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“Beauty is the ultimate defense against complexity” -- 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.
On all set and map containers of the icl, there is an operator
< that implements a strict
weak ordering. The semantics of operator
< is the same as for an stl's
SortedAssociativeContainer,
specifically stl::set
and 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
segmental 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.
On the fundamental 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 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);