sofs

  Functions for manipulating sets of sets.

  This module provides operations on finite sets and relations
  represented as sets. Intuitively, a set is a collection of
  elements; every element belongs to the set, and the set contains
  every element.

  The data representing sofs as used by this module is to be
  regarded as opaque by other modules. In abstract terms, the
  representation is a composite type of existing Erlang terms. See
  note on data types. Any code assuming knowledge of the format is
  running on thin ice.

  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 module; 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 contains.

     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 × 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 × 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

      ￮ 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.

   • function F is a relation, a subset of 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 formulated 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 relation 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 composite 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.

   • 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 constitute 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)
     relation, and say that the relation is a subset of the
     Cartesian product X[1] × ... × 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] × ... × 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[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 creates 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 function 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 relation
  from X to Y, where X is defined as all sets Set(x) for some
  element 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

   • 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, substitution/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 constant 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_function, 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.

See Also

  dict, digraph, orddict, ordsets, sets