General-purpose functional programming language
Standard ML (SML) is a general-purpose, high-level, modular, functional programming language with compile-time type checking and type inference. It is popular for writing compilers, for programming language research, and for developing theorem provers.
Standard ML is a modern dialect of ML, the language used in the Logic for Computable Functions (LCF) theorem-proving project. It is distinctive among widely used languages in that it has a formal specification, given as typing rules and operational semantics in "The Definition of Standard ML".
Language.
Standard ML is a functional programming language with some impure features. Programs written in Standard ML consist of expressions in contrast to statements or commands, although some expressions of type unit are only evaluated for their side-effects.
Functions.
Like all functional languages, a key feature of Standard ML is the function, which is used for abstraction. The factorial function can be expressed as follows:
fun factorial n = 
 if n = 0 then 1 else n * factorial (n - 1)
Type inference.
An SML compiler must infer the static type without user-supplied type annotations. It has to deduce that is only used with integer expressions, and must therefore itself be an integer, and that all terminal expressions are integer expressions.
Declarative definitions.
The same function can be expressed with clausal function definitions where the "if"-"then"-"else" conditional is replaced with templates of the factorial function evaluated for specific values:
fun factorial 0 = 1
 | factorial n = n * factorial (n - 1)
Imperative definitions.
or iteratively:
fun factorial n = let val i = ref n and acc = ref 1 in
 while !i ; 0 do (acc := !acc * !i; i := !i - 1); !acc
end
Lambda functions.
or as a lambda function:
val rec factorial = fn 0 = ; 1 | n = ; n * factorial (n - 1)
Here, the keyword introduces a binding of an identifier to a value, introduces an anonymous function, and allows the definition to be self-referential.
Local definitions.
The encapsulation of an invariant-preserving tail-recursive tight loop with one or more accumulator parameters within an invariant-free outer function, as seen here, is a common idiom in Standard ML.
Using a local function, it can be rewritten in a more efficient tail-recursive style:
local
 fun loop (0, acc) = acc
 | loop (m, acc) = loop (m - 1, m * acc)
in
 fun factorial n = loop (n, 1)
end
Type synonyms.
A type synonym is defined with the keyword . Here is a type synonym for points on a plane, and functions computing the distances between two points, and the area of a triangle with the given corners as per Heron's formula. (These definitions will be used in subsequent examples).
type loc = real * real
fun square (x : real) = x * x
fun dist (x, y) (x', y') =
 Math.sqrt (square (x' - x) + square (y' - y))
fun heron (a, b, c) = let
 val x = dist a b
 val y = dist b c
 val z = dist a c
 val s = (x + y + z) / 2.0
 in
 Math.sqrt (s * (s - x) * (s - y) * (s - z))
 end
Algebraic datatypes.
Standard ML provides strong support for algebraic datatypes (ADT). A data type can be thought of as a disjoint union of tuples (or a "sum of products"). They are easy to define and easy to use, largely because of pattern matching, and most Standard ML implementations' pattern-exhaustiveness checking and pattern redundancy checking.
In object-oriented programming languages, a disjoint union can be expressed as class hierarchies. However, in contrast to class hierarchies, ADTs are closed. Thus, the extensibility of ADTs is orthogonal to the extensibility of class hierarchies. Class hierarchies can be extended with new subclasses which implement the same interface, while the functions of ADTs can be extended for the fixed set of constructors. See expression problem.
A datatype is defined with the keyword , as in:
datatype shape
 = Circle of loc * real (* center and radius *)
 | Square of loc * real (* upper-left corner and side length; axis-aligned *)
 | Triangle of loc * loc * loc (* corners *)
Note that a type synonym cannot be recursive; datatypes are necessary to define recursive constructors. (This is not at issue in this example.)
Pattern matching.
Patterns are matched in the order in which they are defined. C programmers can use tagged unions, dispatching on tag values, to do what ML does with datatypes and pattern matching. Nevertheless, while a C program decorated with appropriate checks will, in a sense, be as robust as the corresponding ML program, those checks will of necessity be dynamic; ML's static checks provide strong guarantees about the correctness of the program at compile time.
Function arguments can be defined as patterns as follows:
fun area (Circle (_, r)) = Math.pi * square r
 | area (Square (_, s)) = square s
 | area (Triangle p) = heron p (* see above *)
The so-called "clausal form" of function definition, where arguments are defined as patterns, is merely syntactic sugar for a case expression:
fun area shape = case shape of
 Circle (_, r) = ; Math.pi * square r
 | Square (_, s) = ; square s
 | Triangle p = ; heron p
Exhaustiveness checking.
Pattern-exhaustiveness checking will make sure that each constructor of the datatype is matched by at least one pattern. 
The following pattern is not exhaustive:
fun center (Circle (c, _)) = c
 | center (Square ((x, y), s)) = (x + s / 2.0, y + s / 2.0)
There is no pattern for the case in the function. The compiler will issue a warning that the case expression is not exhaustive, and if a is passed to this function at runtime, will be raised.
Redundancy checking.
The pattern in the second clause of the following (meaningless) function is redundant:
fun f (Circle ((x, y), r)) = x + y
 | f (Circle _) = 1.0
 | f _ = 0.0
Any value that would match the pattern in the second clause would also match the pattern in the first clause, so the second clause is unreachable. Therefore, this definition as a whole exhibits redundancy, and causes a compile-time warning.
The following function definition is exhaustive and not redundant:
val hasCorners = fn (Circle _) = ; false | _ = ; true
If control gets past the first pattern (), we know the shape must be either a or a . In either of those cases, we know the shape has corners, so we can return without discerning the actual shape.
Higher-order functions.
Functions can consume functions as arguments:
fun map f (x, y) = (f x, f y)
Functions can produce functions as return values:
fun constant k = (fn _ = ; k)
Functions can also both consume and produce functions:
fun compose (f, g) = (fn x = ; f (g x))
The function from the basis library is one of the most commonly used higher-order functions in Standard ML:
fun map _ [] = []
 | map f (x :: xs) = f x :: map f xs
A more efficient implementation with tail-recursive :
fun map f = List.rev o List.foldl (fn (x, acc) = ; f x :: acc) []
Exceptions.
Exceptions are raised with the keyword and handled with the pattern matching construct. The exception system can implement non-local exit; this optimization technique is suitable for functions like the following.
local
 exception Zero;
 val p = fn (0, _) = ; raise Zero | (a, b) = ; a * b
in
 fun prod xs = List.foldl p 1 xs handle Zero = ; 0
end
When is raised, control leaves the function altogether. Consider the alternative: the value 0 would be returned, it would be multiplied by the next integer in the list, the resulting value (inevitably 0) would be returned, and so on. The raising of the exception allows control to skip over the entire chain of frames and avoid the associated computation. Note the use of the underscore () as a wildcard pattern.
The same optimization can be obtained with a tail call.
local
 fun p a (0 :: _) = 0
 | p a (x :: xs) = p (a * x) xs
 | p a [] = a
in
 val prod = p 1
end
Module system.
Standard ML's advanced module system allows programs to be decomposed into hierarchically organized "structures" of logically related type and value definitions. Modules provide not only namespace control but also abstraction, in the sense that they allow the definition of abstract data types. Three main syntactic constructs comprise the module system: signatures, structures and functors.
Signatures.
A "signature" is an interface, usually thought of as a type for a structure; it specifies the names of all entities provided by the structure, the arity of each type component, the type of each value component, and the signature of each substructure. The definitions of type components are optional; type components whose definitions are hidden are "abstract types".
For example, the signature for a queue may be:
signature QUEUE = sig
 type 'a queue
 exception QueueError;
 val empty : 'a queue
 val isEmpty : 'a queue - ; bool
 val singleton : 'a - ; 'a queue
 val fromList : 'a list - ; 'a queue
 val insert : 'a * 'a queue - ; 'a queue
 val peek : 'a queue - ; 'a
 val remove : 'a queue - ; 'a * 'a queue
end
This signature describes a module that provides a polymorphic type , , and values that define basic operations on queues.
Structures.
A "structure" is a module; it consists of a collection of types, exceptions, values and structures (called "substructures") packaged together into a logical unit.
A queue structure can be implemented as follows:
structure TwoListQueue : ; QUEUE = struct
 type 'a queue = 'a list * 'a list
 exception QueueError;
 val empty = ([], [])
 fun isEmpty ([], []) = true
 | isEmpty _ = false
 fun singleton a = ([], [a])
 fun fromList a = ([], a)
 fun insert (a, ([], [])) = singleton a
 | insert (a, (ins, outs)) = (a :: ins, outs)
 fun peek (_, []) = raise QueueError
 | peek (ins, outs) = List.hd outs
 fun remove (_, []) = raise QueueError
 | remove (ins, [a]) = (a, ([], List.rev ins))
 | remove (ins, a :: outs) = (a, (ins, outs))
end
This definition declares that implements . Furthermore, the "opaque ascription" denoted by states that any types which are not defined in the signature (i.e. ) should be abstract, meaning that the definition of a queue as a pair of lists is not visible outside the module. The structure implements all of the definitions in the signature.
The types and values in a structure can be accessed with "dot notation":
val q : string TwoListQueue.queue = TwoListQueue.empty
val q' = TwoListQueue.insert (Real.toString Math.pi, q)
Functors.
A "functor" is a function from structures to structures; that is, a functor accepts one or more arguments, which are usually structures of a given signature, and produces a structure as its result. Functors are used to implement generic data structures and algorithms.
One popular algorithm for breadth-first search of trees makes use of queues. Here is a version of that algorithm parameterized over an abstract queue structure:
functor BFS (Q: QUEUE) = struct
 datatype 'a tree = E | T of 'a * 'a tree * 'a tree
 local
 fun bfsQ q = if Q.isEmpty q then [] else search (Q.remove q)
 and search (E, q) = bfsQ q
 | search (T (x, l, r), q) = x :: bfsQ (insert (insert q l) r)
 and insert q a = Q.insert (a, q)
 in
 fun bfs t = bfsQ (Q.singleton t)
 end
end
structure QueueBFS = BFS (TwoListQueue)
Within , the representation of the queue is not visible. More concretely, there is no way to select the first list in the two-list queue, if that is indeed the representation being used. This data abstraction mechanism makes the breadth-first search truly agnostic to the queue's implementation. This is in general desirable; in this case, the queue structure can safely maintain any logical invariants on which its correctness depends behind the bulletproof wall of abstraction.
Code examples.
Snippets of SML code are most easily studied by entering them into an interactive top-level.
Hello, world!
The following is a "Hello, World!" program:
Algorithms.
Insertion sort.
Insertion sort for (ascending) can be expressed concisely as follows:
fun insert (x, []) = [x] | insert (x, h :: t) = sort x (h, t)
and sort x (h, t) = if x ; h then [x, h] @ t else h :: insert (x,