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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.
identity_element
.
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>
,
interval_map
<D,S,T>
and split_interval_map
<D,S,T>
where CodomainT
type S
is a model of Set
and Trait
type T
is either partial_absorber
or partial_enricher
.
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=
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= ==
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.