m:mi></m:math> and <m:math><m:mi>red</m:mi></m:math> are
<span class="finaledit">arbitary</span> <i>OpenMath</i>
objects <span class="finaledit">whereas <m:math><m:mi>color</m:mi></m:math> is
a symbol</span>.
An example of the use of a semantic attribution would be to indicate the
type of an object. For example</span>
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 <span class="finaledit">arbitary</span> <i>OpenMath</i>
objects <span class="finaledit">whereas <m:math><m:mi>type</m:mi></m:math> is
a symbol</span>.</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><h2 name="sec_names" id="sec_names">2.3 Names</h2>
<p>The names of symbols, variables and content dictionaries must
conform to the production <small><code>Name</code></small> specified in the following
grammar
(which is identical to that for <acronym>XML</acronym> names in XML 1.1,
<a href="omstd20html-g.xml#xml_04">[16]</a>). Informally speaking, a name is a sequence
of Unicode <a href="omstd20html-g.xml#UNICODE">[12]</a>
characters which begins with a letter and cannot contain
certain punctuation and combining characters. The notation
<small><code>#x...</code></small> represents the hexadecimal value of
the encoding of a Unicode character.
Some of the character values or <i>code points</i> in the
following productions are currently unassigned, but this is
likely to change in the future as Unicode evolves<sup><a href="#xml1">*1</a></sup>.
</p><p class="footnote"><a name="xml1" id="xml1"/><sup>*1</sup>
We note that in XML 1 the name production explicitly listed
the characters that were allowed, so all the characters added in
versions of Unicode after 2.0 (which amounted to tens of thousands of
characters) were not allowed in names.
</p>
<blockquote>
<table>
<tr>
<td>Name </td>
<td> <m:math><m:mo>→</m:mo></m:math> </td>
<td> NameStartChar (NameChar)* </td>
</tr>
<tr>
<td>NameStartChar</td>
<td> <m:math><m:mo>→</m:mo></m:math> </td>
<td> ":" | [A-Z] | "_" | [a-z] | [#xC0-#xD6] | [#xD8-#xF6] |</td></tr>
<tr><td/><td/><td>[#xF8-#x2FF] | [#x370-#x37D] | [#x37F-#x1FFF] |</td></tr>
<tr><td/><td/><td>[#x200C-#x200D] | [#x2070-#x218F] | [#x2C00-#x2FEF] |</td></tr>
<tr><td/><td/><td>[#x3001-#xD7FF] | [#xF900-#xFDCF] | [#xFDF0-#xFFFD] |</td></tr>
<tr><td/><td/><td>[#x10000-#xEFFFF]
</td>
</tr>
<tr>
<td>NameChar</td>
<td> <m:math><m:mo>→</m:mo></m:math> </td>
<td> NameStartChar | "-" | "." | [0-9] | #xB7 | [#x0300-#x036F] |</td></tr>
<tr><td/><td/><td>[#x203F-#x2040] </td>
</tr>
</table>
</blockquote>
<p><b>CD Base</b> A cdbase must conform to the grammar for URIs described in
<a href="omstd20html-g.xml#IETF2396">[7]</a>. Note that if non-ASCII characters are
used in a CD or symbol name then when a URI for that symbol is
constructed it will be necessary to map the non-ASCII characters to a
sequence of octets. The precise mechanism for doing this depends on
the URI scheme.</p>
<p><b>Note on content dictionary names</b>
It is a common convention to store a Content Dictionary in a file of
the same name, which can cause difficulties on many file systems. If
this convention is to be followed then <i>OpenMath</i>
<i>recommends</i> that the name be restricted to the
subset of the above grammar which is a legal POSIX
<a href="omstd20html-g.xml#POSIX">[5]</a> filename, namely:
<blockquote>
<table>
<tr>
<td>Name </td>
<td> <m:math><m:mo>→</m:mo></m:math> </td>
<td> (PosixLetter | '_') (Char)*
</td>
</tr>
<tr>
<td>Char</td>
<td> <m:math><m:mo>→</m:mo></m:math> </td>
<td> PosixLetter | Digit | '.' | '-' | '_'
</td>
</tr>
<tr>
<td>PosixLetter</td>
<td> <m:math><m:mo>→</m:mo></m:math> </td>
<td>
'a' | 'b' | ... | 'z' | 'A' | 'B' | ... | 'Z'
</td>
</tr>
</table>
</blockquote>
</p>
<p><b>Canonical URIs for Symbols</b>
To facilitate the use of <i>OpenMath</i> within a URI-based framework (such as RDF
<a href="omstd20html-g.xml#rdf">[19]</a> or OWL <a href="omstd20html-g.xml#owl">[18]</a>), we provide the
following scheme for constructing a canonical URI
for an <i>OpenMath</i> Symbol:
<blockquote>
<p><small><code>URI = cdbase-value + '/' + cd-value + '#' + name-value</code></small></p>
</blockquote>
So for example the URI for the symbol with cdbase
<small><code>http://www.openmath.org/cd</code></small>, cd
<small><code>transc1</code></small> and name <small><code>sin</code></small>
is:
<blockquote>
<p><small><code>http://www.openmath.org/cd/transc1#sin</code></small></p>
</blockquote>
In particular, this now allows us to refer uniquely to an <i>OpenMath</i> symbol from a
MathML document <a href="omstd20html-g.xml#MathML_2003">[17]</a>:
<div class="literal"><pre>
<mathml:csymbol xmlns:mathml="http://www.w3.org/1998/Math/MathML/"
definitionURL="http://www.openmath.org/cd/transc1#sin">
<mo> sin </mo>
</csymbol>
</pre></div>
</p>
</div>
<div><h2 name="sec_summary" id="sec_summary">2.4 Summary</h2>
<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 may be attributed
with non-<i>OpenMath</i> objects via the use of foreign <i>OpenMath</i> objects.
</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>
</div>
<div class="minitoc"><a href="omstd20html-0.xml">OpenMath 2</a> <a href="omstd20html-0.xml#toc">Table of Contents</a><br/><br/><a href="omstd20html-1.xml">Previous: 1 Introduction to <i>OpenMath</i></a><br/><a href="#cha_obj">This: 2 <i>OpenMath</i> Objects</a><br/><a href="omstd20html-3.xml">Next: 3 <i>OpenMath</i> Encodings</a><br/></div>
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