Arvind Computer Science and Artificial Intelligence Laboratory M.I.T. L06-1 September 26, 2006http:// Type Inference September 26, 2006

L06-2 September 26, 2006http:// Type Inference Type inference is typically presented in two different forms: –Type inference rules: Rules define the type of each expression Clean and concise; needed to study the semantic properties, i.e., soundeness, of the type system –Type inference algorithm: Needed by the compiler writer to deduce the type of each subexpression or to deduce that the expression is ill typed. Often it is nontrivial to derive an inference algorithm for a given set of rules. There can be many different algorithms for a set of typing rules.

L06-3 September 26, 2006http:// first... A Simple Type System (no polymorphism)

L06-4 September 26, 2006http:// -calculus with Constants & Letrec E ::= x | x.E | E E | Cond (E, E, E) | PF k (E 1,...,E k ) | CN 0 | CN k (E 1,...,E k ) | CN k (SE 1,...,SE k ) | let S in E PF 1 ::= negate | not |... | Prj 1 | Prj 2 |... PF 2 ::= + |... CN 0 ::= Number | Boolean CN 2 ::= cons |... Statements S ::=  | x = E | S; S not in initial terms There are no types in the syntax of the language!

L06-5 September 26, 2006http:// A Simple Type System Types ::= base types | t type variables |   ->  2 Function types Type Environments TE ::= Identifiers  Types int, bool,...

L06-6 September 26, 2006http:// Simple Type Inference Rules Typing: TE ├ e :  (App) TE ├ e 1 :  ->  ’ TE ├ e 2 :   TE ├ (e 1 e 2 ) :  ’ (Abs) TE ├ x.e : ->  ’ (Var) TE ├ x :  (Const) TE ├ c :  (Let) TE ├ (let x = e 1 in e 2 ) :  ’ TE + {x : } ├ e :  ’ (x : )TE typeof(c) =  TE+{x : } ├ e 1 : TE+{x : } ├ e 2 :  ’

L06-7 September 26, 2006http:// Simple Type Inference Rules cont (Cond) TE ├ Cond(e B, e 1, e 2 ) :  (PF) TE ├ (e 1 + e 2 ) : int (CN) TE ├ CN(e 1, e 2 ) : CN(     (Pro) TE ├ Prj 1 (e) :   TE ├ e B : bool TE ├ e 1 : TE ├ e 2 :  TE ├ e 1 : int TE ├ e 2 : int TE ├ e 1 :   TE ├ e 2 :   TE ├ e : CN(    ) one such rule for each PF one set of such rules for each data structure type

L06-8 September 26, 2006http:// Simple Inference Algorithm W(TE, e) returns (S,) such that S (TE) ├ e :  The type environment TE records the most general type of each identifier while the substitution S records the changes in the type variables Def W(TE, e) = Case e of x =... x.e =... (e 1 e 2 ) =... let x = e 1 in e 2 =...

L06-9 September 26, 2006http:// Simple Inference Algorithm (cont-1) Def W(TE, e)= Case e of c = ({}, Typeof(c)) x = if (x   Dom(TE)) then Fail else let = TE(x); in ({},  ) x.e = let (S 1,  1 ) = W(TE + { x : u }, e); in (S 1, S 1 (u) ->  1 ) (e 1 e 2 ) = let x = e 1 in e 2 = u’s represent new type variables let (S 1,  1 ) = W(TE, e 1 ); (S 2,  2 ) = W(S 1 (TE), e 2 ); S 3 = Unify(S 2 ( 1 ),  2 -> u); in (S 3 S 2 S 1, S 3 (u)) let (S 1,  1 ) = W(TE + {x : u}, e 1 ); S 2 = Unify(S 1 (u),  1 ); (S 3,  2 ) = W(S 2 S 1 (TE) + {x :   }, e 2 ); in (S 3 S 2 S 1,  2 )

L06-10 September 26, 2006http:// Simple Inference Algorithm (cont-1) Cond(e B,e 1, e 2 ) = (e 1 + e 2 ) = CN(e 1, e 2 ) = Prj 1 (e) = let (S 1,  1 ) = W(TE, e 1 ); (S 2,  2 ) = W(S 1 (TE), e 2 ); S 3 = Unify( 1, int); S 4 = Unify( 2, int); in (S 4 S 3 S 2 S 1, int) let (S 1,  1 ) = W(TE, e 1 ); (S 2,  2 ) = W(S 1 (TE), e 2 ); in (S 2 S 1, CN( 1,  2 )) let (S, ) = W(TE, e); S 1 = Unify(, CN(u1,u2))); in (S 1 S, S 1 (u1)) These should probably be type constants let (S B,  B ) = W(TE, e B ); (S 1,  1 ) = W(S B (TE), e 1 ); (S 2,  2 ) = W(S 1 S B (TE), e 2 ); S 3 = Unify( B, bool); S 4 = Unify( 1, 2 ); in (S 4 S 3 S 2 S 1 S B, S 4 S 3  2 )

L06-11 September 26, 2006http:// Type Inference : Example let f = n.cond ((eq0 n), 1, n*(f (pred n)) in f W( , A) = W({f : u 1 }, n.B) = ( [Int / u 2 ], Bool )W({f : u 1, n : u 2 }, (eq0 n) ) = W({f : u 1, n : u 2 }, B) = ( [ ], Int -> Int ) Unify(u 2, Int) = A B [ Int / u 2 ] W({f : u 1, n : Int}, 1) =( [ ], Int) W({f : u 1, n : Int}, n*(f (pred n))) = W({f : u 1, n : Int}, n) =( [ ], Int) ( [Int -> Int / u 1 ], Int) W({f : Int -> Int}, f ) =( [ ], Int -> Int) ([Int -> Int / u 1, Int / u 2 ], Int) ([Int -> Int / u 1 ], Int -> Int) W({f : u 1, n : Int}, f (pred n)) =( [Int -> u 3 / u 1 ], u 3 ) Unify(u 1, Int -> u 3 ) =[ Int -> u 3 / u 1 ] Unify(Int -> Int -> Int, Int -> u 3 -> Int) =[ Int / u 3 ]

L06-12 September 26, 2006http:// Soundness The proposed type system is said to be sound if e :  then e indeed evaluates to a value in  A method of proving soundness: –The semantics of the language is defined in terms of a value space that has integer values, Boolean values etc. as subspaces. –Any expression that may evaluate to a special value “wrong” will not type check. –There is no type expression that denotes the subspace “wrong”. Such proofs tend to be difficult...

L06-13 September 26, 2006http:// Some observations A type system restricts the class of programs that are considered “legal” It is possible a term in the untyped – calculus may not cause any errors but may not be typeable in a particular type system let id = x. x in... (id True)... (id 1)... This term is not typeable in the simple type system we have discussed so far.

L06-14 September 26, 2006http:// Polymorphic Types Constraints: let id = x. x in... (id True)... (id 1)... id :: t 1 --> t 1 id :: Int --> t 2 id :: Bool --> t 3 Does not unify!! id :: t 1. t 1 --> t 1 Different uses of a generalized type variable may be instantiated differently id 2 : Bool --> Bool id 1 : Int --> Int Solution: Generalize the type variable When can we generalize?

L06-15 September 26, 2006http:// now... The Hindley-Milner Type System

L06-16 September 26, 2006http:// A mini Language to study Hindley-Milner Types There are no types in the syntax of the language! The type of each subexpression is derived by the Hindley-Milner type inference algorithm. Expressions E ::= c constant | x variable | x. E abstraction | (E 1 E 2 ) application | let x = E 1 in E 2 let-block

L06-17 September 26, 2006http:// A Formal Type System Note, all the ’s occur in the beginning of a type scheme, i.e., a type  cannot contain a type scheme  Types ::= base types | t type variables |   ->  2 Function types Type Schemes ::=  |t.   Type Environments TE ::= Identifiers  Type Schemes

L06-18 September 26, 2006http:// Instantiations Type scheme can be instantiated into a type  ’ by substituting types for the bound variables of , i.e.,  ’ = S  for some S s.t. Dom(S)  BV() -  ’ is said to be an instance of (> ’ ) -  ’ is said to be a generic instance of when S maps variables to new variables. =t 1...t n.  Example: =t 1. t 1 -> t 2 t 3 -> t 2 is a generic instance of Int -> t 2 is a non generic instance of 

L06-19 September 26, 2006http:// Generalization aka Closing Generalization introduces polymorphism Quantify type variables that are free in  but not free in the type environment (TE) Captures the notion of new type variables of  Gen(TE,) = t 1...t n.  where { t 1...t n } = FV() - FV(TE)

L06-20 September 26, 2006http:// HM Type Inference Rules Typing: TE ├ e :  (App) TE ├ e 1 :  ->  ’ TE ├ e 2 :   TE ├ (e 1 e 2 ) :  ’ (Abs) TE ├ x.e : ->  ’ (Var) TE ├ x :  (Const) TE ├ c :  (Let) TE ├ (let x = e 1 in e 2 ) :  ’ TE + {x : } ├ e :  ’ (x : )TE    typeof(c)   TE+{x : } ├ e 1 : TE+{x:Gen(TE,)} ├ e 2 :  ’

L06-21 September 26, 2006http:// HM Inference Algorithm Def W(TE, e)= Case e of c = ({}, Typeof(c)) x = x.e = (e 1 e 2 ) = let x = e 1 in e 2 = u’s represent new type variables let (S 1,  1 ) = W(TE, e 1 ); (S 2,  2 ) = W(S 1 (TE), e 2 ); S 3 = Unify(S 2 ( 1 ),  2 -> u); in (S 3 S 2 S 1, S 3 (u)) if (x   Dom(TE)) then Fail else let t 1...t n.= TE(x); in ( { }, [u i / t i ] ) let (S 1,  1 ) = W(TE + { x : u }, e); in (S 1, S 1 (u) ->  1 ) let (S 1,  1 ) = W(TE + {x : u}, e 1 ); S 2 = Unify(S 1 (u),  1 );  = Gen(S 2 S 1 (TE), S 2 ( 1 ) ); (S 3,  2 ) = W(S 2 S 1 (TE) + {x : }, e 2 ); in (S 3 S 2 S 1,  2 )

L06-22 September 26, 2006http:// Hindley-Milner: Example x. let f = y.x in (f 1, f True) W( , A) = W({x : u 1 }, B) = ( [ ], u 1 ) ( [ ], u 3 -> u 1 ) u 3.u 3 -> u 1 TE = {x : u 1, f : u 3.u 3 -> u 1 } ( [ ], u 4 -> u 1 ) W(TE, 1) = ( [ ], Int ) [ Int / u 4, u 1 / u 5 ] ( [ ], u 1 ) ( [ ], (u 1, u 1 ) ) Unify(u 4 -> u 1, Int -> u 5 ) = W({x : u 1, f : u 2, y : u 3 }, x ) = W({x : u 1, f : u 2 }, y.x ) = Gen({x : u 1 }, u 3 -> u 1 ) = W(TE, (f 1) ) = ( [ ], u 1 -> (u 1, u 1 ) ) W(TE, f ) = Unify(u 2, u 3 -> u 1 ) = A B [ (u 3 -> u 1 ) / u 2 ]...

L06-23 September 26, 2006http:// Important Observations Do not generalize over type variables used elsewhere Let is the only way of defining polymorphic constructs Generalize the types of let-bound identifiers only after processing their definitions

L06-24 September 26, 2006http:// Properties of HM Type Inference It is sound with respect to the type system. An inferred type is verifiable. It generates most general types of expressions. Any verifiable type is inferred. Complexity PSPACE-Hard DEXPTIME-Complete Nested let blocks

L06-25 September 26, 2006http:// Extensions Type Declarations Sanity check; can relax restrictions Incremental Type checking The whole program is not given at the same time, sound inferencing when types of some functions are not known Typing references to mutable objects Hindley-Milner system is unsound for a language with refs (mutable locations) Overloading Resolution

L06-26 September 26, 2006http:// HM Limitations: -bound vs Let-bound Variables Only let-bound identifiers can be instantiated differently. let twice f x = f (f x) in twice twice succ 4 vs. let twice f x = f (f x) foo g = (g g succ) 4 in foo twice foo is not type correct ! Generic vs. Non-generic type variables

L06-27 September 26, 2006http:// Puzzle: Another set of Inference rules (Gen) TE ├ e : t  FV(TE) TE ├ e : t. (Spec) TE ├ e : t. TE ├ e : [t’/t] (Var)(x : )  TE TE ├ x :  (Let)TE+{x:} ├ e 1 : TE+{x:} ├ e 2 : ’ TE ├ (let x = e 1 in e 2 ) :  ’ (App) and (Abs) rules remain unchanged. Sound but no inference algorithm !

L06-28 September 26, 2006http:// next time --- Overloading