boost/libs/icl/doc/semantics.qbk

[/
    Copyright (c) 2008-2009 Joachim Faulhaber

    Distributed under the Boost Software License, Version 1.0.
    (See accompanying file LICENSE_1_0.txt or copy at
    http://www.boost.org/LICENSE_1_0.txt)
]

[section Semantics]

["Beauty is the ultimate defense against complexity] -- [/David Gelernter]
[@http://en.wikipedia.org/wiki/David_Gelernter David Gelernter]

In the *icl* we follow the notion, that the semantics of a ['*concept*] or
['*abstract data type*] can be expressed by ['*laws*]. 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.
``
Commutativity<T,+>:
T a, b; a + b == b + a;
``
This can of course be coded as a proper c++ class template which has 
been done for the validation of the *icl*. For sake of simplicity 
we will use pseudocode here.

The laws that describe the semantics of the *icl's* class templates
were validated using the Law based Test Automaton ['*LaBatea*],
a tool that generates instances for the law's variables and then
tests it's validity.  
Since the *icl* deals with sets, maps and relations, that are
well known objects from mathematics, the laws that we are
using are mostly /recycled/ ones. Also some of those laws are grouped
in notions like e.g. /orderings/ or /algebras/.

[section Orderings and Equivalences]

[h4 Lexicographical Ordering and Equality]

On all set and map containers of the icl, there is an `operator <`
that implements a
[@http://www.sgi.com/tech/stl/StrictWeakOrdering.html strict weak ordering].
[/ (see also [@http://en.wikipedia.org/wiki/Strict_weak_ordering here]).]
The semantics of `operator <` is the same as for an stl's
[@http://www.sgi.com/tech/stl/SortedAssociativeContainer.html SortedAssociativeContainer], 
specifically
[@http://www.sgi.com/tech/stl/set.html stl::set] 
and 
[@http://www.sgi.com/tech/stl/map.html stl::map]:
``
Irreflexivity<T,< > : T a;     !(a<a)
Asymmetry<T,< >     : T a,b;   a<b implies !(b<a)
Transitivity<T,< >  : T a,b,c; a<b && b<c implies a<c
``

`Operator <` depends on the icl::container's template parameter
`Compare` that implements a ['strict weak ordering] for the container's
`domain_type`.
For a given `Compare` ordering, `operator <` implements a 
lexicographical comparison on icl::containers, that uses the 
`Compare` order to establish a unique sequence of values in 
the container. 

The induced equivalence of `operator <` is 
lexicographical equality 
which is implemented as `operator ==`. 
``
//equivalence induced by strict weak ordering <
!(a<b) && !(b<a) implies a == b;
``
Again this
follows the semantics of the *stl*.
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 __bi_iterative__ aspect. Of all the different sequences that are valid 
for unordered sets and maps, one such sequence is
selected by the `Compare` order of elements. Based on
this selection a unique iteration is possible.

[h4 Subset Ordering and Element Equality]

On the __conceptual__ aspect only membership of elements
matters, not their sequence. So there are functions
`contained_in` and `element_equal` that implement
the subset relation and the equality on elements.
Yet, `contained_in` and `is_element_equal` functions are not
really working on the level of elements. They also
work on the basis of the containers templates
`Compare` parameter. In practical terms we need to
distinguish between lexicographical equality 
`operator ==` and equality of elements `is_element_equal`,
if we work with interval splitting interval containers:
``
split_interval_set<time> w1, w2; //Pseudocode
w1 = {[Mon       ..       Sun)}; //split_interval_set containing a week
w2 = {[Mon .. Fri)[Sat .. Sun)}; //Same week split in work and week end parts.
w1 == w2;                        //false: Different segmentation
is_element_equal(w1,w2);         //true:  Same elements contained  
``   

For a constant `Compare` order on key elements,
member function `contained_in` that is defined for all 
icl::containers implements a
[@http://en.wikipedia.org/wiki/Partially_ordered_set partial order]
on icl::containers.

``
with <= for contained_in,
    =e= for is_element_equal:
Reflexivity<T,<= >     : T a;     a <= a
Antisymmetry<T,<=,=e=> : T a,b;   a <= b && b <= a implies a =e= b 
Transitivity<T,<= >    : T a,b,c; a <= b && b <= c implies a <= c
``

The induced equivalence is the equality of elements that
is implemented via function `is_element_equal`.
``
//equivalence induced by the partial ordering contained_in on icl::container a,b
a.contained_in(b) && b.contained_in(a) implies is_element_equal(a, b);
``

[endsect][/ Orderings and Equivalences]

[section Sets]

For all set types `S` that are models concept `Set`  
(__icl_set__, __itv_set__, __sep_itv_set__ and __spl_itv_set__)
most of the well known mathematical 
[@http://en.wikipedia.org/wiki/Algebra_of_sets laws on sets]
were successfully checked via LaBatea. The next tables 
are giving an overview over the checked laws ordered by
operations. If possible, the laws are formulated with
the stronger lexicographical equality (`operator ==`)
which implies the law's validity for the weaker
element equality `is_element_equal`. Throughout this
chapter we will denote element equality as `=e=` instead
of `is_element_equal` where a short notation is advantageous.

[h5 Laws on set union]

For the operation ['*set union*] available as 
`operator +, +=, |, |=` and the neutral element
`identity_element<S>::value()` which is the empty set `S()`
these laws hold:
``
Associativity<S,+,== >: S a,b,c; a+(b+c) == (a+b)+c
Neutrality<S,+,== >   : S a;       a+S() == a
Commutativity<S,+,== >: S a,b;       a+b == b+a
``

[h5 Laws on set intersection]

For the operation ['*set intersection*] available as 
`operator &, &=` these laws were validated:

``
Associativity<S,&,== >: S a,b,c; a&(b&c) == (a&b)&c
Commutativity<S,&,== >: S a,b;       a&b == b&a
``

[/ FYI
Neutrality has *not* been validated to avoid 
additional requirements on the sets template
parameters.] 

[h5 Laws on set difference]

For set difference there are only these laws. It is
not associative and not commutative. It's neutrality
is non symmetrical.

``
RightNeutrality<S,-,== > : S a;   a-S() == a
Inversion<S,-,== >:        S a;   a - a == S()
``

Summarized in the next table are laws that use `+`, `&` and `-`
as a single operation. For all validated laws,
the left and right hand sides of the equations
are lexicographically equal, as denoted by `==` in the cells
of the table.

``
                 +    &   -
Associativity    ==   == 
Neutrality       ==       ==
Commutativity    ==   ==
Inversion                 ==
``

[h5 Distributivity Laws]

Laws, like distributivity, that use more than one operation can
sometimes be instantiated for different sequences of operators
as can be seen below. In the two instantiations of the distributivity
laws operators `+` and `&` are swapped. So we can have small operator
signatures like `+,&` and `&,+` to describe such instantiations,
which will be used below.
Not all instances of distributivity laws hold for lexicographical equality.
Therefore they are denoted using a /variable/ equality `=v=` below.

``
     Distributivity<S,+,&,=v= > : S a,b,c; a + (b & c) =v= (a + b) & (a + c)
     Distributivity<S,&,+,=v= > : S a,b,c; a & (b + c) =v= (a & b) + (a & c)
RightDistributivity<S,+,-,=v= > : S a,b,c; (a + b) - c =v= (a - c) + (b - c)
RightDistributivity<S,&,-,=v= > : S a,b,c; (a & b) - c =v= (a - c) & (b - c)
``

The next table shows the relationship between 
law instances, 
[link boost_icl.introduction.interval_combining_styles interval combining style]
and the 
used equality relation.

``
                                  +,&    &,+
     Distributivity  joining      ==     ==
                     separating   ==     ==
                     splitting    =e=    =e=
                     
                                  +,-    &,-
RightDistributivity  joining      ==     ==
                     separating   ==     ==
                     splitting    =e=    ==
``

The table gives an overview over 12 instantiations of
the four distributivity laws and shows the equalities 
which the instantiations holds for. For instance
`RightDistributivity` with operator signature `+,-`
instantiated for __spl_itv_sets__ holds only for
element equality (denoted as `=e=`):
``
RightDistributivity<S,+,-,=e= > : S a,b,c; (a + b) - c =e= (a - c) + (b - c)
``
The remaining five instantiations of `RightDistributivity`
are valid for lexicographical equality (demoted as `==`) as well.

[link boost_icl.introduction.interval_combining_styles Interval combining styles]
correspond to containers according to
``
style       set
joining     interval_set
separating  separate_interval_set
splitting   split_interval_set
``


Finally there are two laws that combine all three major set operations:
De Mogans Law and Symmetric Difference.

[h5 DeMorgan's Law]

De Morgans Law is better known in an incarnation where the unary
complement operation `~` is used. `~(a+b) == ~a * ~b`. The version
below is an adaption for the binary set difference `-`, which is 
also called ['*relative complement*].
``
DeMorgan<S,+,&,=v= > : S a,b,c; a - (b + c) =v= (a - b) & (a - c)
DeMorgan<S,&,+,=v= > : S a,b,c; a - (b & c) =v= (a - b) + (a - c)
``

``
                         +,&     &,+
DeMorgan  joining        ==      ==
          separating     ==      =e=
          splitting      ==      =e=
``

Again not all law instances are valid for lexicographical equality.
The second instantiations only holds for element equality, if
the interval sets are non joining.

[h5 Symmetric Difference]

``
SymmetricDifference<S,== > : S a,b,c; (a + b) - (a & b) == (a - b) + (b - a)
``

Finally Symmetric Difference holds for all of icl set types and
lexicographical equality.

[/ pushout laws]

[endsect][/ Sets]

[section Maps]

By definition a map is set of pairs. So we would expect maps to
obey the same laws that are valid for sets. Yet
the semantics of the *icl's* maps may be a different one, because
of it's aggregating facilities, where the aggregating combiner
operations are passed to combine the map's associated values.
It turns out, that the aggregation on overlap principle
induces semantic properties to icl maps in such a way,
that the set of equations that are valid will depend on
the semantics of the type `CodomainT` of the map's associated
values.

This is less magical as it might seem at first glance.
If, for instance, we instantiate an __itv_map__ to 
collect and concatenate
`std::strings` associated to intervals,

``    
interval_map<int,std::string> cat_map;
cat_map += make_pair(interval<int>::rightopen(1,5),std::string("Hello"));
cat_map += make_pair(interval<int>::rightopen(3,7),std::string(" World"));
cout << "cat_map: " << cat_map << endl;
``

we won't be able to apply `operator -=` 
``
// This will not compile because string::operator -= is missing.
cat_map -= make_pair(interval<int>::rightopen(3,7),std::string(" World"));
``
because, as std::sting does not implement `-=` itself, this won't compile.
So all *laws*, that rely on `operator -=` or `-` not only will not be valid they 
can not even be stated.
This reduces the set of laws that can be valid for a richer `CodomainT`
type to a smaller set of laws and thus to a less restricted semantics. 

Currently we have investigated and validated two major instantiations 
of icl::Maps,

* ['*Maps of Sets*] that will be called ['*Collectors*]  and
* ['*Maps of Numbers*] which will be called ['*Quantifiers*]

both of which seem to have many interesting use cases for practical
applications. The semantics associated with the term /Numbers/ 
is a 
[@http://en.wikipedia.org/wiki/Monoid commutative monoid] 
for unsigned numbers
and a 
[@http://en.wikipedia.org/wiki/Abelian_group commutative or abelian group]
for signed numbers.
From a practical point of view we can think
of numbers as counting or quantifying the key values 
of the map.

Icl ['*Maps of Sets*] or ['*Collectors*]
are models of concept `Set`. This implies that all laws that have been
stated as a semantics for `icl::Sets` in the previous chapter also hold for
`Maps of Sets`.
Icl ['*Maps of Numbers*] or ['*Quantifiers*] on the contrary are not models 
of concept `Set`.
But there is a substantial intersection of laws that apply both for
`Collectors` and `Quantifiers`.

[table
[[Kind of Map]    [Alias]     [Behavior]                             ]
[[Maps of Sets]   [Collector] [Collects items *for* key values]      ]
[[Maps of Numbers][Quantifier][Counts or quantifies *the* key values]]
]

In the next two sections the law based semantics of ['*Collectors*]
and ['*Quantifiers*] will be described in more detail.

[endsect][/ Maps]

[section Collectors: Maps of Sets]

Icl `Collectors`, behave like `Sets`. 
This can be understood easily, if we consider, that
every map of sets can be transformed to an equivalent
set of pairs.
For instance in the pseudocode below map `m`
``
icl::map<int,set<int> >  m = {(1->{1,2}), (2->{1})}; 
``
is equivalent to set `s`
``
icl::set<pair<int,int> > s = {(1,1),(1,2),   //representing 1->{1,2}
                              (2,1)       }; //representing 2->{1}
`` 

Also the results of add, subtract and other operations on `map m` and `set s`
preserves the equivalence of the containers ['*almost*] perfectly:
``
m += (1,3); 
m == {(1->{1,2,3}), (2->{1})}; //aggregated on collision of key value 1
s += (1,3);
s == {(1,1),(1,2),(1,3),   //representing 1->{1,2,3}
      (2,1)             }; //representing 2->{1}
``

The equivalence of `m` and `s` is only violated if an 
empty set occurres in `m` by subtraction of a value pair:
``
m -= (2,1); 
m == {(1->{1,2,3}), (2->{})}; //aggregated on collision of key value 2
s -= (2,1);
s == {(1,1),(1,2),(1,3)   //representing 1->{1,2,3}
                       }; //2->{} is not represented in s
``

This problem can be dealt with in two ways. 

# Deleting value pairs form the Collector, if it's associated value 
  becomes a neutral value or `identity_element`.
# Using a different equality, called distinct equality in the laws
  to validate. Distinct equality only 
  accounts for value pairs that that carry values unequal to the `identity_element`. 

Solution (1) led to the introduction of map traits, particularly trait
['*partial_absorber*], which is the default setting in all icl's map
templates.

Solution (2), is applied to check the semantics of icl::Maps for the
partial_enricher trait that does not delete value pairs that carry
identity elements. Distinct equality is implemented by a non member function
called `is_distinct_equal`. Throughout this chapter
distinct equality in pseudocode and law denotations is denoted
as `=d=` operator.

The validity of the sets of laws that make up `Set` semantics
should now be quite evident. So the following text shows the
laws that are validated for all `Collector` types `C`. Which are
__icl_map__`<D,S,T>`, __itv_map__`<D,S,T>` and __spl_itv_map__`<D,S,T>`
where `CodomainT` type `S` is a model of `Set` and `Trait` type `T` is either 
__pabsorber__ or __penricher__.


[h5 Laws on set union, set intersection and set difference]

``
Associativity<C,+,== >: C a,b,c; a+(b+c) == (a+b)+c
Neutrality<C,+,== >   : C a;       a+C() == a
Commutativity<C,+,== >: C a,b;       a+b == b+a

Associativity<C,&,== >: C a,b,c; a&(b&c) ==(a&b)&c
Commutativity<C,&,== >: C a,b;       a&b == b&a

RightNeutrality<C,-,== >: C a;   a-C() ==  a
Inversion<C,-,=v= >     : C a;   a - a =v= C()
``

All the fundamental laws could be validated for all 
icl Maps in their instantiation as Maps of Sets or Collectors.
As expected, Inversion only holds for distinct equality,
if the map is not a `partial_absorber`.

``
                             +    &    -
Associativity                ==   == 
Neutrality                   ==        ==
Commutativity                ==   ==
Inversion partial_absorber             ==
          partial_enricher             =d=
``

[h5 Distributivity Laws]

``
     Distributivity<C,+,&,=v= > : C a,b,c; a + (b & c) =v= (a + b) & (a + c)
     Distributivity<C,&,+,=v= > : C a,b,c; a & (b + c) =v= (a & b) + (a & c)
RightDistributivity<C,+,-,=v= > : C a,b,c; (a + b) - c =v= (a - c) + (b - c)
RightDistributivity<C,&,-,=v= > : C a,b,c; (a & b) - c =v= (a - c) & (b - c)
``

Results for the distributivity laws are almost identical to 
the validation of sets except that
for a `partial_enricher map` the law `(a & b) - c == (a - c) & (b - c)`
holds for lexicographical equality.

``
                                                   +,&    &,+
     Distributivity  joining                       ==     ==
                     splitting   partial_absorber  =e=    =e=
                                 partial_enricher  =e=    ==                     
                     
                                                   +,-    &,-
RightDistributivity  joining                       ==     ==
                     splitting                     =e=    ==
``

[h5 DeMorgan's Law and Symmetric Difference]

``
DeMorgan<C,+,&,=v= > : C a,b,c; a - (b + c) =v= (a - b) & (a - c)
DeMorgan<C,&,+,=v= > : C a,b,c; a - (b & c) =v= (a - b) + (a - c)
``

``
                         +,&     &,+
DeMorgan  joining        ==      ==
          splitting      ==      =e=
``

``
SymmetricDifference<C,== > : C a,b,c; (a + b) - (a * b) == (a - b) + (b - a)
``

Reviewing the validity tables above shows, that the sets of valid laws for
`icl Sets` and `icl Maps of Sets` that are /identity absorbing/ are exactly the same.
As expected, only for Maps of Sets that represent empty sets as associated values,
called /identity enrichers/, there are marginal semantic differences.

[endsect][/ Collectors]

[section Quantifiers: Maps of Numbers]

[h5 Subtraction on Quantifiers]

With `Sets` and `Collectors` the semantics of  
`operator -` is
that of /set difference/ which means, that you can
only subtract what has been put into the container before.
With `Quantifiers` that ['*count*] or ['*quantify*]
their key values in some way, 
the semantics of `operator -` may be different.

The question is how subtraction should be defined here?
``
//Pseudocode:
icl::map<int,some_number> q = {(1->1)};
q -= (2->1);
`` 
If type `some_number` is `unsigned` a /set difference/ kind of
subtraction make sense 
``
icl::map<int,some_number> q = {(1->1)};
q -= (2->1);   // key 2 is not in the map so  
q == {(1->1)}; // q is unchanged by 'aggregate on collision'
`` 
If `some_number` is a `signed` numerical type
the result can also be this
``
icl::map<int,some_number> q = {(1->1)};
q -= (2->1);             // subtracting works like  
q == {(1->1), (2-> -1)}; // adding the inverse element
`` 
As commented in the example, subtraction of a key value
pair `(k,v)` can obviously be defined as adding the ['*inverse element*]
for that key `(k,-v)`, if the key is not yet stored in the map.

[h4 Partial and Total Quantifiers and Infinite Vectors]

Another concept, that we can think of, is that in a `Quantifier` 
every `key_value` is initially quantified `0`-times, where `0` stands
for the neutral element of the numeric `CodomainT` type.
Such a `Quantifier` would be totally defined on all values of
it's `DomainT` type and can be
conceived as an `InfiniteVector`. 

To create an infinite vector
that is totally defined on it's domain we can set
the map's `Trait` parameter to the value __tabsorber__.
The __tabsorber__ trait fits specifically well with
a `Quantifier` if it's `CodomainT` has an inverse 
element, like all signed numerical type have.
As we can see later in this section this kind of 
a total `Quantifier` has the basic properties that
elements of a 
[@http://en.wikipedia.org/wiki/Vector_space vector space] 
do provide.


[h5 Intersection on Quantifiers]

Another difference between `Collectors` and `Quantifiers`
is the semantics of `operator &`, that has the meaning of
set intersection for `Collectors`.

For the /aggregate on overlap principle/ the operation `&`
has to be passed to combine associated values on overlap
of intervals or collision of keys. This can not be done
for `Quantifiers`, since numeric types do not implement
intersection.

For `CodomainT` types that are not models of `Sets` 
`operator & ` is defined as ['aggregation on the intersection
of the domains]. Instead of the `codomain_intersect` functor
`codomain_combine` is used as aggregation operation: 
``
//Pseudocode example for partial Quantifiers p, q:
interval_map<int,int> p, q;
p     = {[1     3)->1   };
q     = {   ([2    4)->1};
p & q =={    [2 3)->2   };
``
So an addition or aggregation of associated values is
done like for `operator +` but value pairs that have
no common keys are not added to the result.

For a `Quantifier` that is a model of an `InfiniteVector`
and which is therefore defined for every key value of
the `DomainT` type, this definition of `operator &`
degenerates to the same sematics that `operaotor +`
implements:
``
//Pseudocode example for total Quantifiers p, q:
interval_map<int,int> p, q;
p   = {[min   1)[1      3)[3         max]};
          ->0      ->1         ->0
q   = {[min        2)[2      4)[4    max]};
            ->0         ->1       ->0
p&q =={[min   1)[1  2)[2 3)[3 4)[4   max]};
          ->0    ->1   ->2  ->1    ->0
``

[h4 Laws for Quantifiers of unsigned Numbers]

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 `CodomainT` parameter.

Again, we are presenting the investigated sets of laws, this time for
`Quantifier` types `Q` which are
__icl_map__`<D,N,T>`, __itv_map__`<D,N,T>` and __spl_itv_map__`<D,N,T>`
where `CodomainT` type `N` is a `Number` and `Trait` type `T` is one of
the icl's map traits. 

``
Associativity<Q,+,== >: Q a,b,c; a+(b+c) == (a+b)+c
Neutrality<Q,+,== >   : Q a;       a+Q() == a
Commutativity<Q,+,== >: Q a,b;       a+b == b+a

Associativity<Q,&,== >: Q a,b,c; a&(b&c) ==(a&b)&c
Commutativity<Q,&,== >: Q a,b;       a&b == b&a

RightNeutrality<Q,-,== >: Q a;   a-Q() ==  a
Inversion<Q,-,=v= >     : Q a;   a - a =v= Q()
``

For an `unsigned Quantifier`, an icl Map of `unsigned numbers`,
the same basic laws apply that are
valid for `Collectors`:

``
                               +    &    -
Associativity                  ==   == 
Neutrality                     ==        ==
Commutativity                  ==   ==
Inversion absorbs_identities             ==
          enriches_identities            =d=
``

The subset of laws, that relates to `operator +` and the neutral
element `Q()` is that of a commutative monoid. This is the same
concept, that applies for the `CodomainT` type. This gives 
rise to the assumption that an icl `Map` over a `CommutativeModoid`
is again a `CommutativeModoid`.

Other laws that were valid for `Collectors` are not valid
for an `unsigned Quantifier`. 

 
[h4 Laws for Quantifiers of signed Numbers]

For `Quantifiers` of signed numbers, or 
`signed Quantifiers`, the pattern of valid
laws is somewhat different:
``
                               +    &    -
Associativity                  =v=  =v=  
Neutrality                     ==        ==
Commutativity                  ==   ==
Inversion absorbs_identities             ==
          enriches_identities            =d=
``

The differences are tagged as `=v=` indicating, that
the associativity law is not uniquely valid for a single
equality relation `==` as this was the case for
`Collector` and `unsigned Quntifier` maps.

The differences are these:
``
                                   +  
Associativity         icl::map     ==   
                  interval_map     ==
            split_interval_map     =e=
``
For `operator +` the associativity on __spl_itv_maps__ is only valid
with element equality `=e=`, which is not a big constrained, because
only element equality is required.

For `operator &` the associativity is broken for all maps
that are partial absorbers. For total absorbers associativity
is valid for element equality. All maps having the /identity enricher/
Trait are associative wrt. lexicographical equality `==`.
``                               
Associativity                        &
   absorbs_identities && !is_total   false
   absorbs_identities &&  is_total   =e=
   enriches_identities               ==                  
``

Note, that all laws that establish a commutative
monoid for `operator +` and identity element `Q()` are valid
for `signed Quantifiers`.
In addition symmetric difference that does not
hold for `unsigned Qunatifiers` is valid
for `signed Qunatifiers`.

``
SymmetricDifference<Q,== > : Q a,b,c; (a + b) - (a & b) == (a - b) + (b - a)
``
For a `signed TotalQuantifier` `Qt` symmetrical difference degenerates to 
a trivial form since `operator &` and `operator +` become identical
``
SymmetricDifference<Qt,== > : Qt a,b,c; (a + b) - (a + b) == (a - b) + (b - a) == Qt()
``

[h5 Existence of an Inverse]

By now `signed Quantifiers` `Q` are 
commutative monoids 
with respect to the
`operator +` and the neutral element `Q()`. 
If the Quantifier's `CodomainT` type has an /inverse element/ 
like e.g. `signed numbers` do,
the `CodomainT` type is a 
['*commutative*] or ['*abelian group*].
In this case a `signed Quantifier` that is also ['*total*] 
has an ['*inverse*] and the following law holds:

``
InverseElement<Qt,== > : Qt a; (0 - a) + a == 0
``

Which means that each `TotalQuantifier` over an
abelian group is an 
abelian group 
itself.

This also implies that a `Quantifier` of `Quantifiers` 
is again a `Quantifiers` and a
`TotalQuantifier` of `TotalQuantifiers`
is also a `TotalQuantifier`. 

`TotalQuantifiers` resemble the notion of a
vector space partially.
The concept could be completed to a vector space, 
if a scalar multiplication was added. 
[endsect][/ Quantifiers]


[section Concept Induction]

Obviously we can observe the induction of semantics
from the `CodomainT` parameter into the instantiations
of icl maps.

[table
[[]                          [is model of]        [if]                         [example]]
[[`Map<D,Monoid>`]           [`Modoid`]           []                           [`interval_map<int,string>`]]
[[`Map<D,Set,Trait>`]        [`Set`]              [`Trait::absorbs_identities`][`interval_map<int,std::set<int> >`]]
[[`Map<D,CommutativeMonoid>`][`CommutativeMonoid`][]                           [`interval_map<int,unsigned int>`]]
[[`Map<D,CommutativeGroup>`] [`CommutativeGroup`] [`Trait::is_total`]    [`interval_map<int,int,total_absorber>`]]
]

[endsect][/ Concept Induction]

[endsect][/ Semantics]