Post on 07-Jan-2016
description
transcript
cs3180 (Prasad) PolyTypes 1
Type Checking and Type Inference
cs3180 (Prasad) PolyTypes 2
Motivation• Application Programmers
– Reliability• Logical and typographical errors manifest themselves as
type errors that can be caught mechanically, thereby increasing our confidence in the code execution.
• Language Implementers– Storage Allocation (temporaries)
– Generating coercion code
– Optimizations
cs3180 (Prasad) PolyTypes 3
Evolution of Type System• Typeless
– Assembly language• Any instruction can be
run on any data “bit pattern”
• Implicit typing and coercion
– FORTRAN
• Explicit type declarations
– Pascal• Type equivalence
• Weak typing– C
• Arrays (bounds not checked), Union type
• Actuals not checked against formals.
• Data Abstraction– CLU
• Type is independent of representation details.
• Generic Types– Ada
• Compile-time facility for “container” classes.
• Reduces source code duplication.
cs3180 (Prasad) PolyTypes 4
• Languages– Strongly typed (“Type errors always caught.”)
• Statically typed (e.g., ML, Ada, Eiffel, and Scala)– Compile-time type checking : Efficient.
• Dynamically typed (e.g., Scheme, Python, and Smalltalk)
– Run-time type checking : Flexible.
– Weakly typed (e.g., C)– Unreliable Casts (int to/from pointer).
– Typing in Object-Oriented Languages • OOPLs, such as Eiffel and Java, impose restrictions
that guarantee type safety and efficiency, but bind the code to function names at run-time.
cs3180 (Prasad) PolyTypes 5
Type inference is abstract interpretation.
( 1 + 4 ) / 2.5 int * int int 5 / 2.5 (ML-
error) real * real real 2.0
( int + int ) / real int / real real
cs3180 (Prasad) PolyTypes 6
Expression Grammar: Type Inference Example
E -> E + E | E * E | x | y | i | j
• Arithmetic Evaluationx, y in {…, -1.1, …, 2.3, …}i, j in {…, -1,0,1,…}
+, * : “infinite table”
• Type Inference
x, y : real i, j : int
cs3180 (Prasad) PolyTypes 7
+,* int real int int real real real real
int int real real
Values can be abstracted as type names and arithmetic operations can be abstracted as operations on these type names.
cs3180 (Prasad) PolyTypes 8
if true then 5 else 0.5
• Not type correct, but runs fine.
if true then 1.0/0.0 else 3.5
• Type correct, but causes run-time error.
Type correctness is neither necessary nor sufficient for programs to run.
cs3180 (Prasad) PolyTypes 9
Assigning types to expressions (ML)• Uniquely determined
fn s => s ^ “.\n”; val it = fn : string -> string
• Over-constrained (type error in ML) (2.5 + 2)
• Under-constrained – Overloadingfn x => fn y => x + y; (* resolvable *)fn record => #name(record); (* error *)
– Polymorphism fn x => 1 ;
val it = fn : 'a -> int
cs3180 (Prasad) PolyTypes 10
Type Signatures : Curried functions
• fun rdivc x y = x / y rdivc : real -> real -> real• fun rdivu (x,y) = x / y rdivu : real * real -> real•fun plusi x y = x + y plusi : int -> int -> int• fun plusr (x:real,y) = x + y plusr : real * real -> real
cs3180 (Prasad) PolyTypes 11
Polymorphic Types
• Semantics of operations on data structures such as stacks, queues, lists, and tables are independent of the component type.
• Polymorphic type system provides a natural representation of generic data structures without sacrificing type safety.
• Polymorphism fun I x = x; I 5; I “x”; for all types I:
cs3180 (Prasad) PolyTypes 12
Programming with Lists in ML
Polymorphic Types, Type Inference
and Pattern Matching
cs3180 (Prasad) PolyTypes 13
Lists is a type listlist is a type
(* Homogeneous lists. *)
– E.g., (true, [fn i:int => "i"])
: bool * (int -> string) list;– E.g., [1, 2, 3], 1::2::3::[] : int list;– E.g., (op ::) : ’a * ’a list ->’a list;
– List constructors [] and :: can be used in patterns.
cs3180 (Prasad) PolyTypes 14
Built-in operations on lists
hd : ’a list -> ’a tl : ’a list -> ’a list
null: ’a list -> bool
op @ : ’a list * ’a list -> ’a list (* append operation; infix operator *)
length : ’a list -> int (* sets vs lists -- multiplicity; ordering *)
cs3180 (Prasad) PolyTypes 15
Catalog of List functionsinit [1,2,3] = [1,2]last [1,2,3] = 3
• Specs:init (xs @ [x]) = xslast (xs @ [x]) = x
• Definitions: fun init (x::[]) = [] | init (x::xs) = x :: init xs; fun last (x::[]) = x | last (x::xs) = last xs;
cs3180 (Prasad) PolyTypes 16
take 3 [1,2,3,4] = [1,2,3] drop 2 [1,2,3] = [3]
• Definition: fun take 0 xs = [] | take n [] = [] | take n (x::xs) = x::take (n-1) xs;
fun drop 0 xs = xs | drop n [] = [] | drop n (x::xs) = drop (n-1) xs;
cs3180 (Prasad) PolyTypes 17
• Role of patterns– For testing type (“discrimination”)– For picking sub-expressions apart
• Inferred Signatures (Captures correct usage)
init: ’a list -> ’a list
last: ’a list -> ’a
take, drop : int -> ’a list -> ’a list
List.take, List.drop : ’a list * int -> ’a list
cs3180 (Prasad) PolyTypes 18
takewhile even [2,4,1,6,2] = [2,4]
Definition:
fun takewhile p [] = [] | takewhile p (x::xs) = if p x then x :: takewhile p xs else [];
takewhile: (’a -> bool) -> ’a list -> ’a list
cs3180 (Prasad) PolyTypes 19
dropwhile even [2,3,8] = [3,8]
•Definition: fun dropwhile p [] = [] | dropwhile p (x::xs) = if p x then dropwhile p xs else x::xs;
dropwhile : (’a -> bool) -> ’a list -> ’a list
cs3180 (Prasad) PolyTypes 20
Composition
val h = f o g; fun comp f g = let fun h x = f (g x) in h;
comp: – Generality + Correct Usage– Equality constraints
cs3180 (Prasad) PolyTypes 21
map-functionfun map f [] = [] | map f (x::xs) = f x :: map f xs
map listlist
map (fn x => “n”) [1, 2, 3]map (fn x => x::[]) [“a”, “b”]
• list patterns; term matching. • definition by cases; ordering of rules
cs3180 (Prasad) PolyTypes 22
Conventions
• Function application is left-associative.
f g h = ( ( f g ) h )• -> is right-associative. int->real->bool = int->(real->bool)• :: is right-associative. a::b::c::[] = a::(b::(c::[])
• Function application binds stronger than ::. f x :: xs = ( f x ) :: xs