The SET signature

Overview

The SET signature is an interface for a set type, which is an unordered collection of items. Sets do not contain duplicates, and are not polymorphic — the type of their elements is fixed by the Key substructure. We use a number of notational conventions which can be seen here. For example, we write $|x|$ for the number of elements in a set $x$, and the empty set is denoted either $\{\}$ or $\emptyset$.

Interface

structure Key : EQKEY
structure Seq : SEQUENCE

type t
type set = t

val size : set → int
val toString : set → string
val toSeq : set → Key.t Seq.t

val empty : unit → set
val singleton : Key.t → set
val fromSeq : Key.t Seq.t → set

val find : set → Key.t → bool
val insert : set * Key.t → set
val delete : set * Key.t → set

val filterKey : (Key.t → bool) → set → set

val reduceKey : (Key.t * Key.t → Key.t) → Key.t → set → Key.t
val iterateKey : (α * Key.t → α) → α → set → α

val union : set * set → set
val intersection : set * set → set
val difference : set * set → set

val $ : Key.t → set

Substructures

structure Key : EQKEY

The Key substructure defines the type of elements in a set, providing equality and other useful functions.

structure Seq : SEQUENCE

The Seq substructure defines the underlying sequence type to and from which sets can be converted.

Types

type t

The abstract set type.

type set = t

An alias for t, for readability.

Values

val size : set → int

(size x) evaluates to $|x|$, the number of elements in the set $x$.

val toString : set → string

(toString x) evaluates to a string representation of $x$. Each element of $x$ is converted to a string by Key.toString. For example, the set $\{1,2,3\}$ would be represented by the string “{1,2,3}”.

val toSeq : set → Key.t Seq.t

Return the sequence containing all elements of a set. The ordering of the elements in the returned sequence is implementation defined.

val empty : unit → set

Construct the empty set, $\{\}$.

val singleton : Key.t → set

(singleton x) evaluates to $\{x\}$, the singleton set containing only the element $x$.

val fromSeq : Key.t Seq.t → set

Return the set containing all elements of a sequence.

val find : set → Key.t → bool

(find x k) returns whether or not $k$ is a member of the set $x$.

val insert : set * Key.t → set

(insert (x, k)) evaluates to the set $x \cup \{k\}$.

val delete : set * Key.t → set

(delete (x, k)) evaluates to the set $x \setminus \{k\}$.

val filterKey : (Key.t → bool) → set → set

(filterKey p x) evaluates to the subset of $x$ containing every key $k$ which satisfies $p(k)$.

val reduceKey : (Key.t * Key.t → Key.t) → Key.t → set → Key.t

(reduceKey f b x) is logically equivalent to (Seq.reduce f b (toSeq x)).

val iterateKey : (α * Key.t → α) → α → set → α

(iterateKey f b x) is logically equivalent to (Seq.iterate f b (toSeq x)).

val union : set * set → set

(union (x, y)) evaluates to the set $x \cup y$.

val intersection : set * set → set

(intersection (x, y)) evaluates to the set $x \cap y$.

val difference : set * set → set

(difference (x, y)) evaluates to the set $x \setminus y$.

val $ : Key.t → set

An alias for singleton.