a fourth look at ml

A Fourth Look At MLA Fourth Look At MLChapter 11Type-Safe Data Structures

Chapter Eleven Modern Programming Languages 1

Type DefinitionsType DefinitionsPredefined, but not primitive in

Type constructor for lists:

Defined for ML in ML

datatype bool = true | false;

datatype 'element list = nil | :: of 'element * 'element list

OutlineOutlineEnumerationsData constructors with

parametersType constructors with

parametersRecursively defined type

constructorsFarewell to ML

◦We hardly knew ye!

Defining Your Own TypesDefining Your Own TypesNew types can be defined using

the keyword datatypeThese declarations define both:

◦type constructors for making new (possibly polymorphic) types

◦data constructors for making values of those new types

ExampleExample

day is the new type constructor and Mon, Tue, etc. are the new data constructors

Why “constructors”? In a moment we will see how both can have parameters…

- datatype day = Mon | Tue | Wed | Thu | Fri | Sat | Sun;datatype day = Fri | Mon | Sat | Sun | Thu | Tue | Wed- fun isWeekDay x = not (x = Sat orelse x = Sun);val isWeekDay = fn : day -> bool- isWeekDay Mon;val it = true : bool- isWeekDay Sat;val it = false : bool

No ParametersNo Parameters

The type constructor day takes no parameters: it is not polymorphic, there is only one day type

The data constructors Mon, Tue, etc. take no parameters: they are constant values of the day type

Capitalize the names of data constructors

- datatype day = Mon | Tue | Wed | Thu | Fri | Sat | Sun;datatype day = Fri | Mon | Sat | Sun | Thu | Tue | Wed

Strict TypingStrict Typing

ML is strict about these new types, just as you would expect

Unlike C enum, no implementation details are exposed to the programmer

- datatype flip = Heads | Tails;datatype flip = Heads | Tails- fun isHeads x = (x = Heads);val isHeads = fn : flip -> bool- isHeads Tails;val it = false : bool- isHeads Mon;Error: operator and operand don't agree [tycon mismatch] operator domain: flip operand: day

Data Constructors In Data Constructors In PatternsPatterns

You can use the data constructors in patterns

In this simple case, they are like constantsBut we will see more general cases next…

fun isWeekDay Sat = false| isWeekDay Sun = false| isWeekDay _ = true;

The The orderorder Datatype DatatypeUsed with pre-defined compare

functions◦Int.compare( )◦Real, String, Char, Time, Date, …

datatype order = LESS | GREATER | EQUAL;

- Real.compare(1.0,2.0);

val it = LESS : order- Char.compare(#"b", #"a");

val it = GREATER : order

WrappersWrappersYou can add a parameter of any type to

a data constructor, using the keyword of:datatype exint = Value of int | PlusInf | MinusInf;

In effect, such a constructor is a wrapper that contains a data item of the given type

PlusInfValue

MinusInf Value26

Value38

Some thingsof type exint:

Value is a data constructor that takes a parameter: the value of the int to store

It looks like a function that takes an int and returns an exint containing that int

- datatype exint = Value of int | PlusInf | MinusInf;datatype exint = MinusInf | PlusInf | Value of int- PlusInf;val it = PlusInf : exint- MinusInf;val it = MinusInf : exint- Value;val it = fn : int -> exint (* Function notation! *)- Value 3;val it = Value 3 : exint

A A ValueValue Is Not An Is Not An intint

Value 5 is an exintIt is not an int, though it contains

oneHow can we get the int out again?By pattern matching…

- val x = Value 5;val x = Value 5 : exint- x+x;Error: overloaded variable not defined at type symbol: + type: exint

Patterns With Data Patterns With Data ConstructorsConstructors

To recover a data constructor’s parameters, use pattern matching

So Value is no ordinary function: ordinary functions can't be pattern-matched this way

- val (Value y) = x;Warning: binding not exhaustive Value y = ...val y = 5 : int

An Exhaustive PatternAn Exhaustive Pattern

Like most uses of the match construct, you get a warning if it is not exhaustive

An exint can be a PlusInf, a MinusInf, or a Value

This one says what to do in all cases

- val s = case x of = PlusInf => "infinity" |= MinusInf => "-infinity" | = Value y => Int.toString y;val s = "5" : string

Pattern-Matching FunctionPattern-Matching Function

Pattern-matching function definitions are especially important when working with your own datatypes

- fun square PlusInf = PlusInf= | square MinusInf = PlusInf= | square (Value x) = Value (x*x);val square = fn : exint -> exint- square MinusInf;val it = PlusInf : exint- square (Value 3);val it = Value 9 : exint

Exception Handling (A Exception Handling (A Peek)Peek)

Patterns are also used in ML for exception handling, as in this example

- fun square PlusInf = PlusInf= | square MinusInf = PlusInf= | square (Value x) = Value (x*x)= handle Overflow => PlusInf;val square = fn : exint -> exint- square (Value 10000);val it = Value 100000000 : exint- square (Value 100000);val it = PlusInf : exint

Partial FunctionsPartial FunctionsA function that is not defined for

all values of its domain type is called a partial function

Examples:◦1/x; functions that expect a non-null

pointerIn ML, the option data type

indicates undefined valuesdatatype ‘a option = NONE |

SOME of ‘a;Chapter Eleven Modern Programming Languages 19

Type Constructors With Type Constructors With ParametersParameters

Type constructors can also use parameters: datatype 'a option = NONE | SOME of 'a;

The parameters of a type constructor are type variables, which are used in the data constructors

The result: a new polymorphic type

SOME"Hello"

Values of typestring option

SOME"world"

SOME1.5

Values of type real option

SOME123.4

Parameter Before NameParameter Before Name

Type constructor parameter comes before the type constructor name:datatype 'a option = NONE | SOME of 'a;

We have types 'a option and int option, just like 'a list and int list

- SOME 4;val it = SOME 4 : int option- SOME 1.2;val it = SOME 1.2 : real option - SOME "pig";val it = SOME "pig" : string option

Uses For Uses For optionoptionPredefined type constructor in MLUsed by predefined functions (or your

own) when the result is not always defined

- fun optdiv a b == if b = 0 then NONE else SOME (a div b);val optdiv = fn : int -> int -> int option- optdiv 7 2;val it = SOME 3 : int option- optdiv 7 0;val it = NONE : int option

Working with Working with optionoptionRetrieve the value of an option,

other than NONE, with valOfProvide a default value for NONE

with getOptval x = SOME 2 : int option

- valOf x;

val it = 2 : int

- getOpt (x,0);

val it = 2 : int

- getOpt (NONE, 0);

val it = 0 : int

See fact.smlChapter Eleven Modern Programming Languages 23

Longer Example: Longer Example: bunchbunch

An 'x bunch is either a thing of type 'x, or a list of things of type 'x

As usual, ML infers types:

datatype 'x bunch = One of 'x | Group of 'x list;

- One 1.0;val it = One 1.0 : real bunch- Group [true,false];val it = Group [true,false] : bool bunch

Example: PolymorphismExample: Polymorphism

ML can infer bunch types, but does not always have to resolve them, just as with list types

- fun size (One _) = 1= | size (Group x) = length x;val size = fn : 'a bunch -> int- size (One 1.0);val it = 1 : int- size (Group [true,false]);val it = 2 : int

Example: No PolymorphismExample: No Polymorphism

We applied the + operator (through foldr) to the list elements

So ML knows the parameter type must be int bunch

- fun sum (One x) = x= | sum (Group xlist) = foldr (op +) 0 xlist;val sum = fn : int bunch -> int- sum (One 5);val it = 5 : int- sum (Group [1,2,3]);val it = 6 : int

OutlineOutline

EnumerationsData constructors with parametersType constructors with parametersRecursively defined type

Recursively Defined Type Recursively Defined Type ConstructorsConstructorsThe type constructor being defined may

be used in its own data constructors: datatype intlist = INTNIL | INTCONS of int * intlist;

Some values oftype intlist:

INTNIL

INTCONS

the empty list

INTNIL 1

the list [1]

INTCONS INTNIL 2

INTCONS

the list [1,2]

Constructing Those ValuesConstructing Those Values

- INTNIL;val it = INTNIL : intlist- INTCONS (1,INTNIL);val it = INTCONS (1,INTNIL) : intlist- INTCONS (1,INTCONS(2,INTNIL));val it = INTCONS (1,INTCONS (2,INTNIL)) : intlist

INTNIL

INTCONS

the empty list

INTNIL 1

the list [1]

INTCONS INTNIL 2

INTCONS

the list [1,2]

An An intlistintlist Length Length FunctionFunction

A length functionMuch like you would write for native

listsExcept, of course, that native lists are

not always lists of integers…

fun intlistLength INTNIL = 0 | intlistLength (INTCONS(_,tail)) = 1 + (intListLength tail);

fun listLength nil = 0 | listLength (_::tail) = 1 + (listLength tail);

Parametric List TypeParametric List Type

A parametric list type, almost like the predefined list

ML handles type inference in the usual way:

datatype 'element mylist = NIL | CONS of 'element * 'element mylist;

- CONS(1.0, NIL);val it = CONS (1.0,NIL) : real mylist- CONS(1, CONS(2, NIL));val it = CONS (1,CONS (2,NIL)) : int mylist

Some Some mylistmylist Functions Functions

This now works almost exactly like the predefined list type constructor

Of course, to add up a list you would use foldr…

fun myListLength NIL = 0 | myListLength (CONS(_,tail)) = 1 + myListLength(tail);

fun addup NIL = 0 | addup (CONS(head,tail)) = head + addup tail;

A A foldrfoldr For For mylistmylist

Definition of a function like foldr that works on 'a mylist

Can now add up an int mylist x with: myfoldr (op +) 0 x

One remaining difference: :: is an operator and CONS is not

fun myfoldr f c NIL = c | myfoldr f c (CONS(a,b)) = f(a, myfoldr f c b);

Defining OperatorsDefining Operators

ML allows new operators to be definedLike this:

- infixr 5 CONS;infixr 5 CONS- 1 CONS 2 CONS NIL;val it = 1 CONS 2 CONS NIL : int mylist

Chinese BoxesChinese BoxesNested, square boxes

◦like Russian DollsEach box has a size (length of

side) and a color

datatype color = Red | Green | Blue;datatype 'a cbox = Contents of 'a option | Trio of int * color * 'a cbox;

Chinese Box FunctionsChinese Box FunctionsSee cbox.smlSee cbox.sml

makebox: int * color * 'a cbox -> 'a cbox

boxcount: 'a cbox -> inttracebox: 'a cbox -> unit

◦ prints the box attributes outside-inopenbox: 'a cbox -> 'a option

◦ returns the inner contentsinsert: int * color * 'a cbox -> 'a cbox

makebox2: (int * color) list * 'a option -> 'a cbox

One More FunctionOne More Functiondifflist = fn : 'a cbox -> int list

- difflist b;val it = [1,1,1,1] : int list

fun difflist1 (Trio (s, _, Trio(s2,c,box))) = (s-s2)::difflist1 (Trio(s2,c,box))| difflist1 _ = nil;

Layered Patterns with Layered Patterns with asas

fun difflist (Trio (s,_,b as Trio(s2,_,_))) = (s-s2)::difflist b| difflist _ = nil;

Polymorphic Binary TreePolymorphic Binary Treedatatype 'data tree = Empty | Node of 'data tree * 'data * 'data tree;

Some values oftype int tree:

the empty tree

the tree 2

Empty Empty

Node 3 Empty Empty

Node 1

the tree 2

Empty Empty 2

Constructing Those ValuesConstructing Those Values

- val treeEmpty = Empty;val treeEmpty = Empty : 'a tree- val tree2 = Node(Empty,2,Empty);val tree2 = Node (Empty,2,Empty) : int tree- val tree123 = Node(Node(Empty,1,Empty),= 2,= Node(Empty,3,Empty));

Increment All ElementsIncrement All Elements

fun incall Empty = Empty | incall (Node(x,y,z)) = Node(incall x, y+1, incall z);

- incall tree123;val it = Node (Node (Empty,2,Empty), 3, Node (Empty,4,Empty)) : int tree

Add Up The ElementsAdd Up The Elements

fun sumall Empty = 0 | sumall (Node(x,y,z)) = sumall x + y + sumall z;

- sumall tree123;val it = 6 : int

Convert To List Convert To List (Polymorphic)(Polymorphic)

(* In-order traversal *)

fun listall Empty = nil | listall (Node(x,y,z)) = listall x @ y :: listall z;

- listall tree123;val it = [1,2,3] : int list

Tree SearchTree Search

fun isintree x Empty = false | isintree x (Node(left,y,right)) = x=y orelse isintree x left orelse isintree x right;

- isintree 4 tree123;val it = false : bool- isintree 3 tree123;val it = true : bool

Infinite ListsInfinite Listsaka aka ““StreamsStreams””

It is possible, and often useful, to define infinite lists

The elements are computed on demand

Allows a high degree of separation between software components (functions)

The The unitunit Data Type Data TypeLike void in C++

◦a “nothing” return type, or◦an “empty” parameter list

Denoted by ( ) in ML

- print "hello\n";helloval it = () : unit- fun f () = "goodbye";val f = fn : unit -> string- f();val it = "goodbye" : string

Nodes for Infinite Linked Nodes for Infinite Linked ListsListsRecall the list type definition:

The first node slot is the dataThe second is a “pointer” to the

tailFor infinite lists, we will delay the

evaluation of the tail:

datatype 'element list = nil | :: of 'element * 'element list

datatype 'a stream = Nil | Cons of 'a * (unit -> 'a stream);

An Infinite Sequence of An Infinite Sequence of IntegersIntegers

- fun intsfrom k = Cons(k,fn()=>intsfrom(k+1));val intsfrom = fn : int -> int stream- val nums = intsfrom 5;val nums = Cons (5,fn) : int stream- val Cons(head,_) = nums;val head = 5 : int- val Cons(_,f) = nums; val f = fn : unit -> int stream- val rest = f();val rest = Cons (6,fn) : int stream- val Cons(head,_) = rest;val head = 6 : int

Head and Tail for StreamsHead and Tail for Streams

fun head Nil = raise Empty| head (Cons(h,_)) = h;

fun thunk Nil = raise Empty| thunk (Cons(_,t)) = t;

fun force f = f ();

fun tail s = force (thunk s);(* Like the real list type, just gives the next Cons *)

Traversing the StreamTraversing the Stream

- nums;val it = Cons (5,fn) : int stream- head nums;val it = 5 : int- thunk nums;val it = fn : unit -> int stream- tail nums;val it = Cons (6,fn) : int stream- val rest = tail nums;val rest = Cons (6,fn) : int stream- head rest;val it = 6 : int- tail rest;val it = Cons (7,fn) : int stream

Other Stream FunctionsOther Stream Functions

fun next Nil = raise Empty| next c = head (tail c);

fun nth 0 s = head s| nth n s = nth (n-1) (tail s);

fun filter _ Nil = Nil| filter f (Cons (h,t)) = if (f h) then Cons (h, fn () => filter f (force t)) else filter f (force t);

fun map_ _ Nil = Nil| map_ f (Cons(h,t)) = Cons(f h, fn() => map_ f (force t));

- nums;val it = Cons (5,fn) : int stream- next nums;val it = 6 : int- nth 5 nums;val it = 10 : int- val evens = filter (fn x => x mod 2 = 0) nums;val evens = Cons (6,fn) : int stream- head evens;val it = 6 : int- next evens;val it = 8 : int- val iseven = map (fn x => x mod 2 = 0) nums;val iseven = Cons (false,fn) : bool stream- nth 0 iseven;val it = false : bool- nth 1 iseven;val it = true : bool

More ExamplesMore Examples

- val takeeven = filter (fn x => x mod 2 = 0);val takeeven = fn : int stream -> int stream- val evens = takeeven (intsfrom 1);val evens = Cons (2,fn) : int stream- printStrm 10 evens;2468101214161820

A A printStrmprintStrm Function for Function for intsints

fun printStrm _ Nil = ()| printStrm 0 _ = ()| printStrm n (Cons(x,rest)) = ( print(Int.toString x ^ "\n"); printStrm (n-1) (force rest) );

val printStrm = fn : int -> int stream -> unit

Obtaining Obtaining ““AllAll”” Prime Prime NumbersNumbers

- val primes = filter isPrime (intsfrom 2);val primes = Cons (2,fn) : int stream- strm2list 1000 primes;val it = [2,3,5,7,11,13,17,19,23,29,31,37,...] : int list- nth 1000 primes;val it = 7927 : int- nth 10000 primes;val it = 104743 : int-nth 100000 primes; val it = 1299721 : int (* took 1 second *)- nth 1000000 primes;val it = 15485867 : int (* took 21 seconds *)

Implementing Implementing foldlfoldlRemember foldl f c xs =

f(xn,f(xn-1,…,f(x2,f(x1,c))…))

fun foldl _ c nil = c

| foldl f c (x::xs) = foldl f (f (x,c)) xs;

foldl for Streamsfoldl for Streams

fun foldl_ _ c Nil = Cons(c, fn()=>Nil)| foldl_ f c (Cons(a,rest)) = let val v = f (a,c) in Cons(v, fn () => foldl_ f v (force rest)) end;

More ExamplesMore ExamplesComposing Even More StreamsComposing Even More Streams

-fun sum strm = foldl_ (op +) 0 strm;val sum = fn : int stream -> int stream- val addevens = sum (takeeven (intsfrom 1));val addevens = Cons (2,fn) : int stream- printStrm 10 addevens;2612203042567290110

Yet More!Yet More!

- fun drop _ Nil = Nil| drop 0 strm = strm| drop n (Cons(x,rest)) = drop (n-1) (force rest);val drop = fn : int -> 'a stream -> 'a stream- val less5 = drop 5 nums;val less5 = Cons (10,fn) : int stream- printStrm 5 less5;1011121314val it = () : unit- fun strm2list _ Nil = nil| strm2list 0 _ = nil| strm2list n (Cons(x,rest)) = x::(strm2list (n-1) (force rest));val strm2list = fn : int -> 'a stream -> 'a list- strm2list 5 less5;val it = [10,11,12,13,14] : int list

Even More ExamplesEven More ExamplesFrom MathematicsFrom Mathematics

See:◦iterate.sml◦newton.sml◦taylor.sml

Streams in PythonStreams in PythonUses generators

◦functions that can be resumed◦thereby producing a sequence of

values

See stream.py, iterate.py, newton.py

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