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A Fourth Look At ML

A Fourth Look At ML. Chapter 11 Type-Safe Data Structures. 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. Outline. Enumerations

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A Fourth Look At ML

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  1. A Fourth Look At ML Chapter 11 Type-Safe Data Structures Modern Programming Languages

  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 Modern Programming Languages

  3. Outline • Enumerations • Data constructors with parameters • Type constructors with parameters • Recursively defined type constructors • Farewell to ML  • We hardly knew ye! Modern Programming Languages

  4. 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 Modern Programming Languages

  5. Example - 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… Modern Programming Languages

  6. No Parameters - datatype day = Mon | Tue | Wed | Thu | Fri | Sat | Sun;datatype day = Fri | Mon | Sat | Sun | Thu | Tue | Wed • The type constructor day takes no parameters: it is not polymorphic, there is only oneday type • The data constructors Mon, Tue, etc. take no parameters: they are constant values of the day type • Capitalize the names of data constructors Modern Programming Languages

  7. Strict Typing - 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 • ML is strict about these new types, just as you would expect • Unlike C enum, no implementation details are exposed to the programmer Modern Programming Languages

  8. Data Constructors In Patterns • 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; Modern Programming Languages

  9. The order Datatype • Used 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 Modern Programming Languages

  10. Outline • Enumerations • Data constructors with parameters • Type constructors with parameters • Recursively defined type constructors • Farewell to ML Modern Programming Languages

  11. 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 Value PlusInf Value 38 36 Some things of type exint: MinusInf Value 26 Modern Programming Languages

  12. - 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 • 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 Modern Programming Languages

  13. A Value Is Not An int • 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 Modern Programming Languages

  14. Patterns With Data Constructors • 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 Modern Programming Languages

  15. An Exhaustive Pattern - val s = case x of = PlusInf => "infinity" |= MinusInf => "-infinity" | = Value y => Int.toString y;val s = "5" : string • 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 Modern Programming Languages

  16. Pattern-Matching Function - 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 definitions are especially important when working with your own datatypes Modern Programming Languages

  17. Exception Handling (A Peek) - 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 • Patterns are also used in ML for exception handling, as in this example Modern Programming Languages

  18. Outline • Enumerations • Data constructors with parameters • Type constructors with parameters • Recursively defined type constructors • Farewell to ML Modern Programming Languages

  19. Partial Functions • A 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 pointer • In ML, the option data type indicates undefined valuesdatatype ‘a option = NONE | SOME of ‘a; Modern Programming Languages

  20. 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 SOME SOME 1.5 "Hello" SOME NONE SOME NONE "world" 123.4 Values of typestring option Values of typereal option Modern Programming Languages

  21. Parameter Before Name - 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 • 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 Modern Programming Languages

  22. 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 Modern Programming Languages

  23. Working with option • Retrieve the value of an option, other than NONE, with valOf • Provide a default value for NONE with getOpt val 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.sml Modern Programming Languages

  24. Longer Example: bunch • 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 Modern Programming Languages

  25. Example: 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 Modern Programming Languages

  26. Example: No Polymorphism - 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 • We applied the + operator (through foldr) to the list elements • So ML knows the parameter type must be int bunch Modern Programming Languages

  27. Outline • Enumerations • Data constructors with parameters • Type constructors with parameters • Recursively defined type constructors • Farewell to ML Modern Programming Languages

  28. 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 of type intlist: Modern Programming Languages

  29. Constructing 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 Modern Programming Languages

  30. An intlist Length Function fun intlistLength INTNIL = 0 | intlistLength (INTCONS(_,tail)) = 1 + (intListLength tail);fun listLength nil = 0 | listLength (_::tail) = 1 + (listLength tail); • A length function • Much like you would write for native lists • Except, of course, that native lists are not always lists of integers… Modern Programming Languages

  31. Parametric List Type datatype 'element mylist = NIL | CONS of 'element * 'element mylist; • A parametric list type, almost like the predefined list • ML handles type inference in the usual way: - 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 Modern Programming Languages

  32. Some mylist Functions fun myListLength NIL = 0 | myListLength (CONS(_,tail)) = 1 + myListLength(tail); • This now works almost exactly like the predefined list type constructor • Of course, to add up a list you would use foldr… fun addup NIL = 0 | addup (CONS(head,tail)) = head + addup tail; Modern Programming Languages

  33. A foldr For mylist • 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); Modern Programming Languages

  34. Defining Operators • 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 Modern Programming Languages

  35. Chinese Boxes • Nested, square boxes • like Russian Dolls • Each 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; Modern Programming Languages

  36. Chinese Box FunctionsSee cbox.sml • makebox: int * color * 'a cbox -> 'a cbox • boxcount: 'a cbox -> int • tracebox: 'a cbox -> unit • prints the box attributes outside-in • openbox: 'a cbox -> 'a option • returns the inner contents • insert: int * color * 'a cbox -> 'a cbox • makebox2: (int * color) list * 'a option -> 'a cbox Modern Programming Languages

  37. One More Function • difflist = 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; Modern Programming Languages

  38. Layered Patterns with as fun difflist (Trio (s,_,b as Trio(s2,_,_))) = (s-s2)::difflist b | difflist _ = nil; Modern Programming Languages

  39. Polymorphic Binary Tree datatype 'data tree = Empty | Node of 'data tree * 'data * 'data tree; Some values of type int tree: Modern Programming Languages

  40. Constructing 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)); Modern Programming Languages

  41. Increment 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 Modern Programming Languages

  42. Add Up The Elements fun sumall Empty = 0 | sumall (Node(x,y,z)) = sumall x + y + sumall z; - sumall tree123;val it = 6 : int Modern Programming Languages

  43. Convert To List (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 Modern Programming Languages

  44. Tree 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 Modern Programming Languages

  45. Infinite Listsaka “Streams” • 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) Modern Programming Languages

  46. The unit Data Type • Like void in C++ • a “nothing” return type, or • an “empty” parameter list • Denoted by ( ) in ML - print "hello\n"; hello val it = () : unit - fun f () = "goodbye"; val f = fn : unit -> string - f(); val it = "goodbye" : string Modern Programming Languages

  47. Nodes for Infinite Linked Lists • Recall the list type definition: • The first node slot is the data • The second is a “pointer” to the tail • For 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); Modern Programming Languages

  48. An Infinite Sequence of Integers - 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 Modern Programming Languages

  49. Head 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 *) Modern Programming Languages

  50. Traversing 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 Modern Programming Languages

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