Misplaced Pages

Standard ML: Difference between revisions

Article snapshot taken from[REDACTED] with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Browse history interactively← Previous editContent deleted Content addedVisualWikitext
Revision as of 14:23, 14 June 2005 edit137.205.192.27 (talk)No edit summary← Previous edit Latest revision as of 02:01, 13 June 2024 edit undoCitation bot (talk | contribs)Bots5,455,831 edits Removed parameters. | Use this bot. Report bugs. | Suggested by Headbomb | Category:CS1 maint: DOI inactive as of June 2024 | #UCB_Category 268/305 
(507 intermediate revisions by more than 100 users not shown)
Line 1: Line 1:
{{Short description|General-purpose functional programming language}}
The '''SML''' programming language is a modern descendant of the ] programming language used in the ] theorem-proving project. It is unique among widely used languages in that it has a formal specification, given as an ] in ''The Definition of Standard ML''.
{{Infobox programming language
| name = Standard ML
| logo =
| paradigm = ]: ], ], ]<ref name="intro"/>
| family = ]
| designer =
| released = {{Start date and age|1983|df=yes}}<ref name="smlnj"/>
| latest release version = Standard ML '97<ref name="smlnj"/>
| latest release date = {{Start date and age|1997|df=yes}}
| typing = ], ], ]
| implementations = ], ], ]
| dialects = ], ], ]
| influenced by = ], ], ]
| influenced = ], ], ], ], ], ],<ref name="itertools"/> ],<ref>{{Cite web |url=https://doc.rust-lang.org/reference/influences.html |title=Influences - The Rust Reference |website=The Rust Reference |access-date=2023-12-31}}</ref> ]
| file ext = .sml
| website =
}}


'''Standard ML''' ('''SML''') is a ], ], ], ] ] with compile-time ] and ]. It is popular for writing ]s, for ], and for developing ].
==Implementations==


Standard ML is a modern dialect of ], the language used in the ] (LCF) theorem-proving project. It is distinctive among widely used languages in that it has a ], given as ]s and ] in ''The Definition of Standard ML''.<ref name="revision"/>
Some SML implementations include:


==Language==
* ] is a whole-program optimizing compiler that produces extremely fast code compared to other ML implementations.
{{multiple issues|section=y|
{{How-to|section|date=November 2021}}
{{Unreferenced section|date=November 2021}}
}}


Standard ML is a functional ] with some impure features. Programs written in Standard ML consist of ] in contrast to statements or commands, although some expressions of type ] are only evaluated for their ].
* ] is a light-weight implementation, based on the ] runtime engine. It implements the full SML language, including SML Modules, and much of the SML Basis Library.


===Functions===
* Poly/ML is a full implementation of Standard ML.
Like all functional languages, a key feature of Standard ML is the ], which is used for abstraction. The factorial function can be expressed as follows:


<syntaxhighlight lang="sml">
* ] (abbreviated SML/NJ) is a full compiler, with associated libraries, tools, an interactive shell, and documentation.
fun factorial n =
if n = 0 then 1 else n * factorial (n - 1)
</syntaxhighlight>


===Type inference===
* SML2c is a batch compiler and compiles only module-level declarations (i.e. signatures, structures and functors) into ]. Based on SML/NJ version 0.67 and shares front end and most of its run-time system, but does not support SML/NJ style debugging and profiling. Module-level programs that run on SML/NJ can be compiled by sml2c without any changes.
An SML compiler must infer the static type {{code|lang=sml|val factorial : int -> int}} without user-supplied type annotations. It has to deduce that {{code|n}} is only used with integer expressions, and must therefore itself be an integer, and that all terminal expressions are integer expressions.


===Declarative definitions===
* ] Is an ML implementation based partly on ]
The same function can be expressed with ]s where the ''if''-''then''-''else'' conditional is replaced with templates of the factorial function evaluated for specific values:


<syntaxhighlight lang="sml">
All of the implementations above are ] and freely available. There are no commercial SML implementations.
fun factorial 0 = 1
| factorial n = n * factorial (n - 1)
</syntaxhighlight>

===Imperative definitions===
or iteratively:

<syntaxhighlight lang="sml">
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
</syntaxhighlight>

===Lambda functions===
or as a lambda function:

<syntaxhighlight lang="sml">
val rec factorial = fn 0 => 1 | n => n * factorial (n - 1)
</syntaxhighlight>

Here, the keyword {{code|lang=sml|val}} introduces a binding of an identifier to a value, {{code|lang=sml|fn}} introduces an ], and {{code|lang=sml|rec}} 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:

<syntaxhighlight lang="sml">
local
fun loop (0, acc) = acc
| loop (m, acc) = loop (m - 1, m * acc)
in
fun factorial n = loop (n, 1)
end
</syntaxhighlight>

===Type synonyms===
A type synonym is defined with the keyword {{code|lang=sml|type}}. Here is a type synonym for points on a ], and functions computing the distances between two points, and the area of a triangle with the given corners as per ]. (These definitions will be used in subsequent examples).

<syntaxhighlight lang="sml">
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
</syntaxhighlight>

===Algebraic datatypes===
Standard ML provides strong support for ]s (ADT). A ] can be thought of as a ] of tuples (or a "sum of products"). They are easy to define and easy to use, largely because of ], and most Standard ML implementations' ] checking and pattern redundancy checking.

In ] languages, a disjoint union can be expressed as ] hierarchies. However, in contrast to ], ADTs are ]. 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 ].

A datatype is defined with the keyword {{code|lang=sml|datatype}}, as in:

<syntaxhighlight lang="sml">
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 *)
</syntaxhighlight>

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. ] programmers can use ]s, 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 ] provide strong guarantees about the correctness of the program at compile time.

Function arguments can be defined as patterns as follows:

<syntaxhighlight lang="sml">
fun area (Circle (_, r)) = Math.pi * square r
| area (Square (_, s)) = square s
| area (Triangle p) = heron p (* see above *)
</syntaxhighlight>

The so-called "clausal form" of function definition, where arguments are defined as patterns, is merely ] for a case expression:

<syntaxhighlight lang="sml">
fun area shape = case shape of
Circle (_, r) => Math.pi * square r
| Square (_, s) => square s
| Triangle p => heron p
</syntaxhighlight>

====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:

<syntaxhighlight lang="sml">
fun center (Circle (c, _)) = c
| center (Square ((x, y), s)) = (x + s / 2.0, y + s / 2.0)
</syntaxhighlight>

There is no pattern for the {{code|Triangle}} case in the {{code|center}} function. The compiler will issue a warning that the case expression is not exhaustive, and if a {{code|Triangle}} is passed to this function at runtime, {{code|lang=sml|exception Match}} will be raised.

====Redundancy checking====
The pattern in the second clause of the following (meaningless) function is redundant:

<syntaxhighlight lang="sml">
fun f (Circle ((x, y), r)) = x + y
| f (Circle _) = 1.0
| f _ = 0.0
</syntaxhighlight>

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:

<syntaxhighlight lang="sml">
val hasCorners = fn (Circle _) => false | _ => true
</syntaxhighlight>

If control gets past the first pattern ({{code|Circle}}), we know the shape must be either a {{code|Square}} or a {{code|Triangle}}. In either of those cases, we know the shape has corners, so we can return {{code|lang=sml|true}} without discerning the actual shape.

===Higher-order functions===
Functions can consume functions as arguments:
<syntaxhighlight lang="sml">fun map f (x, y) = (f x, f y)</syntaxhighlight>

Functions can produce functions as return values:
<syntaxhighlight lang="sml">fun constant k = (fn _ => k)</syntaxhighlight>

Functions can also both consume and produce functions:
<syntaxhighlight lang="sml">fun compose (f, g) = (fn x => f (g x))</syntaxhighlight>

The function {{code|List.map}} from the basis ] is one of the most commonly used higher-order functions in Standard ML:
<syntaxhighlight lang="sml">
fun map _ =
| map f (x :: xs) = f x :: map f xs
</syntaxhighlight>

A more efficient implementation with tail-recursive {{code|List.foldl}}:
<syntaxhighlight lang="sml">
fun map f = List.rev o List.foldl (fn (x, acc) => f x :: acc)
</syntaxhighlight>

===Exceptions===
Exceptions are raised with the keyword {{code|lang=sml|raise}} and handled with the pattern matching {{code|lang=sml|handle}} construct. The exception system can implement ]; this optimization technique is suitable for functions like the following.

<syntaxhighlight lang="sml">
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
</syntaxhighlight>

When {{code|lang=sml|exception Zero}} is raised, control leaves the function {{code|lang=sml|List.foldl}} 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 ({{code|_}}) as a wildcard pattern.

The same optimization can be obtained with a ].

<syntaxhighlight lang="sml">
local
fun p a (0 :: _) = 0
| p a (x :: xs) = p (a * x) xs
| p a = a
in
val prod = p 1
end
</syntaxhighlight>

===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 ] control but also abstraction, in the sense that they allow the definition of ]s. Three main syntactic constructs comprise the module system: signatures, structures and functors.

====Signatures====
A ''signature'' is an ], usually thought of as a type for a structure; it specifies the names of all entities provided by the structure, the ] 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 ] may be:

<syntaxhighlight lang="sml">
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
</syntaxhighlight>

This signature describes a module that provides a polymorphic type {{code|lang=sml|'a queue}}, {{code|lang=sml|exception QueueError}}, 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:

<syntaxhighlight lang="sml">
structure TwoListQueue :> QUEUE = struct
type 'a queue = 'a list * 'a list

exception QueueError;

val empty = (, )

fun isEmpty (, ) = true
| isEmpty _ = false

fun singleton 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, (, List.rev ins))
| remove (ins, a :: outs) = (a, (ins, outs))
end
</syntaxhighlight>

This definition declares that {{code|lang=sml|structure TwoListQueue}} implements {{code|lang=sml|signature QUEUE}}. Furthermore, the ''opaque ascription'' denoted by {{code|lang=sml|:>}} states that any types which are not defined in the signature (i.e. {{code|lang=sml|type 'a queue}}) 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":

<syntaxhighlight lang="sml">
val q : string TwoListQueue.queue = TwoListQueue.empty
val q' = TwoListQueue.insert (Real.toString Math.pi, q)
</syntaxhighlight>

====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 ] data structures and algorithms.

One popular algorithm<ref name="bfs"/> for ] of trees makes use of queues. Here is a version of that algorithm parameterized over an abstract queue structure:

<syntaxhighlight lang="sml">
(* after Okasaki, ICFP, 2000 *)
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)
</syntaxhighlight>

Within {{code|lang=sml|functor BFS}}, 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 ] 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==
{{wikibook|Standard ML Programming}}
{{unreferenced section|date=June 2013}}
Snippets of SML code are most easily studied by entering them into an ].

===Hello, world!===
The following is a ]:

{| class="wikitable"
|-
! hello.sml
|-
|
<syntaxhighlight lang="sml" line>
print "Hello, world!\n";
</syntaxhighlight>
|-
! bash
|-
|
<syntaxhighlight lang="console">
$ mlton hello.sml
$ ./hello
Hello, world!
</syntaxhighlight>
|}

===Algorithms===
====Insertion sort====
Insertion sort for {{code|lang=sml|int list}} (ascending) can be expressed concisely as follows:

<syntaxhighlight lang="sml">
fun insert (x, ) = | insert (x, h :: t) = sort x (h, t)
and sort x (h, t) = if x < h then @ t else h :: insert (x, t)
val insertionsort = List.foldl insert
</syntaxhighlight>

====Mergesort====
{{main|Merge sort}}

Here, the classic mergesort algorithm is implemented in three functions: split, merge and mergesort. Also note the absence of types, with the exception of the syntax {{code|lang=sml|op ::}} and {{code|lang=sml|}} which signify lists. This code will sort lists of any type, so long as a consistent ordering function {{code|cmp}} is defined. Using ], the types of all variables can be inferred, even complicated types such as that of the function {{code|cmp}}.

'''Split'''

{{code|lang=sml|fun split}} is implemented with a ] closure which alternates between {{code|true}} and {{code|false}}, ignoring the input:

<syntaxhighlight lang="sml">
fun alternator {} = let val state = ref true
in fn a => !state before state := not (!state) end

(* Split a list into near-halves which will either be the same length,
* or the first will have one more element than the other.
* Runs in O(n) time, where n = |xs|.
*)
fun split xs = List.partition (alternator {}) xs
</syntaxhighlight>

'''Merge'''

Merge uses a local function loop for efficiency. The inner {{code|loop}} is defined in terms of cases: when both lists are non-empty ({{code|lang=sml|x :: xs}}) and when one list is empty ({{code|lang=sml|}}).

This function merges two sorted lists into one sorted list. Note how the accumulator {{code|acc}} is built backwards, then reversed before being returned. This is a common technique, since {{code|lang=sml|'a list}} is represented as a ]; this technique requires more clock time, but the ] are not worse.

<syntaxhighlight lang="sml">
(* Merge two ordered lists using the order cmp.
* Pre: each list must already be ordered per cmp.
* Runs in O(n) time, where n = |xs| + |ys|.
*)
fun merge cmp (xs, ) = xs
| merge cmp (xs, y :: ys) = let
fun loop (a, acc) (xs, ) = List.revAppend (a :: acc, xs)
| loop (a, acc) (xs, y :: ys) =
if cmp (a, y)
then loop (y, a :: acc) (ys, xs)
else loop (a, y :: acc) (xs, ys)
in
loop (y, ) (ys, xs)
end
</syntaxhighlight>

'''Mergesort'''

The main function:

<syntaxhighlight lang="sml">
fun ap f (x, y) = (f x, f y)

(* Sort a list in according to the given ordering operation cmp.
* Runs in O(n log n) time, where n = |xs|.
*)
fun mergesort cmp =
| mergesort cmp =
| mergesort cmp xs = (merge cmp o ap (mergesort cmp) o split) xs
</syntaxhighlight>

====Quicksort====
{{main|Quicksort}}

Quicksort can be expressed as follows. {{code|lang=sml|fun part}} is a ] that consumes an order operator {{code|lang=sml|op <<}}.

<syntaxhighlight lang="sml">
infix <<

fun quicksort (op <<) = let
fun part p = List.partition (fn x => x << p)
fun sort =
| sort (p :: xs) = join p (part p xs)
and join p (l, r) = sort l @ p :: sort r
in
sort
end
</syntaxhighlight>

===Expression interpreter===
Note the relative ease with which a small expression language can be defined and processed:

<syntaxhighlight lang="sml">
exception TyErr;

datatype ty = IntTy | BoolTy

fun unify (IntTy, IntTy) = IntTy
| unify (BoolTy, BoolTy) = BoolTy
| unify (_, _) = raise TyErr

datatype exp
= True
| False
| Int of int
| Not of exp
| Add of exp * exp
| If of exp * exp * exp

fun infer True = BoolTy
| infer False = BoolTy
| infer (Int _) = IntTy
| infer (Not e) = (assert e BoolTy; BoolTy)
| infer (Add (a, b)) = (assert a IntTy; assert b IntTy; IntTy)
| infer (If (e, t, f)) = (assert e BoolTy; unify (infer t, infer f))
and assert e t = unify (infer e, t)

fun eval True = True
| eval False = False
| eval (Int n) = Int n
| eval (Not e) = if eval e = True then False else True
| eval (Add (a, b)) = (case (eval a, eval b) of (Int x, Int y) => Int (x + y))
| eval (If (e, t, f)) = eval (if eval e = True then t else f)

fun run e = (infer e; SOME (eval e)) handle TyErr => NONE
</syntaxhighlight>

Example usage on well-typed and ill-typed expressions:
<syntaxhighlight lang="sml">
val SOME (Int 3) = run (Add (Int 1, Int 2)) (* well-typed *)
val NONE = run (If (Not (Int 1), True, False)) (* ill-typed *)
</syntaxhighlight>

===Arbitrary-precision integers===
The {{code|IntInf}} module provides arbitrary-precision integer arithmetic. Moreover, integer literals may be used as arbitrary-precision integers without the programmer having to do anything.

The following program implements an arbitrary-precision factorial function:

{| class="wikitable"
|-
! fact.sml
|-
|
<syntaxhighlight lang="sml" line>
fun fact n : IntInf.int = if n = 0 then 1 else n * fact (n - 1);

fun printLine str = TextIO.output (TextIO.stdOut, str ^ "\n");

val () = printLine (IntInf.toString (fact 120));
</syntaxhighlight>
|-
! bash
|-
|
<syntaxhighlight lang="console">
$ mlton fact.sml
$ ./fact
6689502913449127057588118054090372586752746333138029810295671352301
6335572449629893668741652719849813081576378932140905525344085894081
21859898481114389650005964960521256960000000000000000000000000000
</syntaxhighlight>
|}

===Partial application===
Curried functions have many applications, such as eliminating redundant code. For example, a module may require functions of type {{code|lang=sml|a -> b}}, but it is more convenient to write functions of type {{code|lang=sml|a * c -> b}} where there is a fixed relationship between the objects of type {{code|a}} and {{code|c}}. A function of type {{code|lang=sml|c -> (a * c -> b) -> a -> b}} can factor out this commonality. This is an example of the ].{{Citation needed|date=May 2015}}

In this example, {{code|lang=sml|fun d}} computes the numerical derivative of a given function {{code|f}} at point {{code|x}}:

<syntaxhighlight lang="sml" highlight="1">
- fun d delta f x = (f (x + delta) - f (x - delta)) / (2.0 * delta)
val d = fn : real -> (real -> real) -> real -> real
</syntaxhighlight>

The type of {{code|lang=sml|fun d}} indicates that it maps a "float" onto a function with the type {{code|lang=sml|(real -> real) -> real -> real}}. This allows us to partially apply arguments, known as ]. In this case, function {{code|d}} can be specialised by partially applying it with the argument {{code|delta}}. A good choice for {{code|delta}} when using this algorithm is the cube root of the ].{{Citation needed|date=August 2008}}

<syntaxhighlight lang="sml" highlight="1">
- val d' = d 1E~8;
val d' = fn : (real -> real) -> real -> real
</syntaxhighlight>

The inferred type indicates that {{code|d'}} expects a function with the type {{code|lang=sml|real -> real}} as its first argument. We can compute an approximation to the derivative of <math>f(x) = x^3-x-1</math> at <math>x=3</math>. The correct answer is <math>f'(3) = 27-1 = 26</math>.

<syntaxhighlight lang="sml" highlight="1">
- d' (fn x => x * x * x - x - 1.0) 3.0;
val it = 25.9999996644 : real
</syntaxhighlight>

==Libraries==
===Standard===
The Basis Library<ref>{{Cite web|title=Standard ML Basis Library|url=https://smlfamily.github.io/Basis/index.html|access-date=2022-01-10|website=smlfamily.github.io}}</ref> has been standardized and ships with most implementations. It provides modules for trees, arrays, and other data structures, and ] and system interfaces.

===Third party===
For ], a Matrix module exists (but is currently broken), https://www.cs.cmu.edu/afs/cs/project/pscico/pscico/src/matrix/README.html.

For graphics, cairo-sml is an open source interface to the ] graphics library. For machine learning, a library for graphical models exists.

==Implementations==
Implementations of Standard ML include the following:

'''Standard'''
* : a Standard ML interpreter that aims to be an accurate and accessible reference implementation of the standard
* ] (): a ] compiler which strictly conforms to the Definition and produces very fast code compared to other ML implementations, including ] for ] and C
* : a light-weight implementation, based on the ] Light runtime engine which implements the full Standard ML language, including modules and much of the basis library
* : a full implementation of Standard ML that produces fast code and supports multicore hardware (via Portable Operating System Interface (]) threads); its runtime system performs parallel ] and online sharing of immutable substructures.
* ] (): a full compiler, with associated libraries, tools, an interactive shell, and documentation with support for ]
* : a Standard ML compiler for the ] with extensions for linking with other ] framework code
* {{Webarchive|url=https://web.archive.org/web/20160107005413/http://www.elsman.com/mlkit/ |date=2016-01-07}}: an implementation based very closely on the Definition, integrating a garbage collector (which can be disabled) and ] with automatic inference of regions, aiming to support real-time applications

'''Derivative'''
* ]: an interpreter for Standard ML by Saarland University with support for parallel programming using ], ], ] via ]s and ]
* : an extension of SML providing record polymorphism and C language interoperability. It is a conventional native compiler and its name is ''not'' an allusion to running on the .NET framework
* : an implementation written in ], supporting most of the SML language and select parts of the basis library

'''Research'''
* is a REPL version of ML with formally verified runtime and translation to assembler.
* ] ( {{Webarchive|url=https://web.archive.org/web/20200830080049/http://isabelle.in.tum.de/ |date=2020-08-30 }}) integrates parallel Poly/ML into an interactive theorem prover, with a sophisticated IDE (based on ]) for official Standard ML (SML'97), the Isabelle/ML dialect, and the proof language. Starting with Isabelle2016, there is also a source-level debugger for ML.
* ] implements a version of Standard ML, along with ] and ], allowing mixed language programming; all are implemented in ], which is ].
* is a full certifying compiler for Standard ML which uses typed ] to ] code and ensure correctness, and can compile to ].

All of these implementations are ] and freely available. Most are implemented themselves in Standard ML. There are no longer any commercial implementations; ], now defunct, once produced a commercial IDE and compiler called MLWorks which passed on to ] and was later open-sourced after it was acquired by Ravenbrook Limited on April 26, 2013.

==Major projects using SML==
The ]'s entire ] is implemented in around 100,000 lines of SML, including staff records, payroll, course administration and feedback, student project management, and web-based self-service interfaces.<ref name="sml"/>

The ]s ], ], ], and ] are written in Standard ML. It is also used by ] and ]ers such as ].<ref name="machine code verification"/>


==See also== ==See also==
* ] * ]
* ]
* ]
* ]


==References== ==References==
{{refs|
<ref name="intro">{{cite web
|title=Programming in Standard ML: Hierarchies and Parameterization
|url=https://www.cs.cmu.edu/~rwh/introsml/modules/subfun.htm
|access-date=2020-02-22
}}</ref>

<ref name="sml">{{cite journal
|last=Tofte |first=Mads
|year=2009
|title=Standard ML language
|journal=Scholarpedia
|volume=4
|issue=2
|page=7515
|doi=10.4249/scholarpedia.7515
|bibcode=2009SchpJ...4.7515T
|doi-access=free
}}</ref>

<ref name="smlnj">{{cite web
|title=SML '97
|url=http://www.smlnj.org/sml97.html
|website=www.smlnj.org
}}</ref>

<ref name="revision">{{cite book
|last1=Milner |first1=Robin |author1-link=Robin Milner
|last2=Tofte |first2=Mads
|last3=Harper |first3=Robert
|last4=MacQueen |first4=David
|year=1997
|title=The Definition of Standard ML (Revised)
|publisher=MIT Press
|isbn=0-262-63181-4
}}</ref>

<ref name="itertools">{{cite web
|title=itertools — Functions creating iterators for efficient looping — Python 3.7.1rc1 documentation
|url=https://docs.python.org/3/library/itertools.html
|website=docs.python.org
}}</ref>

<ref name="bfs">{{cite conference
|last=Okasaki |first=Chris
|year=2000
|title=Breadth-First Numbering: Lessons from a Small Exercise in Algorithm Design
|book-title=International Conference on Functional Programming 2000
|publisher=ACM
}}</ref>

<ref name="machine code verification">{{cite conference
|last1=Alglave |first1=Jade |author1-link=Jade Alglave
|last2=Fox |first2=Anthony C. J.
|last3=Ishtiaq |first3=Samin
|last4=Myreen |first4=Magnus O.
|last5=Sarkar |first5=Susmit
|last6=Sewell |first6=Peter
|last7=Nardelli |first7=Francesco Zappa
|date=2009
|title=The Semantics of Power and ARM Multiprocessor Machine Code
|conference=DAMP 2009
|pages=13–24
|url=http://www0.cs.ucl.ac.uk/staff/j.alglave/papers/damp09.pdf |archive-url=https://web.archive.org/web/20170814015026/http://www0.cs.ucl.ac.uk/staff/j.alglave/papers/damp09.pdf |archive-date=2017-08-14 |url-status=live
}}</ref>
}}

==External links==
'''About Standard ML'''
*
* {{Webarchive|url=https://web.archive.org/web/20200220023435/http://sml-family.org/ |date=2020-02-20}}
*
*

'''About successor ML'''
* : evolution of ML using Standard ML as a starting point
* : reference implementation for successor ML

'''Practical'''
*
*


'''Academic'''
* ], ], R. Harper and D. MacQueen. ''The Definition of Standard ML (Revised)''. ISBN 0262631814.
*
*


{{ML programming}}
]
{{Programming languages}}
]
{{Authority control}}
]
]
]


]
] ]
] ]
] ]
] ]

Latest revision as of 02:01, 13 June 2024

General-purpose functional programming language
Standard ML
ParadigmMulti-paradigm: functional, imperative, modular
FamilyML
First appeared1983; 42 years ago (1983)
Stable releaseStandard ML '97 / 1997; 28 years ago (1997)
Typing disciplineInferred, static, strong
Filename extensions.sml
Websitesmlfamily.github.io
Major implementations
SML/NJ, MLton, Poly/ML
Dialects
Alice, Concurrent ML, Dependent ML
Influenced by
ML, Hope, Pascal
Influenced
Elm, F#, F*, Haskell, OCaml, Python, Rust, Scala

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

This section has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages)
This section contains instructions, advice, or how-to content. Please help rewrite the content so that it is more encyclopedic or move it to Wikiversity, Wikibooks, or Wikivoyage. (November 2021)
This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2021) (Learn how and when to remove this message)
(Learn how and when to remove this message)

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 val factorial : int -> int without user-supplied type annotations. It has to deduce that n 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 val introduces a binding of an identifier to a value, fn introduces an anonymous function, and rec 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 type. 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 datatype, 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 Triangle case in the center function. The compiler will issue a warning that the case expression is not exhaustive, and if a Triangle is passed to this function at runtime, exception Match 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 (Circle), we know the shape must be either a Square or a Triangle. In either of those cases, we know the shape has corners, so we can return true 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 List.map 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 List.foldl:

fun map f = List.rev o List.foldl (fn (x, acc) => f x :: acc) 

Exceptions

Exceptions are raised with the keyword raise and handled with the pattern matching handle 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 exception Zero is raised, control leaves the function List.foldl 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 'a queue, exception QueueError, 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 = (, )
    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, (, List.rev ins))
      | remove (ins, a :: outs) = (a, (ins, outs))
end

This definition declares that structure TwoListQueue implements signature QUEUE. Furthermore, the opaque ascription denoted by :> states that any types which are not defined in the signature (i.e. type 'a queue) 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:

(* after Okasaki, ICFP, 2000 *)
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 functor BFS, 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

This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (June 2013) (Learn how and when to remove this message)

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:

hello.sml
print "Hello, world!\n";
bash
$ mlton hello.sml
$ ./hello
Hello, world!

Algorithms

Insertion sort

Insertion sort for int list (ascending) can be expressed concisely as follows:

fun insert (x, ) =  | insert (x, h :: t) = sort x (h, t)
and sort x (h, t) = if x < h then  @ t else h :: insert (x, t)
val insertionsort = List.foldl insert 

Mergesort

Main article: Merge sort

Here, the classic mergesort algorithm is implemented in three functions: split, merge and mergesort. Also note the absence of types, with the exception of the syntax op :: and which signify lists. This code will sort lists of any type, so long as a consistent ordering function cmp is defined. Using Hindley–Milner type inference, the types of all variables can be inferred, even complicated types such as that of the function cmp.

Split

fun split is implemented with a stateful closure which alternates between true and false, ignoring the input:

fun alternator {} = let val state = ref true
    in fn a => !state before state := not (!state) end
(* Split a list into near-halves which will either be the same length,
 * or the first will have one more element than the other.
 * Runs in O(n) time, where n = |xs|.
 *)
fun split xs = List.partition (alternator {}) xs

Merge

Merge uses a local function loop for efficiency. The inner loop is defined in terms of cases: when both lists are non-empty (x :: xs) and when one list is empty ().

This function merges two sorted lists into one sorted list. Note how the accumulator acc is built backwards, then reversed before being returned. This is a common technique, since 'a list is represented as a linked list; this technique requires more clock time, but the asymptotics are not worse.

(* Merge two ordered lists using the order cmp.
 * Pre: each list must already be ordered per cmp.
 * Runs in O(n) time, where n = |xs| + |ys|.
 *)
fun merge cmp (xs, ) = xs
  | merge cmp (xs, y :: ys) = let
    fun loop (a, acc) (xs, ) = List.revAppend (a :: acc, xs)
      | loop (a, acc) (xs, y :: ys) =
        if cmp (a, y)
        then loop (y, a :: acc) (ys, xs)
        else loop (a, y :: acc) (xs, ys)
    in
        loop (y, ) (ys, xs)
    end

Mergesort

The main function:

fun ap f (x, y) = (f x, f y)
(* Sort a list in according to the given ordering operation cmp.
 * Runs in O(n log n) time, where n = |xs|.
 *)
fun mergesort cmp  = 
  | mergesort cmp  = 
  | mergesort cmp xs = (merge cmp o ap (mergesort cmp) o split) xs

Quicksort

Main article: Quicksort

Quicksort can be expressed as follows. fun part is a closure that consumes an order operator op <<.

infix <<
fun quicksort (op <<) = let
    fun part p = List.partition (fn x => x << p)
    fun sort  = 
      | sort (p :: xs) = join p (part p xs)
    and join p (l, r) = sort l @ p :: sort r
    in
        sort
    end

Expression interpreter

Note the relative ease with which a small expression language can be defined and processed:

exception TyErr;
datatype ty = IntTy | BoolTy
fun unify (IntTy, IntTy) = IntTy
  | unify (BoolTy, BoolTy) = BoolTy
  | unify (_, _) = raise TyErr
datatype exp
    = True
    | False
    | Int of int
    | Not of exp
    | Add of exp * exp
    | If  of exp * exp * exp
fun infer True = BoolTy
  | infer False = BoolTy
  | infer (Int _) = IntTy
  | infer (Not e) = (assert e BoolTy; BoolTy)
  | infer (Add (a, b)) = (assert a IntTy; assert b IntTy; IntTy)
  | infer (If (e, t, f)) = (assert e BoolTy; unify (infer t, infer f))
and assert e t = unify (infer e, t)
fun eval True = True
  | eval False = False
  | eval (Int n) = Int n
  | eval (Not e) = if eval e = True then False else True
  | eval (Add (a, b)) = (case (eval a, eval b) of (Int x, Int y) => Int (x + y))
  | eval (If (e, t, f)) = eval (if eval e = True then t else f)
fun run e = (infer e; SOME (eval e)) handle TyErr => NONE

Example usage on well-typed and ill-typed expressions:

val SOME (Int 3) = run (Add (Int 1, Int 2)) (* well-typed *)
val NONE = run (If (Not (Int 1), True, False)) (* ill-typed *)

Arbitrary-precision integers

The IntInf module provides arbitrary-precision integer arithmetic. Moreover, integer literals may be used as arbitrary-precision integers without the programmer having to do anything.

The following program implements an arbitrary-precision factorial function:

fact.sml
fun fact n : IntInf.int = if n = 0 then 1 else n * fact (n - 1);
fun printLine str = TextIO.output (TextIO.stdOut, str ^ "\n");
val () = printLine (IntInf.toString (fact 120));
bash
$ mlton fact.sml
$ ./fact
6689502913449127057588118054090372586752746333138029810295671352301
6335572449629893668741652719849813081576378932140905525344085894081
21859898481114389650005964960521256960000000000000000000000000000

Partial application

Curried functions have many applications, such as eliminating redundant code. For example, a module may require functions of type a -> b, but it is more convenient to write functions of type a * c -> b where there is a fixed relationship between the objects of type a and c. A function of type c -> (a * c -> b) -> a -> b can factor out this commonality. This is an example of the adapter pattern.

In this example, fun d computes the numerical derivative of a given function f at point x:

- fun d delta f x = (f (x + delta) - f (x - delta)) / (2.0 * delta)
val d = fn : real -> (real -> real) -> real -> real

The type of fun d indicates that it maps a "float" onto a function with the type (real -> real) -> real -> real. This allows us to partially apply arguments, known as currying. In this case, function d can be specialised by partially applying it with the argument delta. A good choice for delta when using this algorithm is the cube root of the machine epsilon.

- val d' = d 1E~8;
val d' = fn : (real -> real) -> real -> real

The inferred type indicates that d' expects a function with the type real -> real as its first argument. We can compute an approximation to the derivative of f ( x ) = x 3 x 1 {\displaystyle f(x)=x^{3}-x-1} at x = 3 {\displaystyle x=3} . The correct answer is f ( 3 ) = 27 1 = 26 {\displaystyle f'(3)=27-1=26} .

- d' (fn x => x * x * x - x - 1.0) 3.0;
val it = 25.9999996644 : real

Libraries

Standard

The Basis Library has been standardized and ships with most implementations. It provides modules for trees, arrays, and other data structures, and input/output and system interfaces.

Third party

For numerical computing, a Matrix module exists (but is currently broken), https://www.cs.cmu.edu/afs/cs/project/pscico/pscico/src/matrix/README.html.

For graphics, cairo-sml is an open source interface to the Cairo graphics library. For machine learning, a library for graphical models exists.

Implementations

Implementations of Standard ML include the following:

Standard

  • HaMLet: a Standard ML interpreter that aims to be an accurate and accessible reference implementation of the standard
  • MLton (mlton.org): a whole-program optimizing compiler which strictly conforms to the Definition and produces very fast code compared to other ML implementations, including backends for LLVM and C
  • Moscow ML: a light-weight implementation, based on the Caml Light runtime engine which implements the full Standard ML language, including modules and much of the basis library
  • Poly/ML: a full implementation of Standard ML that produces fast code and supports multicore hardware (via Portable Operating System Interface (POSIX) threads); its runtime system performs parallel garbage collection and online sharing of immutable substructures.
  • Standard ML of New Jersey (smlnj.org): a full compiler, with associated libraries, tools, an interactive shell, and documentation with support for Concurrent ML
  • SML.NET: a Standard ML compiler for the Common Language Runtime with extensions for linking with other .NET framework code
  • ML Kit Archived 2016-01-07 at the Wayback Machine: an implementation based very closely on the Definition, integrating a garbage collector (which can be disabled) and region-based memory management with automatic inference of regions, aiming to support real-time applications

Derivative

Research

All of these implementations are open-source and freely available. Most are implemented themselves in Standard ML. There are no longer any commercial implementations; Harlequin, now defunct, once produced a commercial IDE and compiler called MLWorks which passed on to Xanalys and was later open-sourced after it was acquired by Ravenbrook Limited on April 26, 2013.

Major projects using SML

The IT University of Copenhagen's entire enterprise architecture is implemented in around 100,000 lines of SML, including staff records, payroll, course administration and feedback, student project management, and web-based self-service interfaces.

The proof assistants HOL4, Isabelle, LEGO, and Twelf are written in Standard ML. It is also used by compiler writers and integrated circuit designers such as ARM.

See also

References

  1. ^ "Programming in Standard ML: Hierarchies and Parameterization". Retrieved 2020-02-22.
  2. ^ "SML '97". www.smlnj.org.
  3. ^ "itertools — Functions creating iterators for efficient looping — Python 3.7.1rc1 documentation". docs.python.org.
  4. "Influences - The Rust Reference". The Rust Reference. Retrieved 2023-12-31.
  5. ^ Milner, Robin; Tofte, Mads; Harper, Robert; MacQueen, David (1997). The Definition of Standard ML (Revised). MIT Press. ISBN 0-262-63181-4.
  6. ^ Okasaki, Chris (2000). "Breadth-First Numbering: Lessons from a Small Exercise in Algorithm Design". International Conference on Functional Programming 2000. ACM.
  7. "Standard ML Basis Library". smlfamily.github.io. Retrieved 2022-01-10.
  8. ^ Tofte, Mads (2009). "Standard ML language". Scholarpedia. 4 (2): 7515. Bibcode:2009SchpJ...4.7515T. doi:10.4249/scholarpedia.7515.
  9. ^ Alglave, Jade; Fox, Anthony C. J.; Ishtiaq, Samin; Myreen, Magnus O.; Sarkar, Susmit; Sewell, Peter; Nardelli, Francesco Zappa (2009). The Semantics of Power and ARM Multiprocessor Machine Code (PDF). DAMP 2009. pp. 13–24. Archived (PDF) from the original on 2017-08-14.

External links

About Standard ML

About successor ML

Practical

Academic

ML programming
Software
Implementations,
dialects
Caml
Standard ML
Dependent ML
Programming tools
  • Alt-Ergo°
  • Astrée
  • Camlp4°
  • FFTW°
  • Frama-C°
  • Haxe°
  • Marionnet°
  • MTASC°
  • Poplog°
  • Semgrep°
  • SLAM project
  • Theorem provers,
    proof assistants
    Community
    Designers
  • Lennart Augustsson (Lazy ML)
  • Damien Doligez (OCaml)
  • Gérard Huet (Caml)
  • Xavier Leroy (Caml, OCaml)
  • Robin Milner (ML)
  • Don Sannella (Extended ML)
  • Don Syme (F#)
  • Italics = discontinued
  • ° = Open-source software
    Book Category:Family:ML Category:Family:OCaml Category:Software:OCaml
  • Programming languages
    Categories:
    Standard ML: Difference between revisions Add topic