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A Fourth Look At ML 1

A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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Page 1: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

1

A Fourth Look At ML

Page 2: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

2

Type Definitions

• Predefined, but not primitive in ML:

• Type constructor for lists:

• Defined for ML in ML

datatype bool = true | false;

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

Page 3: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

3

Outline

• Enumerations• Data constructors with parameters• Type constructors with parameters• Recursively defined type constructors• Farewell to ML

Page 4: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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Defining Your Own Types

• New types can be defined using the keyword datatype

• These declarations define both:– type constructors for making new (possibly

polymorphic) types– data constructors for making values of those new

types

Page 5: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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- 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

• 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…

Example

Page 6: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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• 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

No Parameters

Page 7: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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• 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

Strict Typing

Page 8: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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• You can use the data constructors in patterns

• In this simple case, they are like constants• But we will see more general cases next

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

Data Constructors In Patterns

Page 9: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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Outline

• Enumerations• Data constructors with parameters• Type constructors with parameters• Recursively defined type constructors• Farewell to ML

Page 10: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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Wrappers

• You 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

36

MinusInf Value

26

Value

38

Some thingsof type exint:

Page 11: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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• 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- Value 3;val it = Value 3 : exint

Page 12: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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• Value 5 is an exint• It is not an int, though it contains one• How 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

A Value Is Not An int

Page 13: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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• 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

Patterns With Data Constructors

Page 14: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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An 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

Page 15: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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• 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

Pattern-Matching Function

Page 16: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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Exception Handling (A 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

Page 17: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

17

Outline

• Enumerations• Data constructors with parameters• Type constructors with parameters• Recursively defined type constructors• Farewell to ML

Page 18: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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Type Constructors With Parameters

• 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

NONE

SOME

"Hello"

Values of typestring option

SOME

"world"NONE

SOME

1.5

Values of type real option

SOME

123.4

Page 19: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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• Type constuctor 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

Parameter Before Name

Page 20: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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Uses For option• Predefined type constructor in ML• Used 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

Page 21: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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

• As usual, ML infers types:

Longer Example: bunchdatatype '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

Page 22: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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• 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: Polymorphism

Page 23: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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Example: 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

Page 24: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

24

Outline

• Enumerations• Data constructors with parameters• Type constructors with parameters• Recursively defined type constructors• Farewell to ML

Page 25: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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Recursively Defined Type Constructors

• The 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

1

the list [1,2]

Page 26: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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- 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

1

the list [1,2]

Constructing Those Values

Page 27: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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• A length function• Much like you would write for native lists• Except, 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);

An intlist Length Function

Page 28: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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Parametric 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

Page 29: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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• 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);

Some mylist Functions

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

Page 30: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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• 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);

A foldr For mylist

Page 31: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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Defining Operators (A Peek)

• ML allows new operators to be defined• Like this:

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

Page 32: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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datatype 'data tree = Empty | Node of 'data tree * 'data * 'data tree;

Some values oftype int tree:

Polymorphic Binary Tree

Empty

Node

the empty tree

2

the tree 2

Empty Empty

Node 3 Empty Empty

Node 1

the tree 2

Empty Empty 2

1 3

Node

Page 33: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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- 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));

Constructing Those Values

Page 34: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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Increment All Elementsfun 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

Page 35: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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Add Up The Elementsfun sumall Empty = 0 | sumall (Node(x,y,z)) = sumall x + y + sumall z;

- sumall tree123;val it = 6 : int

Page 36: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

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

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

Page 37: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

37

Tree Searchfun 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

Page 38: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

38

Outline

• Enumerations• Data constructors with parameters• Type constructors with parameters• Recursively defined type constructors• Farewell to ML

Page 39: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

39

That's All

• There is, of course, a lot more• A few words about the parts we skipped:– records (like tuples with named fields)– arrays, with elements that can be altered– references, for values that can be altered– exception handling

Page 40: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

40

More Parts We Skipped

– support for encapsulation and data hiding:• structures: collections of datatypes, functions, etc.• signatures: interfaces for structures• functors: like functions that operate on structures,

allowing type variables and other things to be instantiated across a whole structure

Page 41: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

41

More Parts We Skipped

– API: the standard basis• predefined functions, types, etc.• Some at the top level but most in structures: Int.maxInt, Real.Math.sqrt, List.nth, etc.

Page 42: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

42

More Parts We Skipped

– eXene: an ML library for applications that work in the X window system

– the Compilation Manager for building large ML projects

• Other dialects besides Standard ML– OCaml– Concurrent ML (CML) extensions

Page 43: A Fourth Look At ML 1. Type Definitions Predefined, but not primitive in ML: Type constructor for lists: Defined for ML in ML datatype bool = true | false;

43

Functional Languages

• ML supports a function-oriented style of programming

• If you like that style, there are many other languages to explore, like Lisp and Haskell


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