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LX: A Language for Flexible Type Analysis
Stephanie WeirichCornell University
joint work with Karl Crary (CMU)
Terms Types
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.
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
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<<2) <> 0
int A[4]
bool B[4]
Monomorphic subscript
sub = Fn a:Type => fn (A:packedarray a,i:int) => typecase a of int => wordsub(A,i)
| bool => (wordsub(A,i div 32) & (1<<(i mod 32))) <> 0
Type Analysis ( iML )
A:array int
B:array bool
type packedarray(a:Type) = typecase a of int => array int
| bool => array int
A Problem
int bool
int
SourceLanguage
TargetLanguage
type compilation
truefalse
0 1
term compilation
What if, during typed compilation, two source types map to the same target type?
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<<(i mod 32))) <> 0
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
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
Example
datakind S = SInt | SBool
interp : S -> Typeinterp = 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<<(i mod 32))) <> 0
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
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)
Examplefun 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) *)
LXvcaseRtype-erasuresemantics
LX
analyzeconstructed types
iMLtype-passing
semantics
analyzenative types
Type-Analyzing Languages
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]
sub = Fn a:Type => fn (A:packedarray a,i:int) => typecase a of int => wordsub(A,i)
| bool => (wordsub(A,i div 32) & (1<<(i mod 32))) <> 0
Type Passing Example ( i
ML )
type packedarray(a:Type) = typecase a of int => array int
| bool => array int
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
Rint => wordsub(A,i) | Rbool => (wordsub(A,i div 32)
& (1<<(i mod 32))) <> 0
Example
datakind S = SInt | SBool
interp : S Type
interp = fn a:S =>
case a of SInt => int
| SBool => int
datatype SRep = RInt | RBool
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<<(i mod 32))) <> 0
Second Try
datatype SRep [a:S ] = RInt of
(case a of SInt => unit
| SBool => void)
| RBool of
(case a of SInt => void
| SBool => unit)
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<<(i mod 32))) <> 0
Related Work
Inductive TypesMendler 87, Werner 94, Howard 92, 96,
Gordon 94
Type AnalysisHarper/Morrisett 95, Duggan 98, etc…
Type Erasure Crary/Weirich/Morrisett 98
Typed CompilationTIL - FLINT - TAL - Church