1. LX: A Language for Flexible Type Analysis Stephanie Weirich Cornell University joint work with Karl Crary (CMU)

2. TermsTypes Source IL Machine Typed Compilation A series of translations between typed languages, propagating a set of invariants throughout the entire compilation process. Types are used for a variety of optimizations and provide safety assurances about the output of compiler.

3. 06/08/9 9 ICFP '98 Because any array may be passed to a polymorphic function, all arrays must look the same, no matter the type of their elements. A: array int B: array bool sub = Fn a: Type => fn (A: array a,i: int ) => wordsub(A,i) Polymorphic Subscript

4. 06/08/9 9 In languages such as C, the type of an array is always known at compilation. We can pack boolean values into integer arrays. A[2] B[2] intsub(A,2) intsub(B,0)&(1 0 int A[4] bool B[4] Monomorphic subscript

5. 06/08/9 9 sub = Fn a: Type => fn (A: packedarray a,i: int ) => typecase a of int => wordsub(A,i) | bool => (wordsub(A,i div 32) & (1 0 Type Analysis ( i ML ) A: array int B: array bool type packedarray(a: Type ) = typecase a of int => array int | bool => array int

6. A Problem intbool int Source Language Target Language type compilation truefalse 01 term compilation What if, during typed compilation, two source types map to the same target type?

7. An initial attempt Target language contains both source language and target language types, and has a built-in type constructor interp to translate between them. sub = Fn a: S => fn (A: array (interp a),i: int ) => typecase a of [ int] S => wordsub(A,i) | [ bool] S => (wordsub(A,i div 32) & (1 0

8. Issues in Compilation How do we preserve the meaning of typecase when the types themselves change? –type translation may not be injective –in TALx86, int  int may be compiled into a variety of types depending on the calling convention, register allocation, etc –In closure conversion, a  b converted to  c. (a * c  b) * c larger type takes longer to analyze typecase is no longer exhaustive

9. Goal Need a facility to describe the types of another language, and describe a translation of those types into the types of the current language. Need a way to examine those representations the term level

10. Example datakind S = SInt | SBool interp : S -> Type interp = fn a: S => case a of SInt => int | SBool => int sub = Fn a: S => fn (A: array (interp a),i: int ) => ccase a of SInt => wordsub(A,i) | SBool => (wordsub(A,i div 32) & (1 0

11. LX Language Type analysis is just a programming idiom System F augmented with building blocks for datakinds –tuples –sums –primitive recursion Strongly Normalizing so that type checking is decidable Term-level ccase

12. Another Example datakind M = Int | Prod of M * M | Arrow of M * M interp : M -> Type fun interp ( a: M ) = case a of Int => int | Prod (c1,c2)) => interp(c1) * interp(c2) | Arrow (c1,c2)) => interp(c1)  interp(c2)

13. Example fun printf [ a : M ] (x: interp a ) => ccase a of Int => (* x is of type int *) print_int x Prod(b,c) => print “<“; printf [ b ] (fst x); print “,”; printf [ c ] (snd x); print “>” Arrow(b,c) => (* x is of type interp(b)  interp(c) *) print “fun” (* x is of type interp(Prod(b,c)) *)(* x is of type interp(b) * interp(c) *)

14. LXvcase R type-erasure semantics LX analyze constructed types i ML type-passing semantics analyze native types Type-Analyzing Languages

15. Type-Passing Semantics –Types are necessary at run time –Requires sophisticated machinery to describe low level languages Abstract kinds and translucent sums for polymorphic typed closure conversion [MMH 96] Type-Erasure Semantics –Types may be erased prior to run time –Standard type theory constructs suffice Simpler typed closure conversion [ MWCG 97][CWM 98]

16. sub = Fn a: Type => fn (A: packedarray a,i: int ) => typecase a of int => wordsub(A,i) | bool => (wordsub(A,i div 32) & (1 0 Type Passing Example ( i ML ) type packedarray(a: Type ) = typecase a of int => array int | bool => array int

17. Type Erasure Example ( R ) type packedarray(a: Type ) = typecase a of int => array int | bool => array int sub = Fn a: Type => fn (A: packedarray a, i: int, rx: R(a) ) => typecase rx of R int => wordsub(A,i) | R bool => (wordsub(A,i div 32) & (1 0

18. Example datakind S = SInt | SBool interp : S  Type interp = fn a: S => case a of SInt => int | SBool => int datatype SRep = RInt | RBool

19. Example sub = Fn a: S => fn (A: array (interp a),i: int, rx: SRep ) => case rx of RInt => ccase a of SInt => wordsub(A,i) | SBool => impossible | RBool => ccase a of SInt => impossible | SBool => (wordsub(A,i div 32) & (1 0

20. Second Try datatype SRep [ a : S ] = RInt of (case a of SInt => unit | SBool => void) | RBool of (case a of SInt => void | SBool => unit)

21. Second Try sub = Fn a: S => fn (A: array (interp a),i: int, rx: SRep(a) ) => case rx of RInt y => vcase a of SInt => wordsub(A,i) | SBool => dead y | RBool y => vcase a of Sint => dead y | SBool => (wordsub(A,i div 32) & (1 0

22. Related Work Inductive Types Mendler 87, Werner 94, Howard 92, 96, Gordon 94 Type Analysis Harper/Morrisett 95, Duggan 98, etc… Type Erasure Crary/Weirich/Morrisett 98 Typed Compilation TIL - FLINT - TAL - Church