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- </div>
- <div class="section">
- <div class="titlepage"><div><div><h2 class="title" style="clear: both">
- <a name="boost_icl.semantics"></a><a class="link" href="semantics.html" title="Semantics">Semantics</a>
- </h2></div></div></div>
- <div class="toc"><dl class="toc">
- <dt><span class="section"><a href="semantics.html#boost_icl.semantics.orderings_and_equivalences">Orderings
- and Equivalences</a></span></dt>
- <dt><span class="section"><a href="semantics/sets.html">Sets</a></span></dt>
- <dt><span class="section"><a href="semantics/maps.html">Maps</a></span></dt>
- <dt><span class="section"><a href="semantics/collectors__maps_of_sets.html">Collectors:
- Maps of Sets</a></span></dt>
- <dt><span class="section"><a href="semantics/quantifiers__maps_of_numbers.html">Quantifiers:
- Maps of Numbers</a></span></dt>
- <dt><span class="section"><a href="semantics/concept_induction.html">Concept Induction</a></span></dt>
- </dl></div>
- <p>
- <span class="quote">“<span class="quote">Beauty is the ultimate defense against complexity</span>”</span> -- <a href="http://en.wikipedia.org/wiki/David_Gelernter" target="_top">David Gelernter</a>
- </p>
- <p>
- In the <span class="bold"><strong>icl</strong></span> we follow the notion, that the
- semantics of a <span class="emphasis"><em><span class="bold"><strong>concept</strong></span></em></span>
- or <span class="emphasis"><em><span class="bold"><strong>abstract data type</strong></span></em></span>
- can be expressed by <span class="emphasis"><em><span class="bold"><strong>laws</strong></span></em></span>.
- We formulate laws over interval containers that can be evaluated for a given
- instantiation of the variables contained in the law. The following pseudocode
- gives a shorthand notation of such a law.
- </p>
- <pre class="programlisting"><span class="identifier">Commutativity</span><span class="special"><</span><span class="identifier">T</span><span class="special">,+>:</span>
- <span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span> <span class="special">==</span> <span class="identifier">b</span> <span class="special">+</span> <span class="identifier">a</span><span class="special">;</span>
- </pre>
- <p>
- This can of course be coded as a proper c++ class template which has been done
- for the validation of the <span class="bold"><strong>icl</strong></span>. For sake of
- simplicity we will use pseudocode here.
- </p>
- <p>
- The laws that describe the semantics of the <span class="bold"><strong>icl's</strong></span>
- class templates were validated using the Law based Test Automaton <span class="emphasis"><em><span class="bold"><strong>LaBatea</strong></span></em></span>, a tool that generates instances
- for the law's variables and then tests it's validity. Since the <span class="bold"><strong>icl</strong></span>
- deals with sets, maps and relations, that are well known objects from mathematics,
- the laws that we are using are mostly <span class="emphasis"><em>recycled</em></span> ones. Also
- some of those laws are grouped in notions like e.g. <span class="emphasis"><em>orderings</em></span>
- or <span class="emphasis"><em>algebras</em></span>.
- </p>
- <div class="section">
- <div class="titlepage"><div><div><h3 class="title">
- <a name="boost_icl.semantics.orderings_and_equivalences"></a><a class="link" href="semantics.html#boost_icl.semantics.orderings_and_equivalences" title="Orderings and Equivalences">Orderings
- and Equivalences</a>
- </h3></div></div></div>
- <h5>
- <a name="boost_icl.semantics.orderings_and_equivalences.h0"></a>
- <span class="phrase"><a name="boost_icl.semantics.orderings_and_equivalences.lexicographical_ordering_and_equality"></a></span><a class="link" href="semantics.html#boost_icl.semantics.orderings_and_equivalences.lexicographical_ordering_and_equality">Lexicographical
- Ordering and Equality</a>
- </h5>
- <p>
- On all set and map containers of the icl, there is an <code class="computeroutput"><span class="keyword">operator</span>
- <span class="special"><</span></code> that implements a <a href="http://www.sgi.com/tech/stl/StrictWeakOrdering.html" target="_top">strict
- weak ordering</a>. The semantics of <code class="computeroutput"><span class="keyword">operator</span>
- <span class="special"><</span></code> is the same as for an stl's
- <a href="http://www.sgi.com/tech/stl/SortedAssociativeContainer.html" target="_top">SortedAssociativeContainer</a>,
- specifically <a href="http://www.sgi.com/tech/stl/set.html" target="_top">stl::set</a>
- and <a href="http://www.sgi.com/tech/stl/map.html" target="_top">stl::map</a>:
- </p>
- <pre class="programlisting"><span class="identifier">Irreflexivity</span><span class="special"><</span><span class="identifier">T</span><span class="special">,<</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">T</span> <span class="identifier">a</span><span class="special">;</span> <span class="special">!(</span><span class="identifier">a</span><span class="special"><</span><span class="identifier">a</span><span class="special">)</span>
- <span class="identifier">Asymmetry</span><span class="special"><</span><span class="identifier">T</span><span class="special">,<</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">;</span> <span class="identifier">a</span><span class="special"><</span><span class="identifier">b</span> <span class="identifier">implies</span> <span class="special">!(</span><span class="identifier">b</span><span class="special"><</span><span class="identifier">a</span><span class="special">)</span>
- <span class="identifier">Transitivity</span><span class="special"><</span><span class="identifier">T</span><span class="special">,<</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span><span class="special"><</span><span class="identifier">b</span> <span class="special">&&</span> <span class="identifier">b</span><span class="special"><</span><span class="identifier">c</span> <span class="identifier">implies</span> <span class="identifier">a</span><span class="special"><</span><span class="identifier">c</span>
- </pre>
- <p>
- </p>
- <p>
- <code class="computeroutput"><span class="identifier">Operator</span> <span class="special"><</span></code>
- depends on the icl::container's template parameter <code class="computeroutput"><span class="identifier">Compare</span></code>
- that implements a <span class="emphasis"><em>strict weak ordering</em></span> for the container's
- <code class="computeroutput"><span class="identifier">domain_type</span></code>. For a given
- <code class="computeroutput"><span class="identifier">Compare</span></code> ordering, <code class="computeroutput"><span class="keyword">operator</span> <span class="special"><</span></code>
- implements a lexicographical comparison on icl::containers, that uses the
- <code class="computeroutput"><span class="identifier">Compare</span></code> order to establish
- a unique sequence of values in the container.
- </p>
- <p>
- The induced equivalence of <code class="computeroutput"><span class="keyword">operator</span>
- <span class="special"><</span></code> is lexicographical equality which
- is implemented as <code class="computeroutput"><span class="keyword">operator</span> <span class="special">==</span></code>.
- </p>
- <pre class="programlisting"><span class="comment">//equivalence induced by strict weak ordering <</span>
- <span class="special">!(</span><span class="identifier">a</span><span class="special"><</span><span class="identifier">b</span><span class="special">)</span> <span class="special">&&</span> <span class="special">!(</span><span class="identifier">b</span><span class="special"><</span><span class="identifier">a</span><span class="special">)</span> <span class="identifier">implies</span> <span class="identifier">a</span> <span class="special">==</span> <span class="identifier">b</span><span class="special">;</span>
- </pre>
- <p>
- Again this follows the semantics of the <span class="bold"><strong>stl</strong></span>.
- Lexicographical equality is stronger than the equality of elements. Two containers
- that contain the same elements can be lexicographically unequal, if their
- elements are differently sorted. Lexicographical comparison belongs to the
- <span class="emphasis"><em><span class="bold"><strong>segmental</strong></span></em></span> aspect. Of
- all the different sequences that are valid for unordered sets and maps, one
- such sequence is selected by the <code class="computeroutput"><span class="identifier">Compare</span></code>
- order of elements. Based on this selection a unique iteration is possible.
- </p>
- <h5>
- <a name="boost_icl.semantics.orderings_and_equivalences.h1"></a>
- <span class="phrase"><a name="boost_icl.semantics.orderings_and_equivalences.subset_ordering_and_element_equality"></a></span><a class="link" href="semantics.html#boost_icl.semantics.orderings_and_equivalences.subset_ordering_and_element_equality">Subset
- Ordering and Element Equality</a>
- </h5>
- <p>
- On the fundamental aspect only membership of elements matters, not their
- sequence. So there are functions <code class="computeroutput"><span class="identifier">contained_in</span></code>
- and <code class="computeroutput"><span class="identifier">element_equal</span></code> that implement
- the subset relation and the equality on elements. Yet, <code class="computeroutput"><span class="identifier">contained_in</span></code>
- and <code class="computeroutput"><span class="identifier">is_element_equal</span></code> functions
- are not really working on the level of elements. They also work on the basis
- of the containers templates <code class="computeroutput"><span class="identifier">Compare</span></code>
- parameter. In practical terms we need to distinguish between lexicographical
- equality <code class="computeroutput"><span class="keyword">operator</span> <span class="special">==</span></code>
- and equality of elements <code class="computeroutput"><span class="identifier">is_element_equal</span></code>,
- if we work with interval splitting interval containers:
- </p>
- <pre class="programlisting"><span class="identifier">split_interval_set</span><span class="special"><</span><span class="identifier">time</span><span class="special">></span> <span class="identifier">w1</span><span class="special">,</span> <span class="identifier">w2</span><span class="special">;</span> <span class="comment">//Pseudocode</span>
- <span class="identifier">w1</span> <span class="special">=</span> <span class="special">{[</span><span class="identifier">Mon</span> <span class="special">..</span> <span class="identifier">Sun</span><span class="special">)};</span> <span class="comment">//split_interval_set containing a week</span>
- <span class="identifier">w2</span> <span class="special">=</span> <span class="special">{[</span><span class="identifier">Mon</span> <span class="special">..</span> <span class="identifier">Fri</span><span class="special">)[</span><span class="identifier">Sat</span> <span class="special">..</span> <span class="identifier">Sun</span><span class="special">)};</span> <span class="comment">//Same week split in work and week end parts.</span>
- <span class="identifier">w1</span> <span class="special">==</span> <span class="identifier">w2</span><span class="special">;</span> <span class="comment">//false: Different segmentation</span>
- <span class="identifier">is_element_equal</span><span class="special">(</span><span class="identifier">w1</span><span class="special">,</span><span class="identifier">w2</span><span class="special">);</span> <span class="comment">//true: Same elements contained </span>
- </pre>
- <p>
- </p>
- <p>
- For a constant <code class="computeroutput"><span class="identifier">Compare</span></code> order
- on key elements, member function <code class="computeroutput"><span class="identifier">contained_in</span></code>
- that is defined for all icl::containers implements a <a href="http://en.wikipedia.org/wiki/Partially_ordered_set" target="_top">partial
- order</a> on icl::containers.
- </p>
- <p>
- </p>
- <pre class="programlisting"><span class="identifier">with</span> <span class="special"><=</span> <span class="keyword">for</span> <span class="identifier">contained_in</span><span class="special">,</span>
- <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> <span class="keyword">for</span> <span class="identifier">is_element_equal</span><span class="special">:</span>
- <span class="identifier">Reflexivity</span><span class="special"><</span><span class="identifier">T</span><span class="special">,<=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">T</span> <span class="identifier">a</span><span class="special">;</span> <span class="identifier">a</span> <span class="special"><=</span> <span class="identifier">a</span>
- <span class="identifier">Antisymmetry</span><span class="special"><</span><span class="identifier">T</span><span class="special">,<=,=</span><span class="identifier">e</span><span class="special">=></span> <span class="special">:</span> <span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">;</span> <span class="identifier">a</span> <span class="special"><=</span> <span class="identifier">b</span> <span class="special">&&</span> <span class="identifier">b</span> <span class="special"><=</span> <span class="identifier">a</span> <span class="identifier">implies</span> <span class="identifier">a</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> <span class="identifier">b</span>
- <span class="identifier">Transitivity</span><span class="special"><</span><span class="identifier">T</span><span class="special">,<=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special"><=</span> <span class="identifier">b</span> <span class="special">&&</span> <span class="identifier">b</span> <span class="special"><=</span> <span class="identifier">c</span> <span class="identifier">implies</span> <span class="identifier">a</span> <span class="special"><=</span> <span class="identifier">c</span>
- </pre>
- <p>
- </p>
- <p>
- The induced equivalence is the equality of elements that is implemented via
- function <code class="computeroutput"><span class="identifier">is_element_equal</span></code>.
- </p>
- <pre class="programlisting"><span class="comment">//equivalence induced by the partial ordering contained_in on icl::container a,b</span>
- <span class="identifier">a</span><span class="special">.</span><span class="identifier">contained_in</span><span class="special">(</span><span class="identifier">b</span><span class="special">)</span> <span class="special">&&</span> <span class="identifier">b</span><span class="special">.</span><span class="identifier">contained_in</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span> <span class="identifier">implies</span> <span class="identifier">is_element_equal</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">);</span>
- </pre>
- <p>
- </p>
- </div>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2007-2010 Joachim
- Faulhaber<br>Copyright © 1999-2006 Cortex Software
- GmbH<p>
- Distributed under the Boost Software License, Version 1.0. (See accompanying
- file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
- </p>
- </div></td>
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