sofs - Functions for manipulating sets of sets.
Please see following description for synopsis
sofs(3) Erlang Module Definition sofs(3)
NAME
sofs - Functions for manipulating sets of sets.
DESCRIPTION
This module provides operations on finite sets and relations repre-
sented as sets. Intuitively, a set is a collection of elements; every
element belongs to the set, and the set contains every element.
Given a set A and a sentence S(x), where x is a free variable, a new
set B whose elements are exactly those elements of A for which S(x)
holds can be formed, this is denoted B = {x in A : S(x)}. Sentences are
expressed using the logical operators "for some" (or "there exists"),
"for all", "and", "or", "not". If the existence of a set containing all
the specified elements is known (as is always the case in this module),
this is denoted B = {x : S(x)}.
* The unordered set containing the elements a, b, and c is denoted
{a, b, c}. This notation is not to be confused with tuples.
The ordered pair of a and b, with first coordinate a and second
coordinate b, is denoted (a, b). An ordered pair is an ordered set
of two elements. In this module, ordered sets can contain one, two,
or more elements, and parentheses are used to enclose the elements.
Unordered sets and ordered sets are orthogonal, again in this mod-
ule; there is no unordered set equal to any ordered set.
* The empty set contains no elements.
Set A is equal to set B if they contain the same elements, which is
denoted A = B. Two ordered sets are equal if they contain the same
number of elements and have equal elements at each coordinate.
Set B is a subset of set A if A contains all elements that B con-
tains.
The union of two sets A and B is the smallest set that contains all
elements of A and all elements of B.
The intersection of two sets A and B is the set that contains all
elements of A that belong to B.
Two sets are disjoint if their intersection is the empty set.
The difference of two sets A and B is the set that contains all
elements of A that do not belong to B.
The symmetric difference of two sets is the set that contains those
element that belong to either of the two sets, but not both.
The union of a collection of sets is the smallest set that contains
all the elements that belong to at least one set of the collection.
The intersection of a non-empty collection of sets is the set that
contains all elements that belong to every set of the collection.
* The Cartesian product of two sets X and Y, denoted X x Y, is the
set {a : a = (x, y) for some x in X and for some y in Y}.
A relation is a subset of X x Y. Let R be a relation. The fact that
(x, y) belongs to R is written as x R y. As relations are sets, the
definitions of the last item (subset, union, and so on) apply to
relations as well.
The domain of R is the set {x : x R y for some y in Y}.
The range of R is the set {y : x R y for some x in X}.
The converse of R is the set {a : a = (y, x) for some (x, y) in R}.
If A is a subset of X, the image of A under R is the set {y : x R y
for some x in A}. If B is a subset of Y, the inverse image of B is
the set {x : x R y for some y in B}.
If R is a relation from X to Y, and S is a relation from Y to Z,
the relative product of R and S is the relation T from X to Z
defined so that x T z if and only if there exists an element y in Y
such that x R y and y S z.
The restriction of R to A is the set S defined so that x S y if and
only if there exists an element x in A such that x R y.
If S is a restriction of R to A, then R is an extension of S to X.
If X = Y, then R is called a relation in X.
The field of a relation R in X is the union of the domain of R and
the range of R.
If R is a relation in X, and if S is defined so that x S y if x R y
and not x = y, then S is the strict relation corresponding to R.
Conversely, if S is a relation in X, and if R is defined so that x
R y if x S y or x = y, then R is the weak relation corresponding to
S.
A relation R in X is reflexive if x R x for every element x of X,
it is symmetric if x R y implies that y R x, and it is transitive
if x R y and y R z imply that x R z.
* A function F is a relation, a subset of X x Y, such that the domain
of F is equal to X and such that for every x in X there is a unique
element y in Y with (x, y) in F. The latter condition can be formu-
lated as follows: if x F y and x F z, then y = z. In this module,
it is not required that the domain of F is equal to X for a rela-
tion to be considered a function.
Instead of writing (x, y) in F or x F y, we write F(x) = y when F
is a function, and say that F maps x onto y, or that the value of F
at x is y.
As functions are relations, the definitions of the last item
(domain, range, and so on) apply to functions as well.
If the converse of a function F is a function F', then F' is called
the inverse of F.
The relative product of two functions F1 and F2 is called the com-
posite of F1 and F2 if the range of F1 is a subset of the domain of
F2.
* Sometimes, when the range of a function is more important than the
function itself, the function is called a family.
The domain of a family is called the index set, and the range is
called the indexed set.
If x is a family from I to X, then x[i] denotes the value of the
function at index i. The notation "a family in X" is used for such
a family.
When the indexed set is a set of subsets of a set X, we call x a
family of subsets of X.
If x is a family of subsets of X, the union of the range of x is
called the union of the family x.
If x is non-empty (the index set is non-empty), the intersection of
the family x is the intersection of the range of x.
In this module, the only families that are considered are families
of subsets of some set X; in the following, the word "family" is
used for such families of subsets.
* A partition of a set X is a collection S of non-empty subsets of X
whose union is X and whose elements are pairwise disjoint.
A relation in a set is an equivalence relation if it is reflexive,
symmetric, and transitive.
If R is an equivalence relation in X, and x is an element of X, the
equivalence class of x with respect to R is the set of all those
elements y of X for which x R y holds. The equivalence classes con-
stitute a partitioning of X. Conversely, if C is a partition of X,
the relation that holds for any two elements of X if they belong to
the same equivalence class, is an equivalence relation induced by
the partition C.
If R is an equivalence relation in X, the canonical map is the
function that maps every element of X onto its equivalence class.
* Relations as defined above (as sets of ordered pairs) are from now
on referred to as binary relations.
We call a set of ordered sets (x[1], ..., x[n]) an (n-ary) rela-
tion, and say that the relation is a subset of the Cartesian prod-
uct X[1] x ... x X[n], where x[i] is an element of X[i], 1 <= i <=
n.
The projection of an n-ary relation R onto coordinate i is the set
{x[i] : (x[1], ..., x[i], ..., x[n]) in R for some x[j] in X[j], 1
<= j <= n and not i = j}. The projections of a binary relation R
onto the first and second coordinates are the domain and the range
of R, respectively.
The relative product of binary relations can be generalized to n-
ary relations as follows. Let TR be an ordered set (R[1], ...,
R[n]) of binary relations from X to Y[i] and S a binary relation
from (Y[1] x ... x Y[n]) to Z. The relative product of TR and S is
the binary relation T from X to Z defined so that x T z if and only
if there exists an element y[i] in Y[i] for each 1 <= i <= n such
that x R[i] y[i] and (y[1], ..., y[n]) S z. Now let TR be a an
ordered set (R[1], ..., R[n]) of binary relations from X[i] to Y[i]
and S a subset of X[1] x ... x X[n]. The multiple relative product
of TR and S is defined to be the set {z : z = ((x[1], ..., x[n]),
(y[1],...,y[n])) for some (x[1], ..., x[n]) in S and for some
(x[i], y[i]) in R[i], 1 <= i <= n}.
The natural join of an n-ary relation R and an m-ary relation S on
coordinate i and j is defined to be the set {z : z = (x[1], ...,
x[n], y[1], ..., y[j-1], y[j+1], ..., y[m]) for some (x[1], ...,
x[n]) in R and for some (y[1], ..., y[m]) in S such that x[i] =
y[j]}.
* The sets recognized by this module are represented by elements of
the relation Sets, which is defined as the smallest set such that:
* For every atom T, except '_', and for every term X, (T, X)
belongs to Sets (atomic sets).
* (['_'], []) belongs to Sets (the untyped empty set).
* For every tuple T = {T[1], ..., T[n]} and for every tuple X =
{X[1], ..., X[n]}, if (T[i], X[i]) belongs to Sets for every 1 <=
i <= n, then (T, X) belongs to Sets (ordered sets).
* For every term T, if X is the empty list or a non-empty sorted
list [X[1], ..., X[n]] without duplicates such that (T, X[i])
belongs to Sets for every 1 <= i <= n, then ([T], X) belongs to
Sets (typed unordered sets).
An external set is an element of the range of Sets.
A type is an element of the domain of Sets.
If S is an element (T, X) of Sets, then T is a valid type of X, T
is the type of S, and X is the external set of S. from_term/2 cre-
ates a set from a type and an Erlang term turned into an external
set.
The sets represented by Sets are the elements of the range of func-
tion Set from Sets to Erlang terms and sets of Erlang terms:
* Set(T,Term) = Term, where T is an atom
* Set({T[1], ..., T[n]}, {X[1], ..., X[n]}) = (Set(T[1], X[1]),
..., Set(T[n], X[n]))
* Set([T], [X[1], ..., X[n]]) = {Set(T, X[1]), ..., Set(T, X[n])}
* Set([T], []) = {}
When there is no risk of confusion, elements of Sets are identified
with the sets they represent. For example, if U is the result of
calling union/2 with S1 and S2 as arguments, then U is said to be
the union of S1 and S2. A more precise formulation is that Set(U)
is the union of Set(S1) and Set(S2).
The types are used to implement the various conditions that sets must
fulfill. As an example, consider the relative product of two sets R and
S, and recall that the relative product of R and S is defined if R is a
binary relation to Y and S is a binary relation from Y. The function
that implements the relative product, relative_product/2, checks that
the arguments represent binary relations by matching [{A,B}] against
the type of the first argument (Arg1 say), and [{C,D}] against the type
of the second argument (Arg2 say). The fact that [{A,B}] matches the
type of Arg1 is to be interpreted as Arg1 representing a binary rela-
tion from X to Y, where X is defined as all sets Set(x) for some ele-
ment x in Sets the type of which is A, and similarly for Y. In the same
way Arg2 is interpreted as representing a binary relation from W to Z.
Finally it is checked that B matches C, which is sufficient to ensure
that W is equal to Y. The untyped empty set is handled separately: its
type, ['_'], matches the type of any unordered set.
A few functions of this module (drestriction/3, family_projection/2,
partition/2, partition_family/2, projection/2, restriction/3, substitu-
tion/2) accept an Erlang function as a means to modify each element of
a given unordered set. Such a function, called SetFun in the following,
can be specified as a functional object (fun), a tuple {external, Fun},
or an integer:
* If SetFun is specified as a fun, the fun is applied to each element
of the given set and the return value is assumed to be a set.
* If SetFun is specified as a tuple {external, Fun}, Fun is applied
to the external set of each element of the given set and the return
value is assumed to be an external set. Selecting the elements of
an unordered set as external sets and assembling a new unordered
set from a list of external sets is in the present implementation
more efficient than modifying each element as a set. However, this
optimization can only be used when the elements of the unordered
set are atomic or ordered sets. It must also be the case that the
type of the elements matches some clause of Fun (the type of the
created set is the result of applying Fun to the type of the given
set), and that Fun does nothing but selecting, duplicating, or
rearranging parts of the elements.
* Specifying a SetFun as an integer I is equivalent to specifying
{external, fun(X) -> element(I, X) end}, but is to be preferred, as
it makes it possible to handle this case even more efficiently.
Examples of SetFuns:
fun sofs:union/1
fun(S) -> sofs:partition(1, S) end
{external, fun(A) -> A end}
{external, fun({A,_,C}) -> {C,A} end}
{external, fun({_,{_,C}}) -> C end}
{external, fun({_,{_,{_,E}=C}}) -> {E,{E,C}} end}
2
The order in which a SetFun is applied to the elements of an unordered
set is not specified, and can change in future versions of this module.
The execution time of the functions of this module is dominated by the
time it takes to sort lists. When no sorting is needed, the execution
time is in the worst case proportional to the sum of the sizes of the
input arguments and the returned value. A few functions execute in con-
stant time: from_external/2, is_empty_set/1, is_set/1, is_sofs_set/1,
to_external/1 type/1.
The functions of this module exit the process with a badarg, bad_func-
tion, or type_mismatch message when given badly formed arguments or
sets the types of which are not compatible.
When comparing external sets, operator ==/2 is used.
DATA TYPES
anyset() = ordset() | a_set()
Any kind of set (also included are the atomic sets).
binary_relation() = relation()
A binary relation.
external_set() = term()
An external set.
family() = a_function()
A family (of subsets).
a_function() = relation()
A function.
ordset()
An ordered set.
relation() = a_set()
An n-ary relation.
a_set()
An unordered set.
set_of_sets() = a_set()
An unordered set of unordered sets.
set_fun() =
integer() >= 1 |
{external, fun((external_set()) -> external_set())} |
fun((anyset()) -> anyset())
A SetFun.
spec_fun() =
{external, fun((external_set()) -> boolean())} |
fun((anyset()) -> boolean())
type() = term()
A type.
tuple_of(T)
A tuple where the elements are of type T.
EXPORTS
a_function(Tuples) -> Function
a_function(Tuples, Type) -> Function
Types:
Function = a_function()
Tuples = [tuple()]
Type = type()
Creates a function. a_function(F, T) is equivalent to
from_term(F, T) if the result is a function. If no type is
explicitly specified, [{atom, atom}] is used as the function
type.
canonical_relation(SetOfSets) -> BinRel
Types:
BinRel = binary_relation()
SetOfSets = set_of_sets()
Returns the binary relation containing the elements (E, Set)
such that Set belongs to SetOfSets and E belongs to Set. If
SetOfSets is a partition of a set X and R is the equivalence
relation in X induced by SetOfSets, then the returned relation
is the canonical map from X onto the equivalence classes with
respect to R.
1> Ss = sofs:from_term([[a,b],[b,c]]),
CR = sofs:canonical_relation(Ss),
sofs:to_external(CR).
[{a,[a,b]},{b,[a,b]},{b,[b,c]},{c,[b,c]}]
composite(Function1, Function2) -> Function3
Types:
Function1 = Function2 = Function3 = a_function()
Returns the composite of the functions Function1 and Function2.
1> F1 = sofs:a_function([{a,1},{b,2},{c,2}]),
F2 = sofs:a_function([{1,x},{2,y},{3,z}]),
F = sofs:composite(F1, F2),
sofs:to_external(F).
[{a,x},{b,y},{c,y}]
constant_function(Set, AnySet) -> Function
Types:
AnySet = anyset()
Function = a_function()
Set = a_set()
Creates the function that maps each element of set Set onto Any-
Set.
1> S = sofs:set([a,b]),
E = sofs:from_term(1),
R = sofs:constant_function(S, E),
sofs:to_external(R).
[{a,1},{b,1}]
converse(BinRel1) -> BinRel2
Types:
BinRel1 = BinRel2 = binary_relation()
Returns the converse of the binary relation BinRel1.
1> R1 = sofs:relation([{1,a},{2,b},{3,a}]),
R2 = sofs:converse(R1),
sofs:to_external(R2).
[{a,1},{a,3},{b,2}]
difference(Set1, Set2) -> Set3
Types:
Set1 = Set2 = Set3 = a_set()
Returns the difference of the sets Set1 and Set2.
digraph_to_family(Graph) -> Family
digraph_to_family(Graph, Type) -> Family
Types:
Graph = digraph:graph()
Family = family()
Type = type()
Creates a family from the directed graph Graph. Each vertex a of
Graph is represented by a pair (a, {b[1], ..., b[n]}), where the
b[i]:s are the out-neighbors of a. If no type is explicitly
specified, [{atom, [atom]}] is used as type of the family. It is
assumed that Type is a valid type of the external set of the
family.
If G is a directed graph, it holds that the vertices and edges
of G are the same as the vertices and edges of fam-
ily_to_digraph(digraph_to_family(G)).
domain(BinRel) -> Set
Types:
BinRel = binary_relation()
Set = a_set()
Returns the domain of the binary relation BinRel.
1> R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),
S = sofs:domain(R),
sofs:to_external(S).
[1,2]
drestriction(BinRel1, Set) -> BinRel2
Types:
BinRel1 = BinRel2 = binary_relation()
Set = a_set()
Returns the difference between the binary relation BinRel1 and
the restriction of BinRel1 to Set.
1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
S = sofs:set([2,4,6]),
R2 = sofs:drestriction(R1, S),
sofs:to_external(R2).
[{1,a},{3,c}]
drestriction(R, S) is equivalent to difference(R, restriction(R,
S)).
drestriction(SetFun, Set1, Set2) -> Set3
Types:
SetFun = set_fun()
Set1 = Set2 = Set3 = a_set()
Returns a subset of Set1 containing those elements that do not
give an element in Set2 as the result of applying SetFun.
1> SetFun = {external, fun({_A,B,C}) -> {B,C} end},
R1 = sofs:relation([{a,aa,1},{b,bb,2},{c,cc,3}]),
R2 = sofs:relation([{bb,2},{cc,3},{dd,4}]),
R3 = sofs:drestriction(SetFun, R1, R2),
sofs:to_external(R3).
[{a,aa,1}]
drestriction(F, S1, S2) is equivalent to difference(S1, restric-
tion(F, S1, S2)).
empty_set() -> Set
Types:
Set = a_set()
Returns the untyped empty set. empty_set() is equivalent to
from_term([], ['_']).
extension(BinRel1, Set, AnySet) -> BinRel2
Types:
AnySet = anyset()
BinRel1 = BinRel2 = binary_relation()
Set = a_set()
Returns the extension of BinRel1 such that for each element E in
Set that does not belong to the domain of BinRel1, BinRel2 con-
tains the pair (E, AnySet).
1> S = sofs:set([b,c]),
A = sofs:empty_set(),
R = sofs:family([{a,[1,2]},{b,[3]}]),
X = sofs:extension(R, S, A),
sofs:to_external(X).
[{a,[1,2]},{b,[3]},{c,[]}]
family(Tuples) -> Family
family(Tuples, Type) -> Family
Types:
Family = family()
Tuples = [tuple()]
Type = type()
Creates a family of subsets. family(F, T) is equivalent to
from_term(F, T) if the result is a family. If no type is explic-
itly specified, [{atom, [atom]}] is used as the family type.
family_difference(Family1, Family2) -> Family3
Types:
Family1 = Family2 = Family3 = family()
If Family1 and Family2 are families, then Family3 is the family
such that the index set is equal to the index set of Family1,
and Family3[i] is the difference between Family1[i] and Fam-
ily2[i] if Family2 maps i, otherwise Family1[i].
1> F1 = sofs:family([{a,[1,2]},{b,[3,4]}]),
F2 = sofs:family([{b,[4,5]},{c,[6,7]}]),
F3 = sofs:family_difference(F1, F2),
sofs:to_external(F3).
[{a,[1,2]},{b,[3]}]
family_domain(Family1) -> Family2
Types:
Family1 = Family2 = family()
If Family1 is a family and Family1[i] is a binary relation for
every i in the index set of Family1, then Family2 is the family
with the same index set as Family1 such that Family2[i] is the
domain of Family1[i].
1> FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),
F = sofs:family_domain(FR),
sofs:to_external(F).
[{a,[1,2,3]},{b,[]},{c,[4,5]}]
family_field(Family1) -> Family2
Types:
Family1 = Family2 = family()
If Family1 is a family and Family1[i] is a binary relation for
every i in the index set of Family1, then Family2 is the family
with the same index set as Family1 such that Family2[i] is the
field of Family1[i].
1> FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),
F = sofs:family_field(FR),
sofs:to_external(F).
[{a,[1,2,3,a,b,c]},{b,[]},{c,[4,5,d,e]}]
family_field(Family1) is equivalent to family_union(fam-
ily_domain(Family1), family_range(Family1)).
family_intersection(Family1) -> Family2
Types:
Family1 = Family2 = family()
If Family1 is a family and Family1[i] is a set of sets for every
i in the index set of Family1, then Family2 is the family with
the same index set as Family1 such that Family2[i] is the inter-
section of Family1[i].
If Family1[i] is an empty set for some i, the process exits with
a badarg message.
1> F1 = sofs:from_term([{a,[[1,2,3],[2,3,4]]},{b,[[x,y,z],[x,y]]}]),
F2 = sofs:family_intersection(F1),
sofs:to_external(F2).
[{a,[2,3]},{b,[x,y]}]
family_intersection(Family1, Family2) -> Family3
Types:
Family1 = Family2 = Family3 = family()
If Family1 and Family2 are families, then Family3 is the family
such that the index set is the intersection of Family1:s and
Family2:s index sets, and Family3[i] is the intersection of Fam-
ily1[i] and Family2[i].
1> F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),
F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),
F3 = sofs:family_intersection(F1, F2),
sofs:to_external(F3).
[{b,[4]},{c,[]}]
family_projection(SetFun, Family1) -> Family2
Types:
SetFun = set_fun()
Family1 = Family2 = family()
If Family1 is a family, then Family2 is the family with the same
index set as Family1 such that Family2[i] is the result of call-
ing SetFun with Family1[i] as argument.
1> F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),
F2 = sofs:family_projection(fun sofs:union/1, F1),
sofs:to_external(F2).
[{a,[1,2,3]},{b,[]}]
family_range(Family1) -> Family2
Types:
Family1 = Family2 = family()
If Family1 is a family and Family1[i] is a binary relation for
every i in the index set of Family1, then Family2 is the family
with the same index set as Family1 such that Family2[i] is the
range of Family1[i].
1> FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),
F = sofs:family_range(FR),
sofs:to_external(F).
[{a,[a,b,c]},{b,[]},{c,[d,e]}]
family_specification(Fun, Family1) -> Family2
Types:
Fun = spec_fun()
Family1 = Family2 = family()
If Family1 is a family, then Family2 is the restriction of Fam-
ily1 to those elements i of the index set for which Fun applied
to Family1[i] returns true. If Fun is a tuple {external, Fun2},
then Fun2 is applied to the external set of Family1[i], other-
wise Fun is applied to Family1[i].
1> F1 = sofs:family([{a,[1,2,3]},{b,[1,2]},{c,[1]}]),
SpecFun = fun(S) -> sofs:no_elements(S) =:= 2 end,
F2 = sofs:family_specification(SpecFun, F1),
sofs:to_external(F2).
[{b,[1,2]}]
family_to_digraph(Family) -> Graph
family_to_digraph(Family, GraphType) -> Graph
Types:
Graph = digraph:graph()
Family = family()
GraphType = [digraph:d_type()]
Creates a directed graph from family Family. For each pair (a,
{b[1], ..., b[n]}) of Family, vertex a and the edges (a, b[i])
for 1 <= i <= n are added to a newly created directed graph.
If no graph type is specified, digraph:new/0 is used for creat-
ing the directed graph, otherwise argument GraphType is passed
on as second argument to digraph:new/1.
It F is a family, it holds that F is a subset of digraph_to_fam-
ily(family_to_digraph(F), type(F)). Equality holds if
union_of_family(F) is a subset of domain(F).
Creating a cycle in an acyclic graph exits the process with a
cyclic message.
family_to_relation(Family) -> BinRel
Types:
Family = family()
BinRel = binary_relation()
If Family is a family, then BinRel is the binary relation con-
taining all pairs (i, x) such that i belongs to the index set of
Family and x belongs to Family[i].
1> F = sofs:family([{a,[]}, {b,[1]}, {c,[2,3]}]),
R = sofs:family_to_relation(F),
sofs:to_external(R).
[{b,1},{c,2},{c,3}]
family_union(Family1) -> Family2
Types:
Family1 = Family2 = family()
If Family1 is a family and Family1[i] is a set of sets for each
i in the index set of Family1, then Family2 is the family with
the same index set as Family1 such that Family2[i] is the union
of Family1[i].
1> F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),
F2 = sofs:family_union(F1),
sofs:to_external(F2).
[{a,[1,2,3]},{b,[]}]
family_union(F) is equivalent to family_projection(fun
sofs:union/1, F).
family_union(Family1, Family2) -> Family3
Types:
Family1 = Family2 = Family3 = family()
If Family1 and Family2 are families, then Family3 is the family
such that the index set is the union of Family1:s and Family2:s
index sets, and Family3[i] is the union of Family1[i] and Fam-
ily2[i] if both map i, otherwise Family1[i] or Family2[i].
1> F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),
F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),
F3 = sofs:family_union(F1, F2),
sofs:to_external(F3).
[{a,[1,2]},{b,[3,4,5]},{c,[5,6,7,8]},{d,[9,10]}]
field(BinRel) -> Set
Types:
BinRel = binary_relation()
Set = a_set()
Returns the field of the binary relation BinRel.
1> R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),
S = sofs:field(R),
sofs:to_external(S).
[1,2,a,b,c]
field(R) is equivalent to union(domain(R), range(R)).
from_external(ExternalSet, Type) -> AnySet
Types:
ExternalSet = external_set()
AnySet = anyset()
Type = type()
Creates a set from the external set ExternalSet and the type
Type. It is assumed that Type is a valid type of ExternalSet.
from_sets(ListOfSets) -> Set
Types:
Set = a_set()
ListOfSets = [anyset()]
Returns the unordered set containing the sets of list ListOf-
Sets.
1> S1 = sofs:relation([{a,1},{b,2}]),
S2 = sofs:relation([{x,3},{y,4}]),
S = sofs:from_sets([S1,S2]),
sofs:to_external(S).
[[{a,1},{b,2}],[{x,3},{y,4}]]
from_sets(TupleOfSets) -> Ordset
Types:
Ordset = ordset()
TupleOfSets = tuple_of(anyset())
Returns the ordered set containing the sets of the non-empty
tuple TupleOfSets.
from_term(Term) -> AnySet
from_term(Term, Type) -> AnySet
Types:
AnySet = anyset()
Term = term()
Type = type()
Creates an element of Sets by traversing term Term, sorting
lists, removing duplicates, and deriving or verifying a valid
type for the so obtained external set. An explicitly specified
type Type can be used to limit the depth of the traversal; an
atomic type stops the traversal, as shown by the following exam-
ple where "foo" and {"foo"} are left unmodified:
1> S = sofs:from_term([{{"foo"},[1,1]},{"foo",[2,2]}], [{atom,[atom]}]),
sofs:to_external(S).
[{{"foo"},[1]},{"foo",[2]}]
from_term can be used for creating atomic or ordered sets. The
only purpose of such a set is that of later building unordered
sets, as all functions in this module that do anything operate
on unordered sets. Creating unordered sets from a collection of
ordered sets can be the way to go if the ordered sets are big
and one does not want to waste heap by rebuilding the elements
of the unordered set. The following example shows that a set can
be built "layer by layer":
1> A = sofs:from_term(a),
S = sofs:set([1,2,3]),
P1 = sofs:from_sets({A,S}),
P2 = sofs:from_term({b,[6,5,4]}),
Ss = sofs:from_sets([P1,P2]),
sofs:to_external(Ss).
[{a,[1,2,3]},{b,[4,5,6]}]
Other functions that create sets are from_external/2 and
from_sets/1. Special cases of from_term/2 are a_function/1,2,
empty_set/0, family/1,2, relation/1,2, and set/1,2.
image(BinRel, Set1) -> Set2
Types:
BinRel = binary_relation()
Set1 = Set2 = a_set()
Returns the image of set Set1 under the binary relation BinRel.
1> R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),
S1 = sofs:set([1,2]),
S2 = sofs:image(R, S1),
sofs:to_external(S2).
[a,b,c]
intersection(SetOfSets) -> Set
Types:
Set = a_set()
SetOfSets = set_of_sets()
Returns the intersection of the set of sets SetOfSets.
Intersecting an empty set of sets exits the process with a
badarg message.
intersection(Set1, Set2) -> Set3
Types:
Set1 = Set2 = Set3 = a_set()
Returns the intersection of Set1 and Set2.
intersection_of_family(Family) -> Set
Types:
Family = family()
Set = a_set()
Returns the intersection of family Family.
Intersecting an empty family exits the process with a badarg
message.
1> F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),
S = sofs:intersection_of_family(F),
sofs:to_external(S).
[2]
inverse(Function1) -> Function2
Types:
Function1 = Function2 = a_function()
Returns the inverse of function Function1.
1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
R2 = sofs:inverse(R1),
sofs:to_external(R2).
[{a,1},{b,2},{c,3}]
inverse_image(BinRel, Set1) -> Set2
Types:
BinRel = binary_relation()
Set1 = Set2 = a_set()
Returns the inverse image of Set1 under the binary relation Bin-
Rel.
1> R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),
S1 = sofs:set([c,d,e]),
S2 = sofs:inverse_image(R, S1),
sofs:to_external(S2).
[2,3]
is_a_function(BinRel) -> Bool
Types:
Bool = boolean()
BinRel = binary_relation()
Returns true if the binary relation BinRel is a function or the
untyped empty set, otherwise false.
is_disjoint(Set1, Set2) -> Bool
Types:
Bool = boolean()
Set1 = Set2 = a_set()
Returns true if Set1 and Set2 are disjoint, otherwise false.
is_empty_set(AnySet) -> Bool
Types:
AnySet = anyset()
Bool = boolean()
Returns true if AnySet is an empty unordered set, otherwise
false.
is_equal(AnySet1, AnySet2) -> Bool
Types:
AnySet1 = AnySet2 = anyset()
Bool = boolean()
Returns true if AnySet1 and AnySet2 are equal, otherwise false.
The following example shows that ==/2 is used when comparing
sets for equality:
1> S1 = sofs:set([1.0]),
S2 = sofs:set([1]),
sofs:is_equal(S1, S2).
true
is_set(AnySet) -> Bool
Types:
AnySet = anyset()
Bool = boolean()
Returns true if AnySet is an unordered set, and false if AnySet
is an ordered set or an atomic set.
is_sofs_set(Term) -> Bool
Types:
Bool = boolean()
Term = term()
Returns true if Term is an unordered set, an ordered set, or an
atomic set, otherwise false.
is_subset(Set1, Set2) -> Bool
Types:
Bool = boolean()
Set1 = Set2 = a_set()
Returns true if Set1 is a subset of Set2, otherwise false.
is_type(Term) -> Bool
Types:
Bool = boolean()
Term = term()
Returns true if term Term is a type.
join(Relation1, I, Relation2, J) -> Relation3
Types:
Relation1 = Relation2 = Relation3 = relation()
I = J = integer() >= 1
Returns the natural join of the relations Relation1 and Rela-
tion2 on coordinates I and J.
1> R1 = sofs:relation([{a,x,1},{b,y,2}]),
R2 = sofs:relation([{1,f,g},{1,h,i},{2,3,4}]),
J = sofs:join(R1, 3, R2, 1),
sofs:to_external(J).
[{a,x,1,f,g},{a,x,1,h,i},{b,y,2,3,4}]
multiple_relative_product(TupleOfBinRels, BinRel1) -> BinRel2
Types:
TupleOfBinRels = tuple_of(BinRel)
BinRel = BinRel1 = BinRel2 = binary_relation()
If TupleOfBinRels is a non-empty tuple {R[1], ..., R[n]} of
binary relations and BinRel1 is a binary relation, then BinRel2
is the multiple relative product of the ordered set (R[i], ...,
R[n]) and BinRel1.
1> Ri = sofs:relation([{a,1},{b,2},{c,3}]),
R = sofs:relation([{a,b},{b,c},{c,a}]),
MP = sofs:multiple_relative_product({Ri, Ri}, R),
sofs:to_external(sofs:range(MP)).
[{1,2},{2,3},{3,1}]
no_elements(ASet) -> NoElements
Types:
ASet = a_set() | ordset()
NoElements = integer() >= 0
Returns the number of elements of the ordered or unordered set
ASet.
partition(SetOfSets) -> Partition
Types:
SetOfSets = set_of_sets()
Partition = a_set()
Returns the partition of the union of the set of sets SetOfSets
such that two elements are considered equal if they belong to
the same elements of SetOfSets.
1> Sets1 = sofs:from_term([[a,b,c],[d,e,f],[g,h,i]]),
Sets2 = sofs:from_term([[b,c,d],[e,f,g],[h,i,j]]),
P = sofs:partition(sofs:union(Sets1, Sets2)),
sofs:to_external(P).
[[a],[b,c],[d],[e,f],[g],[h,i],[j]]
partition(SetFun, Set) -> Partition
Types:
SetFun = set_fun()
Partition = Set = a_set()
Returns the partition of Set such that two elements are consid-
ered equal if the results of applying SetFun are equal.
1> Ss = sofs:from_term([[a],[b],[c,d],[e,f]]),
SetFun = fun(S) -> sofs:from_term(sofs:no_elements(S)) end,
P = sofs:partition(SetFun, Ss),
sofs:to_external(P).
[[[a],[b]],[[c,d],[e,f]]]
partition(SetFun, Set1, Set2) -> {Set3, Set4}
Types:
SetFun = set_fun()
Set1 = Set2 = Set3 = Set4 = a_set()
Returns a pair of sets that, regarded as constituting a set,
forms a partition of Set1. If the result of applying SetFun to
an element of Set1 gives an element in Set2, the element belongs
to Set3, otherwise the element belongs to Set4.
1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
S = sofs:set([2,4,6]),
{R2,R3} = sofs:partition(1, R1, S),
{sofs:to_external(R2),sofs:to_external(R3)}.
{[{2,b}],[{1,a},{3,c}]}
partition(F, S1, S2) is equivalent to {restriction(F, S1, S2),
drestriction(F, S1, S2)}.
partition_family(SetFun, Set) -> Family
Types:
Family = family()
SetFun = set_fun()
Set = a_set()
Returns family Family where the indexed set is a partition of
Set such that two elements are considered equal if the results
of applying SetFun are the same value i. This i is the index
that Family maps onto the equivalence class.
1> S = sofs:relation([{a,a,a,a},{a,a,b,b},{a,b,b,b}]),
SetFun = {external, fun({A,_,C,_}) -> {A,C} end},
F = sofs:partition_family(SetFun, S),
sofs:to_external(F).
[{{a,a},[{a,a,a,a}]},{{a,b},[{a,a,b,b},{a,b,b,b}]}]
product(TupleOfSets) -> Relation
Types:
Relation = relation()
TupleOfSets = tuple_of(a_set())
Returns the Cartesian product of the non-empty tuple of sets
TupleOfSets. If (x[1], ..., x[n]) is an element of the n-ary
relation Relation, then x[i] is drawn from element i of TupleOf-
Sets.
1> S1 = sofs:set([a,b]),
S2 = sofs:set([1,2]),
S3 = sofs:set([x,y]),
P3 = sofs:product({S1,S2,S3}),
sofs:to_external(P3).
[{a,1,x},{a,1,y},{a,2,x},{a,2,y},{b,1,x},{b,1,y},{b,2,x},{b,2,y}]
product(Set1, Set2) -> BinRel
Types:
BinRel = binary_relation()
Set1 = Set2 = a_set()
Returns the Cartesian product of Set1 and Set2.
1> S1 = sofs:set([1,2]),
S2 = sofs:set([a,b]),
R = sofs:product(S1, S2),
sofs:to_external(R).
[{1,a},{1,b},{2,a},{2,b}]
product(S1, S2) is equivalent to product({S1, S2}).
projection(SetFun, Set1) -> Set2
Types:
SetFun = set_fun()
Set1 = Set2 = a_set()
Returns the set created by substituting each element of Set1 by
the result of applying SetFun to the element.
If SetFun is a number i >= 1 and Set1 is a relation, then the
returned set is the projection of Set1 onto coordinate i.
1> S1 = sofs:from_term([{1,a},{2,b},{3,a}]),
S2 = sofs:projection(2, S1),
sofs:to_external(S2).
[a,b]
range(BinRel) -> Set
Types:
BinRel = binary_relation()
Set = a_set()
Returns the range of the binary relation BinRel.
1> R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),
S = sofs:range(R),
sofs:to_external(S).
[a,b,c]
relation(Tuples) -> Relation
relation(Tuples, Type) -> Relation
Types:
N = integer()
Type = N | type()
Relation = relation()
Tuples = [tuple()]
Creates a relation. relation(R, T) is equivalent to from_term(R,
T), if T is a type and the result is a relation. If Type is an
integer N, then [{atom, ..., atom}]), where the tuple size is N,
is used as type of the relation. If no type is explicitly speci-
fied, the size of the first tuple of Tuples is used if there is
such a tuple. relation([]) is equivalent to relation([], 2).
relation_to_family(BinRel) -> Family
Types:
Family = family()
BinRel = binary_relation()
Returns family Family such that the index set is equal to the
domain of the binary relation BinRel, and Family[i] is the image
of the set of i under BinRel.
1> R = sofs:relation([{b,1},{c,2},{c,3}]),
F = sofs:relation_to_family(R),
sofs:to_external(F).
[{b,[1]},{c,[2,3]}]
relative_product(ListOfBinRels) -> BinRel2
relative_product(ListOfBinRels, BinRel1) -> BinRel2
Types:
ListOfBinRels = [BinRel, ...]
BinRel = BinRel1 = BinRel2 = binary_relation()
If ListOfBinRels is a non-empty list [R[1], ..., R[n]] of binary
relations and BinRel1 is a binary relation, then BinRel2 is the
relative product of the ordered set (R[i], ..., R[n]) and Bin-
Rel1.
If BinRel1 is omitted, the relation of equality between the ele-
ments of the Cartesian product of the ranges of R[i], range R[1]
x ... x range R[n], is used instead (intuitively, nothing is
"lost").
1> TR = sofs:relation([{1,a},{1,aa},{2,b}]),
R1 = sofs:relation([{1,u},{2,v},{3,c}]),
R2 = sofs:relative_product([TR, R1]),
sofs:to_external(R2).
[{1,{a,u}},{1,{aa,u}},{2,{b,v}}]
Notice that relative_product([R1], R2) is different from rela-
tive_product(R1, R2); the list of one element is not identified
with the element itself.
relative_product(BinRel1, BinRel2) -> BinRel3
Types:
BinRel1 = BinRel2 = BinRel3 = binary_relation()
Returns the relative product of the binary relations BinRel1 and
BinRel2.
relative_product1(BinRel1, BinRel2) -> BinRel3
Types:
BinRel1 = BinRel2 = BinRel3 = binary_relation()
Returns the relative product of the converse of the binary rela-
tion BinRel1 and the binary relation BinRel2.
1> R1 = sofs:relation([{1,a},{1,aa},{2,b}]),
R2 = sofs:relation([{1,u},{2,v},{3,c}]),
R3 = sofs:relative_product1(R1, R2),
sofs:to_external(R3).
[{a,u},{aa,u},{b,v}]
relative_product1(R1, R2) is equivalent to relative_product(con-
verse(R1), R2).
restriction(BinRel1, Set) -> BinRel2
Types:
BinRel1 = BinRel2 = binary_relation()
Set = a_set()
Returns the restriction of the binary relation BinRel1 to Set.
1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
S = sofs:set([1,2,4]),
R2 = sofs:restriction(R1, S),
sofs:to_external(R2).
[{1,a},{2,b}]
restriction(SetFun, Set1, Set2) -> Set3
Types:
SetFun = set_fun()
Set1 = Set2 = Set3 = a_set()
Returns a subset of Set1 containing those elements that gives an
element in Set2 as the result of applying SetFun.
1> S1 = sofs:relation([{1,a},{2,b},{3,c}]),
S2 = sofs:set([b,c,d]),
S3 = sofs:restriction(2, S1, S2),
sofs:to_external(S3).
[{2,b},{3,c}]
set(Terms) -> Set
set(Terms, Type) -> Set
Types:
Set = a_set()
Terms = [term()]
Type = type()
Creates an unordered set. set(L, T) is equivalent to
from_term(L, T), if the result is an unordered set. If no type
is explicitly specified, [atom] is used as the set type.
specification(Fun, Set1) -> Set2
Types:
Fun = spec_fun()
Set1 = Set2 = a_set()
Returns the set containing every element of Set1 for which Fun
returns true. If Fun is a tuple {external, Fun2}, Fun2 is
applied to the external set of each element, otherwise Fun is
applied to each element.
1> R1 = sofs:relation([{a,1},{b,2}]),
R2 = sofs:relation([{x,1},{x,2},{y,3}]),
S1 = sofs:from_sets([R1,R2]),
S2 = sofs:specification(fun sofs:is_a_function/1, S1),
sofs:to_external(S2).
[[{a,1},{b,2}]]
strict_relation(BinRel1) -> BinRel2
Types:
BinRel1 = BinRel2 = binary_relation()
Returns the strict relation corresponding to the binary relation
BinRel1.
1> R1 = sofs:relation([{1,1},{1,2},{2,1},{2,2}]),
R2 = sofs:strict_relation(R1),
sofs:to_external(R2).
[{1,2},{2,1}]
substitution(SetFun, Set1) -> Set2
Types:
SetFun = set_fun()
Set1 = Set2 = a_set()
Returns a function, the domain of which is Set1. The value of an
element of the domain is the result of applying SetFun to the
element.
1> L = [{a,1},{b,2}].
[{a,1},{b,2}]
2> sofs:to_external(sofs:projection(1,sofs:relation(L))).
[a,b]
3> sofs:to_external(sofs:substitution(1,sofs:relation(L))).
[{{a,1},a},{{b,2},b}]
4> SetFun = {external, fun({A,_}=E) -> {E,A} end},
sofs:to_external(sofs:projection(SetFun,sofs:relation(L))).
[{{a,1},a},{{b,2},b}]
The relation of equality between the elements of {a,b,c}:
1> I = sofs:substitution(fun(A) -> A end, sofs:set([a,b,c])),
sofs:to_external(I).
[{a,a},{b,b},{c,c}]
Let SetOfSets be a set of sets and BinRel a binary relation. The
function that maps each element Set of SetOfSets onto the image
of Set under BinRel is returned by the following function:
images(SetOfSets, BinRel) ->
Fun = fun(Set) -> sofs:image(BinRel, Set) end,
sofs:substitution(Fun, SetOfSets).
External unordered sets are represented as sorted lists. So,
creating the image of a set under a relation R can traverse all
elements of R (to that comes the sorting of results, the image).
In image/2, BinRel is traversed once for each element of SetOf-
Sets, which can take too long. The following efficient function
can be used instead under the assumption that the image of each
element of SetOfSets under BinRel is non-empty:
images2(SetOfSets, BinRel) ->
CR = sofs:canonical_relation(SetOfSets),
R = sofs:relative_product1(CR, BinRel),
sofs:relation_to_family(R).
symdiff(Set1, Set2) -> Set3
Types:
Set1 = Set2 = Set3 = a_set()
Returns the symmetric difference (or the Boolean sum) of Set1
and Set2.
1> S1 = sofs:set([1,2,3]),
S2 = sofs:set([2,3,4]),
P = sofs:symdiff(S1, S2),
sofs:to_external(P).
[1,4]
symmetric_partition(Set1, Set2) -> {Set3, Set4, Set5}
Types:
Set1 = Set2 = Set3 = Set4 = Set5 = a_set()
Returns a triple of sets:
* Set3 contains the elements of Set1 that do not belong to
Set2.
* Set4 contains the elements of Set1 that belong to Set2.
* Set5 contains the elements of Set2 that do not belong to
Set1.
to_external(AnySet) -> ExternalSet
Types:
ExternalSet = external_set()
AnySet = anyset()
Returns the external set of an atomic, ordered, or unordered
set.
to_sets(ASet) -> Sets
Types:
ASet = a_set() | ordset()
Sets = tuple_of(AnySet) | [AnySet]
AnySet = anyset()
Returns the elements of the ordered set ASet as a tuple of sets,
and the elements of the unordered set ASet as a sorted list of
sets without duplicates.
type(AnySet) -> Type
Types:
AnySet = anyset()
Type = type()
Returns the type of an atomic, ordered, or unordered set.
union(SetOfSets) -> Set
Types:
Set = a_set()
SetOfSets = set_of_sets()
Returns the union of the set of sets SetOfSets.
union(Set1, Set2) -> Set3
Types:
Set1 = Set2 = Set3 = a_set()
Returns the union of Set1 and Set2.
union_of_family(Family) -> Set
Types:
Family = family()
Set = a_set()
Returns the union of family Family.
1> F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),
S = sofs:union_of_family(F),
sofs:to_external(S).
[0,1,2,3,4]
weak_relation(BinRel1) -> BinRel2
Types:
BinRel1 = BinRel2 = binary_relation()
Returns a subset S of the weak relation W corresponding to the
binary relation BinRel1. Let F be the field of BinRel1. The sub-
set S is defined so that x S y if x W y for some x in F and for
some y in F.
1> R1 = sofs:relation([{1,1},{1,2},{3,1}]),
R2 = sofs:weak_relation(R1),
sofs:to_external(R2).
[{1,1},{1,2},{2,2},{3,1},{3,3}]
SEE ALSO
dict(3), digraph(3), orddict(3), ordsets(3), sets(3)
Ericsson AB stdlib 3.17 sofs(3)