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| <div class="section boost_icl_semantics_quantifiers__maps_of_numbers" lang="en"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="boost_icl.semantics.quantifiers__maps_of_numbers"></a><a class="link" href="quantifiers__maps_of_numbers.html" title="Quantifiers: Maps of Numbers">Quantifiers: |
| Maps of Numbers</a> |
| </h3></div></div></div> |
| <a name="boost_icl.semantics.quantifiers__maps_of_numbers.subtraction_on_quantifiers"></a><h6> |
| <a name="id1093602"></a> |
| <a class="link" href="quantifiers__maps_of_numbers.html#boost_icl.semantics.quantifiers__maps_of_numbers.subtraction_on_quantifiers">Subtraction |
| on Quantifiers</a> |
| </h6> |
| <p> |
| With <code class="computeroutput"><span class="identifier">Sets</span></code> and <code class="computeroutput"><span class="identifier">Collectors</span></code> the semantics of <code class="computeroutput"><span class="keyword">operator</span> <span class="special">-</span></code> |
| is that of <span class="emphasis"><em>set difference</em></span> which means, that you can |
| only subtract what has been put into the container before. With <code class="computeroutput"><span class="identifier">Quantifiers</span></code> that <span class="emphasis"><em><span class="bold"><strong>count</strong></span></em></span> |
| or <span class="emphasis"><em><span class="bold"><strong>quantify</strong></span></em></span> their key |
| values in some way, the semantics of <code class="computeroutput"><span class="keyword">operator</span> |
| <span class="special">-</span></code> may be different. |
| </p> |
| <p> |
| The question is how subtraction should be defined here? |
| </p> |
| <pre class="programlisting"><span class="comment">//Pseudocode: |
| </span><span class="identifier">icl</span><span class="special">::</span><span class="identifier">map</span><span class="special"><</span><span class="keyword">int</span><span class="special">,</span><span class="identifier">some_number</span><span class="special">></span> <span class="identifier">q</span> <span class="special">=</span> <span class="special">{(</span><span class="number">1</span><span class="special">-></span><span class="number">1</span><span class="special">)};</span> |
| <span class="identifier">q</span> <span class="special">-=</span> <span class="special">(</span><span class="number">2</span><span class="special">-></span><span class="number">1</span><span class="special">);</span> |
| </pre> |
| <p> |
| If type <code class="computeroutput"><span class="identifier">some_number</span></code> is <code class="computeroutput"><span class="keyword">unsigned</span></code> a <span class="emphasis"><em>set difference</em></span> |
| kind of subtraction make sense |
| </p> |
| <pre class="programlisting"><span class="identifier">icl</span><span class="special">::</span><span class="identifier">map</span><span class="special"><</span><span class="keyword">int</span><span class="special">,</span><span class="identifier">some_number</span><span class="special">></span> <span class="identifier">q</span> <span class="special">=</span> <span class="special">{(</span><span class="number">1</span><span class="special">-></span><span class="number">1</span><span class="special">)};</span> |
| <span class="identifier">q</span> <span class="special">-=</span> <span class="special">(</span><span class="number">2</span><span class="special">-></span><span class="number">1</span><span class="special">);</span> <span class="comment">// key 2 is not in the map so |
| </span><span class="identifier">q</span> <span class="special">==</span> <span class="special">{(</span><span class="number">1</span><span class="special">-></span><span class="number">1</span><span class="special">)};</span> <span class="comment">// q is unchanged by 'aggregate on collision' |
| </span></pre> |
| <p> |
| If <code class="computeroutput"><span class="identifier">some_number</span></code> is a <code class="computeroutput"><span class="keyword">signed</span></code> numerical type the result can also |
| be this |
| </p> |
| <pre class="programlisting"><span class="identifier">icl</span><span class="special">::</span><span class="identifier">map</span><span class="special"><</span><span class="keyword">int</span><span class="special">,</span><span class="identifier">some_number</span><span class="special">></span> <span class="identifier">q</span> <span class="special">=</span> <span class="special">{(</span><span class="number">1</span><span class="special">-></span><span class="number">1</span><span class="special">)};</span> |
| <span class="identifier">q</span> <span class="special">-=</span> <span class="special">(</span><span class="number">2</span><span class="special">-></span><span class="number">1</span><span class="special">);</span> <span class="comment">// subtracting works like |
| </span><span class="identifier">q</span> <span class="special">==</span> <span class="special">{(</span><span class="number">1</span><span class="special">-></span><span class="number">1</span><span class="special">),</span> <span class="special">(</span><span class="number">2</span><span class="special">-></span> <span class="special">-</span><span class="number">1</span><span class="special">)};</span> <span class="comment">// adding the inverse element |
| </span></pre> |
| <p> |
| As commented in the example, subtraction of a key value pair <code class="computeroutput"><span class="special">(</span><span class="identifier">k</span><span class="special">,</span><span class="identifier">v</span><span class="special">)</span></code> can |
| obviously be defined as adding the <span class="emphasis"><em><span class="bold"><strong>inverse |
| element</strong></span></em></span> for that key <code class="computeroutput"><span class="special">(</span><span class="identifier">k</span><span class="special">,-</span><span class="identifier">v</span><span class="special">)</span></code>, if the key is not yet stored in the map. |
| </p> |
| <a name="boost_icl.semantics.quantifiers__maps_of_numbers.partial_and_total_quantifiers_and_infinite_vectors"></a><h5> |
| <a name="id1094139"></a> |
| <a class="link" href="quantifiers__maps_of_numbers.html#boost_icl.semantics.quantifiers__maps_of_numbers.partial_and_total_quantifiers_and_infinite_vectors">Partial |
| and Total Quantifiers and Infinite Vectors</a> |
| </h5> |
| <p> |
| Another concept, that we can think of, is that in a <code class="computeroutput"><span class="identifier">Quantifier</span></code> |
| every <code class="computeroutput"><span class="identifier">key_value</span></code> is initially |
| quantified <code class="computeroutput"><span class="number">0</span></code>-times, where <code class="computeroutput"><span class="number">0</span></code> stands for the neutral element of the numeric |
| <code class="computeroutput"><span class="identifier">CodomainT</span></code> type. Such a <code class="computeroutput"><span class="identifier">Quantifier</span></code> would be totally defined on |
| all values of it's <code class="computeroutput"><span class="identifier">DomainT</span></code> |
| type and can be conceived as an <code class="computeroutput"><span class="identifier">InfiniteVector</span></code>. |
| </p> |
| <p> |
| To create an infinite vector that is totally defined on it's domain we can |
| set the map's <code class="computeroutput"><span class="identifier">Trait</span></code> parameter |
| to the value <code class="computeroutput"><a class="link" href="../../boost/icl/total_absorber.html" title="Struct total_absorber">total_absorber</a></code>. |
| The <code class="computeroutput"><a class="link" href="../../boost/icl/total_absorber.html" title="Struct total_absorber">total_absorber</a></code> |
| trait fits specifically well with a <code class="computeroutput"><span class="identifier">Quantifier</span></code> |
| if it's <code class="computeroutput"><span class="identifier">CodomainT</span></code> has an |
| inverse element, like all signed numerical type have. As we can see later |
| in this section this kind of a total <code class="computeroutput"><span class="identifier">Quantifier</span></code> |
| has the basic properties that elements of a <a href="http://en.wikipedia.org/wiki/Vector_space" target="_top">vector |
| space</a> do provide. |
| </p> |
| <a name="boost_icl.semantics.quantifiers__maps_of_numbers.intersection_on_quantifiers"></a><h6> |
| <a name="id1094269"></a> |
| <a class="link" href="quantifiers__maps_of_numbers.html#boost_icl.semantics.quantifiers__maps_of_numbers.intersection_on_quantifiers">Intersection |
| on Quantifiers</a> |
| </h6> |
| <p> |
| Another difference between <code class="computeroutput"><span class="identifier">Collectors</span></code> |
| and <code class="computeroutput"><span class="identifier">Quantifiers</span></code> is the semantics |
| of <code class="computeroutput"><span class="keyword">operator</span> <span class="special">&</span></code>, |
| that has the meaning of set intersection for <code class="computeroutput"><span class="identifier">Collectors</span></code>. |
| </p> |
| <p> |
| For the <span class="emphasis"><em>aggregate on overlap principle</em></span> the operation |
| <code class="computeroutput"><span class="special">&</span></code> has to be passed to combine |
| associated values on overlap of intervals or collision of keys. This can |
| not be done for <code class="computeroutput"><span class="identifier">Quantifiers</span></code>, |
| since numeric types do not implement intersection. |
| </p> |
| <p> |
| For <code class="computeroutput"><span class="identifier">CodomainT</span></code> types that |
| are not models of <code class="computeroutput"><span class="identifier">Sets</span></code> <code class="computeroutput"><span class="keyword">operator</span> <span class="special">&</span> </code> |
| is defined as <span class="emphasis"><em>aggregation on the intersection of the domains</em></span>. |
| Instead of the <code class="computeroutput"><span class="identifier">codomain_intersect</span></code> |
| functor <code class="computeroutput"><span class="identifier">codomain_combine</span></code> |
| is used as aggregation operation: |
| </p> |
| <pre class="programlisting"><span class="comment">//Pseudocode example for partial Quantifiers p, q: |
| </span><span class="identifier">interval_map</span><span class="special"><</span><span class="keyword">int</span><span class="special">,</span><span class="keyword">int</span><span class="special">></span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">q</span><span class="special">;</span> |
| <span class="identifier">p</span> <span class="special">=</span> <span class="special">{[</span><span class="number">1</span> <span class="number">3</span><span class="special">)-></span><span class="number">1</span> <span class="special">};</span> |
| <span class="identifier">q</span> <span class="special">=</span> <span class="special">{</span> <span class="special">([</span><span class="number">2</span> <span class="number">4</span><span class="special">)-></span><span class="number">1</span><span class="special">};</span> |
| <span class="identifier">p</span> <span class="special">&</span> <span class="identifier">q</span> <span class="special">=={</span> <span class="special">[</span><span class="number">2</span> <span class="number">3</span><span class="special">)-></span><span class="number">2</span> <span class="special">};</span> |
| </pre> |
| <p> |
| So an addition or aggregation of associated values is done like for <code class="computeroutput"><span class="keyword">operator</span> <span class="special">+</span></code> |
| but value pairs that have no common keys are not added to the result. |
| </p> |
| <p> |
| For a <code class="computeroutput"><span class="identifier">Quantifier</span></code> that is |
| a model of an <code class="computeroutput"><span class="identifier">InfiniteVector</span></code> |
| and which is therefore defined for every key value of the <code class="computeroutput"><span class="identifier">DomainT</span></code> |
| type, this definition of <code class="computeroutput"><span class="keyword">operator</span> |
| <span class="special">&</span></code> degenerates to the same sematics |
| that <code class="computeroutput"><span class="identifier">operaotor</span> <span class="special">+</span></code> |
| implements: |
| </p> |
| <pre class="programlisting"><span class="comment">//Pseudocode example for total Quantifiers p, q: |
| </span><span class="identifier">interval_map</span><span class="special"><</span><span class="keyword">int</span><span class="special">,</span><span class="keyword">int</span><span class="special">></span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">q</span><span class="special">;</span> |
| <span class="identifier">p</span> <span class="special">=</span> <span class="special">{[</span><span class="identifier">min</span> <span class="number">1</span><span class="special">)[</span><span class="number">1</span> <span class="number">3</span><span class="special">)[</span><span class="number">3</span> <span class="identifier">max</span><span class="special">]};</span> |
| <span class="special">-></span><span class="number">0</span> <span class="special">-></span><span class="number">1</span> <span class="special">-></span><span class="number">0</span> |
| <span class="identifier">q</span> <span class="special">=</span> <span class="special">{[</span><span class="identifier">min</span> <span class="number">2</span><span class="special">)[</span><span class="number">2</span> <span class="number">4</span><span class="special">)[</span><span class="number">4</span> <span class="identifier">max</span><span class="special">]};</span> |
| <span class="special">-></span><span class="number">0</span> <span class="special">-></span><span class="number">1</span> <span class="special">-></span><span class="number">0</span> |
| <span class="identifier">p</span><span class="special">&</span><span class="identifier">q</span> <span class="special">=={[</span><span class="identifier">min</span> <span class="number">1</span><span class="special">)[</span><span class="number">1</span> <span class="number">2</span><span class="special">)[</span><span class="number">2</span> <span class="number">3</span><span class="special">)[</span><span class="number">3</span> <span class="number">4</span><span class="special">)[</span><span class="number">4</span> <span class="identifier">max</span><span class="special">]};</span> |
| <span class="special">-></span><span class="number">0</span> <span class="special">-></span><span class="number">1</span> <span class="special">-></span><span class="number">2</span> <span class="special">-></span><span class="number">1</span> <span class="special">-></span><span class="number">0</span> |
| </pre> |
| <p> |
| </p> |
| <a name="boost_icl.semantics.quantifiers__maps_of_numbers.laws_for_quantifiers_of_unsigned_numbers"></a><h5> |
| <a name="id1094932"></a> |
| <a class="link" href="quantifiers__maps_of_numbers.html#boost_icl.semantics.quantifiers__maps_of_numbers.laws_for_quantifiers_of_unsigned_numbers">Laws |
| for Quantifiers of unsigned Numbers</a> |
| </h5> |
| <p> |
| The semantics of icl Maps of Numbers is different for unsigned or signed |
| numbers. So the sets of laws that are valid for Quantifiers will be different |
| depending on the instantiation of an unsigned or a signed number type as |
| <code class="computeroutput"><span class="identifier">CodomainT</span></code> parameter. |
| </p> |
| <p> |
| Again, we are presenting the investigated sets of laws, this time for <code class="computeroutput"><span class="identifier">Quantifier</span></code> types <code class="computeroutput"><span class="identifier">Q</span></code> |
| which are <code class="computeroutput"><a class="link" href="../../boost/icl/map.html" title="Class template map">icl::map</a></code><code class="computeroutput"><span class="special"><</span><span class="identifier">D</span><span class="special">,</span><span class="identifier">N</span><span class="special">,</span><span class="identifier">T</span><span class="special">></span></code>, <code class="computeroutput"><a class="link" href="../../boost/icl/interval_map.html" title="Class template interval_map">interval_map</a></code><code class="computeroutput"><span class="special"><</span><span class="identifier">D</span><span class="special">,</span><span class="identifier">N</span><span class="special">,</span><span class="identifier">T</span><span class="special">></span></code> and <code class="computeroutput"><a class="link" href="../../boost/icl/split_interval_map.html" title="Class template split_interval_map">split_interval_map</a></code><code class="computeroutput"><span class="special"><</span><span class="identifier">D</span><span class="special">,</span><span class="identifier">N</span><span class="special">,</span><span class="identifier">T</span><span class="special">></span></code> where <code class="computeroutput"><span class="identifier">CodomainT</span></code> |
| type <code class="computeroutput"><span class="identifier">N</span></code> is a <code class="computeroutput"><span class="identifier">Number</span></code> and <code class="computeroutput"><span class="identifier">Trait</span></code> |
| type <code class="computeroutput"><span class="identifier">T</span></code> is one of the icl's |
| map traits. |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">Associativity</span><span class="special"><</span><span class="identifier">Q</span><span class="special">,+,==</span> <span class="special">>:</span> <span class="identifier">Q</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">c</span><span class="special">)</span> <span class="special">==</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">c</span> |
| <span class="identifier">Neutrality</span><span class="special"><</span><span class="identifier">Q</span><span class="special">,+,==</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">Q</span> <span class="identifier">a</span><span class="special">;</span> <span class="identifier">a</span><span class="special">+</span><span class="identifier">Q</span><span class="special">()</span> <span class="special">==</span> <span class="identifier">a</span> |
| <span class="identifier">Commutativity</span><span class="special"><</span><span class="identifier">Q</span><span class="special">,+,==</span> <span class="special">>:</span> <span class="identifier">Q</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">Associativity</span><span class="special"><</span><span class="identifier">Q</span><span class="special">,&,==</span> <span class="special">>:</span> <span class="identifier">Q</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">c</span><span class="special">)</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">c</span> |
| <span class="identifier">Commutativity</span><span class="special"><</span><span class="identifier">Q</span><span class="special">,&,==</span> <span class="special">>:</span> <span class="identifier">Q</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">RightNeutrality</span><span class="special"><</span><span class="identifier">Q</span><span class="special">,-,==</span> <span class="special">>:</span> <span class="identifier">Q</span> <span class="identifier">a</span><span class="special">;</span> <span class="identifier">a</span><span class="special">-</span><span class="identifier">Q</span><span class="special">()</span> <span class="special">==</span> <span class="identifier">a</span> |
| <span class="identifier">Inversion</span><span class="special"><</span><span class="identifier">Q</span><span class="special">,-,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">Q</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="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="identifier">Q</span><span class="special">()</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| For an <code class="computeroutput"><span class="keyword">unsigned</span> <span class="identifier">Quantifier</span></code>, |
| an icl Map of <code class="computeroutput"><span class="keyword">unsigned</span> <span class="identifier">numbers</span></code>, |
| the same basic laws apply that are valid for <code class="computeroutput"><span class="identifier">Collectors</span></code>: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"> <span class="special">+</span> <span class="special">&</span> <span class="special">-</span> |
| <span class="identifier">Associativity</span> <span class="special">==</span> <span class="special">==</span> |
| <span class="identifier">Neutrality</span> <span class="special">==</span> <span class="special">==</span> |
| <span class="identifier">Commutativity</span> <span class="special">==</span> <span class="special">==</span> |
| <span class="identifier">Inversion</span> <span class="identifier">absorbs_identities</span> <span class="special">==</span> |
| <span class="identifier">enriches_identities</span> <span class="special">=</span><span class="identifier">d</span><span class="special">=</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| The subset of laws, that relates to <code class="computeroutput"><span class="keyword">operator</span> |
| <span class="special">+</span></code> and the neutral element <code class="computeroutput"><span class="identifier">Q</span><span class="special">()</span></code> is |
| that of a commutative monoid. This is the same concept, that applies for |
| the <code class="computeroutput"><span class="identifier">CodomainT</span></code> type. This |
| gives rise to the assumption that an icl <code class="computeroutput"><span class="identifier">Map</span></code> |
| over a <code class="computeroutput"><span class="identifier">CommutativeModoid</span></code> |
| is again a <code class="computeroutput"><span class="identifier">CommutativeModoid</span></code>. |
| </p> |
| <p> |
| Other laws that were valid for <code class="computeroutput"><span class="identifier">Collectors</span></code> |
| are not valid for an <code class="computeroutput"><span class="keyword">unsigned</span> <span class="identifier">Quantifier</span></code>. |
| </p> |
| <a name="boost_icl.semantics.quantifiers__maps_of_numbers.laws_for_quantifiers_of_signed_numbers"></a><h5> |
| <a name="id1096964"></a> |
| <a class="link" href="quantifiers__maps_of_numbers.html#boost_icl.semantics.quantifiers__maps_of_numbers.laws_for_quantifiers_of_signed_numbers">Laws |
| for Quantifiers of signed Numbers</a> |
| </h5> |
| <p> |
| For <code class="computeroutput"><span class="identifier">Quantifiers</span></code> of signed |
| numbers, or <code class="computeroutput"><span class="keyword">signed</span> <span class="identifier">Quantifiers</span></code>, |
| the pattern of valid laws is somewhat different: |
| </p> |
| <pre class="programlisting"> <span class="special">+</span> <span class="special">&</span> <span class="special">-</span> |
| <span class="identifier">Associativity</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> |
| <span class="identifier">Neutrality</span> <span class="special">==</span> <span class="special">==</span> |
| <span class="identifier">Commutativity</span> <span class="special">==</span> <span class="special">==</span> |
| <span class="identifier">Inversion</span> <span class="identifier">absorbs_identities</span> <span class="special">==</span> |
| <span class="identifier">enriches_identities</span> <span class="special">=</span><span class="identifier">d</span><span class="special">=</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| The differences are tagged as <code class="computeroutput"><span class="special">=</span><span class="identifier">v</span><span class="special">=</span></code> indicating, |
| that the associativity law is not uniquely valid for a single equality relation |
| <code class="computeroutput"><span class="special">==</span></code> as this was the case for |
| <code class="computeroutput"><span class="identifier">Collector</span></code> and <code class="computeroutput"><span class="keyword">unsigned</span> <span class="identifier">Quntifier</span></code> |
| maps. |
| </p> |
| <p> |
| The differences are these: |
| </p> |
| <pre class="programlisting"> <span class="special">+</span> |
| <span class="identifier">Associativity</span> <span class="identifier">icl</span><span class="special">::</span><span class="identifier">map</span> <span class="special">==</span> |
| <span class="identifier">interval_map</span> <span class="special">==</span> |
| <span class="identifier">split_interval_map</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> |
| </pre> |
| <p> |
| For <code class="computeroutput"><span class="keyword">operator</span> <span class="special">+</span></code> |
| the associativity on <code class="computeroutput"><a class="link" href="../../boost/icl/split_interval_map.html" title="Class template split_interval_map">split_interval_maps</a></code> |
| is only valid with element equality <code class="computeroutput"><span class="special">=</span><span class="identifier">e</span><span class="special">=</span></code>, which |
| is not a big constrained, because only element equality is required. |
| </p> |
| <p> |
| For <code class="computeroutput"><span class="keyword">operator</span> <span class="special">&</span></code> |
| the associativity is broken for all maps that are partial absorbers. For |
| total absorbers associativity is valid for element equality. All maps having |
| the <span class="emphasis"><em>identity enricher</em></span> Trait are associative wrt. lexicographical |
| equality <code class="computeroutput"><span class="special">==</span></code>. |
| </p> |
| <pre class="programlisting"><span class="identifier">Associativity</span> <span class="special">&</span> |
| <span class="identifier">absorbs_identities</span> <span class="special">&&</span> <span class="special">!</span><span class="identifier">is_total</span> <span class="keyword">false</span> |
| <span class="identifier">absorbs_identities</span> <span class="special">&&</span> <span class="identifier">is_total</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> |
| <span class="identifier">enriches_identities</span> <span class="special">==</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| Note, that all laws that establish a commutative monoid for <code class="computeroutput"><span class="keyword">operator</span> <span class="special">+</span></code> |
| and identity element <code class="computeroutput"><span class="identifier">Q</span><span class="special">()</span></code> |
| are valid for <code class="computeroutput"><span class="keyword">signed</span> <span class="identifier">Quantifiers</span></code>. |
| In addition symmetric difference that does not hold for <code class="computeroutput"><span class="keyword">unsigned</span> |
| <span class="identifier">Qunatifiers</span></code> is valid for <code class="computeroutput"><span class="keyword">signed</span> <span class="identifier">Qunatifiers</span></code>. |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">SymmetricDifference</span><span class="special"><</span><span class="identifier">Q</span><span class="special">,==</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">Q</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="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">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">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> |
| </pre> |
| <p> |
| For a <code class="computeroutput"><span class="keyword">signed</span> <span class="identifier">TotalQuantifier</span></code> |
| <code class="computeroutput"><span class="identifier">Qt</span></code> symmetrical difference |
| degenerates to a trivial form since <code class="computeroutput"><span class="keyword">operator</span> |
| <span class="special">&</span></code> and <code class="computeroutput"><span class="keyword">operator</span> |
| <span class="special">+</span></code> become identical |
| </p> |
| <pre class="programlisting"><span class="identifier">SymmetricDifference</span><span class="special"><</span><span class="identifier">Qt</span><span class="special">,==</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">Qt</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="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">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">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="special">==</span> <span class="identifier">Qt</span><span class="special">()</span> |
| </pre> |
| <p> |
| </p> |
| <a name="boost_icl.semantics.quantifiers__maps_of_numbers.existence_of_an_inverse"></a><h6> |
| <a name="id1097785"></a> |
| <a class="link" href="quantifiers__maps_of_numbers.html#boost_icl.semantics.quantifiers__maps_of_numbers.existence_of_an_inverse">Existence |
| of an Inverse</a> |
| </h6> |
| <p> |
| By now <code class="computeroutput"><span class="keyword">signed</span> <span class="identifier">Quantifiers</span></code> |
| <code class="computeroutput"><span class="identifier">Q</span></code> are commutative monoids |
| with respect to the <code class="computeroutput"><span class="keyword">operator</span> <span class="special">+</span></code> and the neutral element <code class="computeroutput"><span class="identifier">Q</span><span class="special">()</span></code>. If the Quantifier's <code class="computeroutput"><span class="identifier">CodomainT</span></code> |
| type has an <span class="emphasis"><em>inverse element</em></span> like e.g. <code class="computeroutput"><span class="keyword">signed</span> <span class="identifier">numbers</span></code> |
| do, the <code class="computeroutput"><span class="identifier">CodomainT</span></code> type is |
| a <span class="emphasis"><em><span class="bold"><strong>commutative</strong></span></em></span> or <span class="emphasis"><em><span class="bold"><strong>abelian group</strong></span></em></span>. In this case a <code class="computeroutput"><span class="keyword">signed</span> <span class="identifier">Quantifier</span></code> |
| that is also <span class="emphasis"><em><span class="bold"><strong>total</strong></span></em></span> |
| has an <span class="emphasis"><em><span class="bold"><strong>inverse</strong></span></em></span> and |
| the following law holds: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">InverseElement</span><span class="special"><</span><span class="identifier">Qt</span><span class="special">,==</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">Qt</span> <span class="identifier">a</span><span class="special">;</span> <span class="special">(</span><span class="number">0</span> <span class="special">-</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="number">0</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| Which means that each <code class="computeroutput"><span class="identifier">TotalQuantifier</span></code> |
| over an abelian group is an abelian group itself. |
| </p> |
| <p> |
| This also implies that a <code class="computeroutput"><span class="identifier">Quantifier</span></code> |
| of <code class="computeroutput"><span class="identifier">Quantifiers</span></code> is again a |
| <code class="computeroutput"><span class="identifier">Quantifiers</span></code> and a <code class="computeroutput"><span class="identifier">TotalQuantifier</span></code> of <code class="computeroutput"><span class="identifier">TotalQuantifiers</span></code> |
| is also a <code class="computeroutput"><span class="identifier">TotalQuantifier</span></code>. |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">TotalQuantifiers</span></code> resemble |
| the notion of a vector space partially. The concept could be completed to |
| a vector space, if a scalar multiplication was added. |
| </p> |
| </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> |
| </tr></table> |
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