formulasearchengine - User contributions [en]

Non-linear least squares

2014-12-17T09:31:33Z

89.204.135.20: /* Marquardt parameter */ added 'be'

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Relativity priority dispute

2014-03-01T13:32:59Z

89.204.153.161: Reimann -> Riemann

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Antipope John XXIII

2014-01-30T22:08:35Z

89.204.137.88: Fixed geographical description of flight

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Bogosort

2014-01-24T04:00:33Z

89.204.135.221: +Category:Computer humor

{{Infobox programming language
|name = Standard ML
|logo =
|paradigm = [[multi-paradigm programming language|multi-paradigm]]: [[functional programming language|functional]], [[imperative programming|imperative]]
|year =
|designer =
|typing = [[strong typing|strong]], [[static typing|static]], [[type inference|inferred]]
|implementations = [[MLKit]], [[MLton]], [[MLWorks]], [[Moscow ML]], [[Poly/ML]], [[Standard ML of New Jersey|SML/NJ]], [[MLj]], [[SML.NET]]
|dialects = [[Alice (programming language)|Alice]], [[Dependent ML]]
|influenced_by = [[ML programming language|ML]], [[Hope (programming language)|Hope]]
|influenced = [[Rust (programming language)|Rust]]
|file_ext = .sml
|website =
}}

'''Standard ML''' ('''SML''') is a general-purpose, [[Module (programming)|modular]], [[functional programming language]] with [[compile-time type checking]] and [[type inference]]. It is popular among [[compiler]] writers and [[programming language research]]ers, as well as in the development of [[automated theorem proving|theorem provers]].

SML is a modern descendant of the [[ML programming 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 [[type rules|typing rules]] and [[operational semantics]] in ''The Definition of Standard ML'' (1990, revised and simplified as ''The Definition of Standard ML (Revised)'' in 1997).<ref>{{cite book
| last = Milner
| first = R.
| authorlink = Robin Milner
| coauthors = [[Mads Tofte|M. Tofte]], [[Robert Harper (computer scientist)|R. Harper]] and D. MacQueen.
| title = The Definition of Standard ML (Revised)
| publisher = MIT Press
| year = 1997
| isbn = 0-262-63181-4 }}</ref>

==Language==
Standard ML is a functional programming language with some impure features. Programs written in Standard ML consist of [[expression (programming)|expression]]s to be evaluated, as opposed to statements or commands, although some expressions return a trivial "unit" value and are only evaluated for their side-effects.

Like all functional programming languages, a key feature of Standard ML is the [[function (programming)|function]], which is used for abstraction. For instance, the [[factorial]] function can be expressed as:

'''fun''' factorial n =
'''if''' n = 0 '''then''' 1 '''else''' n * factorial (n-1)

A Standard ML compiler is required to infer the static type <tt>int -> int </tt>of this function without user-supplied type annotations. I.e., it has to deduce that ''n'' is only used with integer expressions, and must therefore itself be an integer, and that all value-producing expressions within the function return integers.

The same function can be expressed with [[clausal function definitions]] where the ''if''-''then''-''else'' conditional is replaced by a sequence of templates of the factorial function evaluated for specific values, separated by '|', which are tried one by one in the order written until a match is found:

'''fun''' factorial 0 = 1
<nowiki> |</nowiki> factorial n = n * factorial (n - 1)<nowiki></nowiki>

This can be rewritten using a case statement like this:

'''val''' '''rec''' factorial =
'''fn''' n => '''case''' n '''of''' 0 => 1
<nowiki> |</nowiki> n => n * factorial (n - 1)<nowiki></nowiki>

or as a lambda function:

'''val''' '''rec''' factorial = '''fn''' 0 => 1 | n => n * factorial(n -1)

Here, the keyword <code>val</code> introduces a binding of an identifier to a value, <code>fn</code> introduces the definition of an [[anonymous function]], and <code>case</code> introduces a sequence of patterns and corresponding expressions.

Using a local function, this function can be rewritten in a more efficient [[tail recursive]] style.
'''fun''' factorial n = '''let'''
'''fun''' lp (0, acc) = acc
<nowiki> |</nowiki> lp (m, acc) = lp (m-1, m*acc)<nowiki></nowiki>
'''in'''
lp (n, 1)
'''end'''

(The value of a '''let'''-expression is that of the expression between '''in''' and '''end'''.) The encapsulation of an invariant-preserving tail-recursive tight loop with one or more accumulator parameters inside an invariant-free outer function, as seen here, is a common idiom in Standard ML, and appears with great frequency in SML code.

===Type synonyms===

A type synonym is defined with the '''type''' keyword. Here is a type synonym for points in the plane, and functions computing the distances between two points, and the area of a triangle with the given corners as per [[Heron's formula]].
'''type''' loc = real * real

'''fun''' dist ((x0, y0), (x1, y1)) = '''let'''
'''val''' dx = x1 - x0
'''val''' dy = y1 - y0
'''in'''
Math.sqrt (dx * dx + dy * dy)
'''end'''

'''fun''' heron (a, b, c) = '''let'''
'''val''' ab = dist (a, b)
'''val''' bc = dist (b, c)
'''val''' ac = dist (a, c)
'''val''' perim = ab + bc + ac
'''val''' s = perim / 2.0
'''in'''
Math.sqrt (s * (s - ab) * (s - bc) * (s - ac))
'''end'''

===Algebraic datatypes and pattern matching===

Standard ML provides strong support for [[algebraic datatypes]]. An ML datatype can be thought of as a [[disjoint union]]. They are easy to define and easy to program with, in large part because of Standard ML's [[pattern matching]] as well as most Standard ML implementations' pattern exhaustiveness checking and pattern redundancy checking.

A datatype is defined with the '''datatype''' keyword, as in
'''datatype''' shape
= Circle '''of''' loc * real (* center and radius *)
<nowiki> | Square </nowiki>'''of''' loc * real (* upper-left corner and side length; axis-aligned *)
<nowiki> | Triangle </nowiki>'''of''' loc * loc * loc (* corners *)
(See above for the definition of <tt>loc</tt>.) Note: datatypes, not type synonyms, are necessary to define recursive constructors. (This is not at issue in the present example.)

Order matters in pattern matching; patterns that are textually first are tried first. Pattern matching can be syntactically embedded in function definitions as follows:
'''fun''' area (Circle (_, r)) = 3.14 * r * r
<nowiki> |</nowiki> area (Square (_, s)) = s * s
<nowiki> |</nowiki> area (Triangle (a, b, c)) = heron (a, b, c) (* see above *)<nowiki></nowiki><nowiki></nowiki>
Note that subcomponents whose values are not needed in a particular computation are ellided with underscores, or so-called wildcard patterns.

The so-called "clausal form" style function definition, where patterns appear immediately after the function name, is merely [[syntactic sugar]] for
'''fun''' area shape =
'''case''' shape
'''of''' Circle (_, r) => 3.14 * r * r
<nowiki> |</nowiki> Square (_, s) => s * s
<nowiki> |</nowiki> Triangle (a, b, c) => heron (a, b, c)<nowiki></nowiki><nowiki></nowiki>

Pattern exhaustiveness checking will make sure each case of the datatype has been accounted for, and will produce a warning if not. The following pattern is inexhaustive:
'''fun''' center (Circle (c, _)) = c
<nowiki> |</nowiki> center (Square ((x, y), s)) = (x + s / 2.0, y + s / 2.0)<nowiki></nowiki>
There is no pattern for the <tt>Triangle</tt> case in the <tt>center</tt> function. The compiler will issue a warning that the pattern is inexhaustive, and if, at runtime, a <tt>Triangle</tt> is passed to this function, the exception <tt>Match</tt> will be raised.

The set of clauses in the following function definition is exhaustive and not redundant:
'''fun''' hasCorners (Circle _) = false
<nowiki> |</nowiki> hasCorners _ = true<nowiki></nowiki>
If control gets past the first pattern (the <tt>Circle</tt>), we know the value must be either a <tt>Square</tt> or a <tt>Triangle</tt>. In either of those cases, we know the shape has corners, so we can return <tt>true</tt> without discriminating which case we are in.

The pattern in second clause the following (meaningless) function is redundant:
'''fun''' f (Circle ((x, y), r)) = x+y
<nowiki> |</nowiki> f (Circle _) = 1.0
<nowiki> | f _ = 0.0</nowiki>
Any value that matches the pattern in the second clause will 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.

C programmers will often use [[tagged union]]s, dispatching on tag values, to accomplish what ML accomplishes with datatypes and pattern matching. Nevertheless, while a C program decorated with appropriate checks will be in a sense as robust as the corresponding ML program, those checks will of necessity be dynamic; ML provides a set of static checks that give the programmer a high degree of confidence in the correctness of the program at compile time.

Note that in object-oriented programming languages, such as Java, a disjoint union can be expressed by designing [[class hierarchies]]. However, as opposed to class hierarchies, ADTs are [[Closed world assumption|closed]]. This makes ADT extensible in a way that is orthogonal to the extensibility of class hierarchies. Class hierarchies can be extended with new subclasses but no new methods, while ADTs can be extended to provide new behavior for all existing constructors, but do not allow defining new constructors.

===Higher-order functions===

Functions can consume functions as arguments:
'''fun''' applyToBoth f x y = (f x, f y)

Functions can produce functions as return values:
'''fun''' constantFn k = '''let'''
'''fun''' const anything = k
'''in'''
const
'''end'''
(alternatively)
'''fun''' constantFn k = ('''fn''' anything => k)

Functions can also both consume and produce functions:
'''fun''' compose (f, g) = '''let'''
'''fun''' h x = f (g x)
'''in'''
h
'''end'''
(alternatively)
'''fun''' compose (f, g) = ('''fn''' x => f (g x))

The function <code>List.map</code> from the basis library is one of the most commonly used higher-order functions in Standard ML:
'''fun''' map _ [] = []
<nowiki> |</nowiki> map f (x::xs) = f x<nowiki></nowiki> :: map f xs
(A more efficient implementation of <code>map</code> would define a tail-recursive inner loop as follows:)
'''fun''' map f xs = '''let'''
'''fun''' m ([], acc) = List.rev acc
<nowiki> |</nowiki> m (x::xs, acc) = m (xs, f x<nowiki></nowiki> :: acc)
'''in'''
m (xs, [])
'''end'''

===Exceptions===

Exceptions are raised with the <code>raise</code> keyword, and handled with pattern matching <code>handle</code> constructs.
'''exception''' Undefined
'''fun''' max [x] = x
<nowiki> |</nowiki> max (x::xs) = <nowiki></nowiki>'''let''' '''val''' m = max xs '''in''' '''if''' x > m '''then''' x '''else''' m '''end'''
| max [] = '''raise''' Undefined
'''fun''' main xs = '''let'''
'''val''' msg = (Int.toString (max xs)) '''handle''' Undefined => "empty list...there is no max!"
'''in'''
print (msg ^ "\n")
'''end'''

The exception system can be exploited to implement [[non-local exit]], an optimization technique suitable for functions like the following.
'''exception''' Zero
'''fun''' listProd ns = '''let'''
'''fun''' p [] = 1
<nowiki> | p (0::_) = </nowiki>'''raise''' Zero
<nowiki> |</nowiki> p (h::t) = h * p t<nowiki></nowiki>
'''in'''
(p ns) '''handle''' Zero => 0
'''end'''
When the exception <code>Zero</code> is raised in the 0 case, control leaves the function <code>p</code> altogether. Consider the alternative: the value 0 would be returned to the most recent awaiting frame, it would be multiplied by the local value of <code>h</code>, the resulting value (inevitably 0) would be returned in turn to the next awaiting frame, and so on. The raising of the exception allows control to leapfrog directly over the entire chain of frames and avoid the associated computation. It has to be noted that the same optimization could have been obtained by using a tail recursion for this example.

===Module system===

Standard ML has an advanced [[Module (programming)|module]] system, allowing programs to be decomposed into hierarchically organized ''structures'' of logically related type and value declarations. SML modules provide not only [[namespace]] control but also abstraction, in the sense that they allow programmers to define [[abstract data type]]s.

Three main syntactic constructs comprise the SML module system: signatures, structures and functors. 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 ''signature'' is an [[Interface (computer science)|interface]], usually thought of as a type for a structure: it specifies the names of all the entities provided by the structure as well as the [[arity|arities]] of type components, the types of value components, and signatures for substructures. The definitions of type components may or may not be exported; type components whose definitions are hidden are ''abstract types''. Finally, 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 programming|generic]] data structures and algorithms.

For example, the signature for a [[Queue (data structure)|queue]] data structure might be:

'''signature''' QUEUE =
'''sig'''
'''type''' 'a queue
'''exception''' Queue
'''val''' empty : 'a queue
'''val''' isEmpty : 'a queue -> bool
'''val''' singleton : 'a -> '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 parameterized type <code>queue</code> of queues, an exception called <code>Queue</code>, and six values (five of which are functions) providing the basic operations on queues. One can now implement the queue data structure by writing a structure with this signature:

'''structure''' TwoListQueue :> QUEUE =
'''struct'''
'''type''' 'a queue = 'a list * 'a list
'''exception''' Queue

'''val''' empty = ([],[])

'''fun''' isEmpty ([],[]) = true
<nowiki> |</nowiki> isEmpty _ = false<nowiki></nowiki><nowiki></nowiki>

'''fun''' singleton a = ([], [a])

'''fun''' insert (a, ([], [])) = ([], [a])
<nowiki> |</nowiki> insert (a, (ins, outs)) = (a::ins, outs)<nowiki></nowiki><nowiki></nowiki>

'''fun''' peek (_,[]) = '''raise''' Queue
<nowiki> |</nowiki> peek (ins, a::outs) = a<nowiki></nowiki><nowiki></nowiki>

'''fun''' remove (_,[]) = '''raise''' Queue
<nowiki> |</nowiki> remove (ins, [a]) = (a, ([], rev ins))<nowiki>
|</nowiki> remove (ins, a::outs) = (a, (ins,outs))<nowiki></nowiki><nowiki></nowiki><nowiki></nowiki>

'''end'''

This definition declares that <code>TwoListQueue</code> is an implementation of the <code>QUEUE</code> signature. Furthermore, the ''opaque ascription'' (denoted by <code>:></code>) states that any type components whose definitions are not provided in the signature (''i.e.,'' <code>queue</code>) should be treated as abstract, meaning that the definition of a queue as a pair of lists is not visible outside the module. The body of the structure provides bindings for all of the components listed in the signature.

To use a structure, one can access its type and value members using "dot notation". For instance, a queue of strings would have type <code>string TwoListQueue.queue</code>, the empty queue is <code>TwoListQueue.empty</code>, and to remove the first element from a queue called <code>q</code> one would write <code>TwoListQueue.remove q</code>.

One popular algorithm<ref>{{cite conference
| last = Okasaki
| first = Chris
| title = Breadth-First Numbering: Lessons from a Small Exercise in Algorithm Design
| booktitle = International Conference on Functional Programming 2000
| publisher = ACM
| year = 2000}}</ref> for [[breadth-first traversal]] of trees makes uses of queues. Here we present a version of that algorithm parameterized over an abstract queue structure:
'''functor''' BFT (Q: QUEUE) = (* after Okasaki, ICFP, 2000 *)
'''struct'''
'''datatype''' 'a tree
= E
| T '''of''' 'a * 'a tree * 'a tree
'''fun''' bftQ (q : 'a tree Q.queue) : 'a list =
'''if''' Q.isEmpty q '''then''' []
'''else''' '''let'''
'''val''' (t, q') = Q.remove q
'''in''' '''case''' t
'''of''' E => bftQ q'
<nowiki> |</nowiki> T (x, l, r) => <nowiki></nowiki>'''let'''
'''val''' q<nowiki>''</nowiki> = Q.insert (r, Q.insert (l, q'))
'''in'''
x :: bftQ q<nowiki>''</nowiki>
'''end'''
'''end'''
'''fun''' bft t = bftQ (Q.singleton t)
'''end'''
Please note that inside the <code>BFT</code> structure, the program has no access to the particular queue representation in play. More concretely, there is no way for the program to, say. select the first list in the two-list queue representation, if that is indeed the representation being used. This [[data abstraction]] mechanism makes the breadth-first code truly agnostic to the queue representation choice.
This is in general desirable; in the present case, the queue structure can safely maintain any of the various logical invariants on which its correctness depends behind the bulletproof wall of abstraction.

==Code examples==
{{unreferenced section|date=June 2013}}
Snippets of SML code are most easily studied by entering them into a "top-level", also known as a [[read-eval-print loop]]. This is an interactive session that prints the inferred types of resulting or defined expressions. Many SML implementations provide an interactive top-level, including SML/NJ:

$ sml
Standard ML of New Jersey v110.52 [built: Fri Jan 21 16:42:10 2005]
-

Code can then be entered at the "-" prompt. For example, to calculate 1+2*3:

- 1 + 2 * 3;
val it = 7 : int

The top-level infers the type of the expression to be "int" and gives the result "7".

===Hello world===

The following program "hello.sml":

print "Hello world!\n";

can be compiled with MLton:

$ mlton hello.sml

and executed:

$ ./hello
Hello world!

===Insertion sort===

Insertion sort for lists of integers (ascending) is expressed concisely as follows:
'''fun''' ins (n, []) = [n]
<nowiki> | ins (n, ns </nowiki>'''as''' h::t) = '''if''' (n<h) '''then''' n::ns '''else''' h::(ins (n, t))
'''val''' insertionSort = List.foldr ins []

This can be made polymorphic by abstracting over the ordering operator. Here we use the symbolic name <code><<</code> for that operator.
'''fun''' ins' << (num, nums) = '''let'''
'''fun''' i (n, []) = [n]
<nowiki> | i (n, ns </nowiki>'''as''' h::t) = '''if''' <<(n,h) '''then''' n::ns '''else''' h::i(n,t)
'''in'''
i (num, nums)
'''end'''
'''fun''' insertionSort' << = List.foldr (ins' <<) []
The type of <code>insertionSort'</code> is <code>('a * 'a -> bool) -> ('a list) -> ('a list)</code>.

===Mergesort===

{{main|Merge sort}}

Here, the classic mergesort algorithm is implemented in three functions: split, merge and mergesort.

The function <code>split</code> is implemented with a local function named <code>loop</code>, which has two additional parameters. The local function <code>loop</code> is written in a [[tail recursion|tail-recursive]] style; as such it can be compiled efficiently. This function makes use of SML's pattern matching syntax to differentiate between non-empty list (<code>x::xs</code>) and empty list (<code>[]</code>) cases. For stability, the input list <code>ns</code> is reversed before being passed to <code>loop</code>.

''(* Split list into two near-halves, returned as a pair.''
'' * The “halves” will either be the same size,''
'' * or the first will have one more element than the second.''
'' * Runs in O(n) time, where n = |xs|. *)''
'''local'''
'''fun''' loop (x::y::zs, xs, ys) = loop (zs, x::xs, y::ys)
| loop (x::[], xs, ys) = (x::xs, ys)
| loop ([], xs, ys) = (xs, ys)
'''in'''
'''fun''' split ns = loop (List.rev ns, [], [])
'''end'''

The local-in-end syntax could be replaced with a let-in-end syntax, yielding the equivalent definition:

'''fun''' split ns = '''let'''
'''fun''' loop (x::y::zs, xs, ys) = loop (zs, x::xs, y::ys)
| loop (x::[], xs, ys) = (x::xs, ys)
| loop ([], xs, ys) = (xs, ys)
'''in'''
loop (List.rev ns, [], [])
'''end'''

As with split, merge also uses a local function loop for efficiency. The inner <code>loop</code> is defined in terms of cases: when two non-empty lists are passed, when one non-empty list is passed, and when two empty lists are passed. Note the use of the underscore (<code>_</code>) as a wildcard pattern.

This function merges two "ascending" lists into one ascending list. Note how the accumulator <code>out</code> is built "backwards", then reversed with <code>List.rev</code> before being returned. This is a common technique—build a list backwards, then reverse it before returning it. In SML, lists are represented as imbalanced binary trees, and thus it is efficient to prepend an element to a list, but inefficient to append an element to a list. The extra pass over the list is a [[linear time]] operation, so while this technique requires more wall clock time, the asymptotics are not any worse.

''(* Merge two ordered lists using the order lt.''
'' * Pre: the given lists xs and ys must already be ordered per lt.''
'' * Runs in O(n) time, where n = |xs| + |ys|. *)''
'''fun''' merge lt (xs, ys) = '''let'''
'''fun''' loop (out, left '''as''' x::xs, right '''as''' y::ys) =
'''if''' lt (x, y) '''then''' loop (x::out, xs, right)
'''else''' loop (y::out, left, ys)
| loop (out, x::xs, []) = loop (x::out, xs, [])
| loop (out, [], y::ys) = loop (y::out, [], ys)
| loop (out, [], []) = List.rev out
'''in'''
loop ([], xs, ys)
'''end'''

The main function.

''(* Sort a list in according to the given ordering operation lt.''
'' * Runs in O(n log n) time, where n = |xs|.''
'' *)''
'''fun''' mergesort lt xs = '''let'''
'''val''' merge' = merge lt
'''fun''' ms [] = []
| ms [x] = [x]
| ms xs = '''let'''
'''val''' (left, right) = split xs
'''in'''
merge' (ms left, ms right)
'''end'''
'''in'''
ms xs
'''end'''

Also note that the code makes no mention of variable types, with the exception of the :: and [] syntax which signify lists. This code will sort lists of any type, so long as a consistent ordering function lt can be defined. Using [[Hindley–Milner type inference]], the compiler is capable of inferring the types of all variables, even complicated types such as that of the lt function.

===Quicksort===

Quicksort can be expressed as follows. This generic quicksort consumes an order operator <code><<</code>.
'''fun''' quicksort << xs = '''let'''
'''fun''' qs [] = []
<nowiki> | qs [x] = [x]
| qs (p::xs) = </nowiki>'''let'''
'''val''' (less, more) = List.partition ('''fn''' x => << (x, p)) xs
'''in'''
qs less @ p :: qs more
'''end'''
'''in'''
qs xs
'''end'''

===Expression language===

Note the relative ease with which a small expression language is defined and processed.

'''exception''' Err

'''datatype''' ty
= IntTy
| BoolTy

'''datatype''' exp
= True
| False
| Int '''of''' int
<nowiki> | Not </nowiki>'''of''' exp
<nowiki> | Add </nowiki>'''of''' exp * exp
<nowiki> | If </nowiki>'''of''' exp * exp * exp

'''fun''' typeOf (True) = BoolTy
<nowiki> |</nowiki> typeOf (False) = BoolTy<nowiki>
|</nowiki> typeOf (Int _) = IntTy<nowiki>
|</nowiki> typeOf (Not e) = <nowiki></nowiki>'''if''' typeOf e = BoolTy '''then''' BoolTy '''else''' '''raise''' Err
<nowiki> |</nowiki> typeOf (Add (e1, e2)) = <nowiki></nowiki>
'''if''' (typeOf e1 = IntTy) '''andalso''' (typeOf e2 = IntTy) '''then''' IntTy '''else''' '''raise''' Err
<nowiki> |</nowiki> typeOf (If (e1, e2, e3)) = <nowiki></nowiki>
'''if''' typeOf e1 <> BoolTy '''then''' '''raise''' Err
'''else''' '''if''' typeOf e2 <> typeOf e3 '''then''' '''raise''' Err
'''else''' typeOf e2

'''fun''' eval (True) = True
<nowiki> |</nowiki> eval (False) = False<nowiki>
|</nowiki> eval (Int n) = Int n<nowiki>
|</nowiki> eval (Not e) =
<nowiki> (</nowiki>'''case''' eval e
'''of''' True => False
<nowiki> | False => True
| _ => </nowiki>'''raise''' Fail "type-checking is broken")
<nowiki> |</nowiki> eval (Add (e1, e2)) = <nowiki></nowiki>'''let'''
'''val''' (Int n1) = eval e1
'''val''' (Int n2) = eval e2
'''in'''
Int (n1 + n2)
'''end'''
| eval (If (e1, e2, e3)) =
'''if''' eval e1 = True '''then''' eval e2 '''else''' eval e3

'''fun''' chkEval e = (ignore (typeOf e); eval e) ''(* will raise Err on type error *)''

===Arbitrary-precision factorial function (libraries)===

In SML, 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 "fact.sml" implements an arbitrary-precision factorial function and prints the factorial of 120:

'''fun''' fact n : IntInf.int =
'''if''' n=0 '''then''' 1 '''else''' n * fact(n - 1)

'''val''' () =
print (IntInf.toString (fact 120) ^ "\n")

and can be compiled and run with:

$ mlton fact.sml
$ ./fact
66895029134491270575881180540903725867527463331380298102956713523016335
57244962989366874165271984981308157637893214090552534408589408121859898
481114389650005964960521256960000000000000000000000000000

===Numerical derivative (higher-order functions)===

Since SML is a functional programming language, it is easy to create and pass around functions in SML programs. This capability has an enormous number of applications. Calculating the numerical derivative of a function is one such application. The following SML function "d" computes the numerical derivative of a given function "f" at a given point "x":

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

This function requires a small value "delta". A good choice for delta when using this algorithm is the cube root of the [[machine epsilon]].{{Citation needed|date=August 2008}}

The type of the function "d" indicates that it maps a "float" onto another function with the type "(real -> real) -> real -> real". This allows us to partially apply arguments. This functional style is known as [[currying]]. In this case, it is useful to partially apply the first argument "delta" to "d", to obtain a more specialised function:

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

Note that the inferred type indicates that the replacement "d" is expecting a function with the type "real -> real" as its first argument. We can compute a numerical approximation to the derivative of <math>f(x) = x^3-x-1</math> at <math>x=3</math> with:

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

The correct answer is <math>f'(x) = 3x^2-1 </math>=><math> f'(3) = 27-1 = 26</math>.

The function "d" is called a "higher-order function" because it accepts another function ("f") as an argument.

Curried and higher-order functions can be used to eliminate redundant code. For example, a library may require functions of type <code>a -> b</code>, but it is more convenient to write functions of type <code>a * c -> b</code> where there is a fixed relationship between the objects of type <code>a</code> and <code>c</code>. A higher order function of type (a * c -> b) -> (a -> b) can factor out this commonality. This is an example of the [[adapter pattern]].

===Discrete wavelet transform (pattern matching)===

The 1D [[Haar wavelet]] [[Discrete wavelet transform|transform]] of an [[integer]]-power-of-two-length list of numbers can be implemented very succinctly in SML and is an excellent example of the use of [[pattern matching]] over lists, taking pairs of elements ("h1" and "h2") off the front and storing their sums and differences on the lists "s" and "d", respectively:

- fun haar l = let
fun aux [s] [] d = s :: d
<nowiki> |</nowiki> aux [] s d = aux s [] d<nowiki>
|</nowiki> aux (h1::h2::t) s d = aux t (h1+h2<nowiki></nowiki> :: s) (h1-h2 :: d)
<nowiki> |</nowiki> aux _ _ _ = raise Empty
<nowiki> in
</nowiki> aux l [] []
<nowiki> end;
val haar = fn</nowiki> : int list -> int list

For example:

- haar [1, 2, 3, 4, ~4, ~3, ~2, ~1];
val it = [0,20,4,4,~1,~1,~1,~1] : int list

Pattern matching is a useful construct that allows complicated transformations to be represented clearly and succinctly. Moreover, SML compilers turn pattern matches into efficient code, resulting in programs that are not only shorter but also faster.

==Implementations==

Many SML implementations exist, including:

* [[MLton]] is a [[Whole program optimization|whole-program optimizing]] compiler that produces very fast code compared to other ML implementations. [http://www.mlton.org]
* [http://www.polyml.org/ Poly/ML] is a full implementation of Standard ML that produces fast code and supports multicore hardware (via Posix threads); its runtime system performs parallel garbage collection and online sharing of immutable substructures.
* [http://isabelle.in.tum.de Isabelle/ML] integrates parallel Poly/ML into an interactive theorem prover, with a sophisticated IDE (based on [[jEdit]]) both for ML and the proof language.
* [[Standard ML of New Jersey]] (abbreviated SML/NJ) is a full compiler, with associated libraries, tools, an interactive shell, and documentation. [http://www.smlnj.org/]
* [[Moscow ML]] is a light-weight implementation, based on the [[Caml|CAML Light]] runtime engine. It implements the full SML language, including SML Modules, and much of the SML Basis Library. [http://www.itu.dk/people/sestoft/mosml.html]
* [http://www.cs.cornell.edu/home/jgm/tilt.html TILT] is a full certifying compiler for SML. It uses typed intermediate languages to optimize code and ensure correctness, and can compile to [[Type system|typed]] [[Assembly language]].
* [http://www.mpi-sws.org/~rossberg/hamlet/ HaMLet] is an SML interpreter that aims to be an accurate and accessible reference implementation of the standard.
* The [http://www.it-c.dk/research/mlkit/ ML Kit] integrates a garbage collector (which can be disabled) and [[region-based memory management]] with automatic inference of regions, aiming realtime applications. Its implementation is based very closely on the Definition.
* [http://www.cl.cam.ac.uk/Research/TSG/SMLNET/ SML.NET] allows compiling to the Microsoft [[Common Language Runtime|CLR]] and has extensions for linking with other [[.NET Framework|.NET]] code.
* SML2c is a batch compiler and compiles only module-level declarations (i.e. signatures, structures, functors) into [[C (programming language)|C]]. It is based on SML/NJ version 0.67 and shares the 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 with no changes.
* The [[Poplog]] system implements a version of SML, with [[POP-11]], and optionally [[Common Lisp]], and [[Prolog]], allowing mixed language programming. For all, the implementation language is POP-11, which is compiled incrementally. It also has an integrated [[Emacs]]-like editor that communicates with the compiler.
* [http://www.pllab.riec.tohoku.ac.jp/smlsharp/ SML#] is 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.
* [[Alice (programming language)|Alice]]: an interpreter for Standard ML by Saarland University adding features for [[lazy evaluation]], [[Concurrency (computer science)|concurrency]] ([[thread (computer science)|multithreading]] and [[distributed computing]] via [[remote procedure call]]s) and [[constraint programming]].
All of these implementations are [[open-source]] and freely available. Most are implemented themselves in SML. There are no longer any commercial SML implementations. [[Harlequin (software company)|Harlequin]] once produced a commercial IDE and compiler for SML called [[MLWorks]]. The company is now defunct. [[MLWorks]] passed on to [[Xanalys]] and was later acquired by [[Ravenbrook Limited]] on 2013-04-26 and open sourced.

==See also==
* [[Alice (programming language)|Alice]]
* [[ML programming language|ML]]
* [[Concurrent ML]]
* [[Dependent ML]]
* [[Extensible ML]]
* [[Extended ML]]
* [[F Sharp (programming language)|F#]]
* [[OCaml]]
* [http://www.impredicative.com/ur/ Ur/Web]
* [http://mythryl.org Mythryl]

==References==

{{reflist}}

==External links==
* [http://www.smlnj.org/sml.html What is SML?]
* [http://www.smlnj.org/sml97.html What is SML '97?]
* [http://www.successor-ml.org successor ML (sML)] is intended to provide a vehicle for the continued evolution of ML, using Standard ML as a starting point.
* {{scholarpedia|title=Standard ML language|urlname=Standard_ML_language|curator=[[Mads Tofte]]}}
* [http://www.classes.cs.uchicago.edu/archive/2007/winter/22610-1/docs/sml-tutorial.pdf Univ. of Chicago - SML tutorial (slides)]
* [http://www.cs.cmu.edu/~rwh/smlbook/book.pdf Carnegie Mellon Univ. - The Book of SML]
* [http://www.cs.cmu.edu/~rwh/smlbook/examples/ Carnegie Mellon Univ. - SML - Examples]
* [http://www.lfcs.inf.ed.ac.uk/reports/97/ECS-LFCS-97-364/ECS-LFCS-97-364.pdf University of Edinburgh - Programming in Standard ML '97: A Tutorial Introduction (pdf)]
* [http://www.dcs.ed.ac.uk/home/stg/NOTES/notes.html University of Edinburgh - SML '97 - Online Tutorial]

[[Category:Procedural programming languages]]
[[Category:ML programming language family]]
[[Category:Functional languages]]
[[Category:Programming languages created in 1990]]

Gustafson's law

2014-01-23T22:08:13Z

89.204.138.35: /* External links */ replaced Category:Rules of thumb by Category:Programming rules of thumb with related Amdahl's law

{| border="1" cellpadding="5" cellspacing="0" align="right"
! style="background:#ffdead;" | [[Feynman diagrams]]
|-
| align="center" | '''t-channel''' [[Image:MollerScattering-t.svg|220px]]
|-
| align="center" | '''u-channel''' [[Image:MollerScattering-u.svg|220px]]
|}
'''Møller scattering''' is the name given to [[electron]]-electron scattering in [[Quantum Field Theory]], named after the Danish physicist [[Christian Møller]]. The electron interaction that is idealized in Møller scattering forms the theoretical basis of many familiar phenomena such as the repulsion of electrons in the Helium nucleus. While formerly many particle colliders were designed specifically for electron-electron collisions, more recently electron-positron colliders have become more common. Nevertheless Møller scattering remains a paradigmatic process within the theory of particle interactions.

We can express this process in the usual notation, often used in [[particle physics]]:

:<math>
e^{-} e^{-} \longrightarrow e^{-} e^{-}
</math>,

In [[quantum electrodynamics]], there are two tree-level [[Feynman diagrams]] describing the process: a [[Mandelstam variables|t-channel]] diagram in which the electrons exchange a [[photon]] and a similar u-channel diagram. [[Crossing symmetry]], one of the tricks often used to evaluate Feynman diagrams, in this case implies that Møller scattering should have the same cross section as [[Bhabha scattering]] (electron-[[positron]] scattering).


In the electroweak theory the process is instead described by four tree-level diagrams: the two from QED and an identical pair in which a [[Z boson]] is exchanged instead of a photon. The weak force is purely left-handed, but the weak and electromagnetic forces mix into the particles we observe. The photon is symmetric by construction, but the Z boson prefers left-handed particles to right-handed particles. Thus the cross sections for left-handed electrons and right-handed differ. The difference was first noticed by the Russian physicist [[Yakov Zel'dovich]] in 1959, but at the time he believed the [[parity (physics)|parity]] violating asymmetry (a few hundred parts per billion) was too small to be observed. This parity violating asymmetry can be measured by firing a polarized beam of electrons through an unpolarized electron target ([[liquid hydrogen]], for instance), as was done by an experiment at the [[Stanford Linear Accelerator Center]], SLAC-E158.<ref>{{cite journal
| title = Precision Measurement of the Weak Mixing Angle in Møller Scattering
| author = Anthony, P. L. and others
| collaboration = SLAC E158 Collaboration
| journal = Phys. Rev. Lett.
| volume = 95
| issue = 8
| pages = 081601
| numpages = 5
|date=Aug 2005
| doi = 10.1103/PhysRevLett.95.081601
| url = http://link.aps.org/doi/10.1103/PhysRevLett.95.081601
| publisher = American Physical Society|arxiv = hep-ex/0504049 |bibcode = 2005PhRvL..95h1601A }}</ref> The asymmetry in Møller scattering is

:<math>
A_{PV}=-m E \frac{G_F}{ \sqrt{2} \pi \alpha } \frac {16 \sin^2 \Theta_{\textrm{cm}}}
{\left(3+\cos^2 \Theta_{\textrm{cm}} \right)^2 } \left( \frac{1}{4} - \sin^2 \theta_W \right)
</math>,

where m is the electron mass, E the energy of the incoming electron (in the reference frame of the other electron), <math>G_F</math> is [[Fermi's interaction|Fermi's constant]], <math>\alpha</math> is the [[fine structure constant]], <math>\Theta_{\textrm{cm}}</math> is the scattering angle in the center of mass frame, and <math>\theta_W</math> is the weak mixing angle, also known as the [[Weinberg angle]].

== QED computation ==




{{Empty section|date=July 2010}}

==References==
{{reflist}}

==External links==
*[http://www-project.slac.stanford.edu/e158/ SLAC E158: Measuring the Electron's WEAK Charge]

{{QED}}

{{DEFAULTSORT:Moller Scattering}}
[[Category:Quantum field theory]]
[[Category:Quantum electrodynamics]]

PAQ

2013-12-28T13:09:37Z

89.204.139.20: /* PAQ derivations */ cite web error (missing url=)

In [[number theory]], the '''Hasse norm theorem''' states that if L/K is a [[cyclic extension]] of [[number field]]s, then if a nonzero element of K is a local norm everywhere, then it is a global norm.
Here to be a global norm means to be an element ''k'' of K such that there is an element ''l'' of L with <math>\mathbf{N}_{L/K}(l) = k</math>; in other words ''k'' is a relative norm of some element of the extension field L. To be a local norm means that for some prime '''p''' of K and some prime '''P''' of L lying over K, then ''k'' is a norm from L'''P'''; here the "prime" '''p''' can be an archimedean valuation, and the theorem is a statement about completions in all valuations, archimedean and non-archimedean.

The theorem is no longer true in general if the extension is abelian but not cyclic. A counter-example is given by the field <math>{\mathbf Q}(\sqrt{13},\sqrt{17})/{\mathbf Q}</math> where every rational square is a local norm everywhere but <math>5^2</math> is not a global norm.

This is an example of a theorem stating a [[local-global principle]], and is due to [[Helmut Hasse]].

The Hasse norm theorem can be deduced from the theorem that an element of the Galois cohomology group H2(''L''/''K'') is trivial if it is trivial locally everywhere, which is in turn equivalent to the deep theorem that the first cohomology of the idele class group vanishes. This is true for all finite Galois extensions of number fields, not just cyclic ones. For cyclic extensions the group H2(''L''/''K'') is isomorphic to the Tate cohomology group H0(''L''/''K'') which describes which elements are norms, so for cyclic extensions it becomes Hasse's theorem that an element is a norm if it is a local norm everywhere.

==See also==

*[[Grunwald–Wang theorem]], about when an element that is a power everywhere locally is a power.

==References==

* H. Hasse, "A history of class field theory", in [[J.W.S. Cassels]] and [[A. Frohlich]] (edd), ''Algebraic number theory'', [[Academic Press]], 1973. Chap.XI.
* G. Janusz, ''Algebraic number fields'', Academic Press, 1973. Theorem V.4.5, p. 156

[[Category:Class field theory]]
[[Category:Theorems in algebraic number theory]]

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2012-07-13T21:58:27Z

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