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| <div class="section" lang="en"> |
| <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> |
| <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">Beauty is the ultimate defense against complexity</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" lang="en"> |
| <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> |
| <a name="boost_icl.semantics.orderings_and_equivalences.lexicographical_ordering_and_equality"></a><h5> |
| <a name="id1078610"></a> |
| <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>. |
| </p> |
| <p> |
| 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> |
| <a name="boost_icl.semantics.orderings_and_equivalences.subset_ordering_and_element_equality"></a><h5> |
| <a name="id1079099"></a> |
| <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> |
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| <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> |
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