Type Analysis and Typed Compilation

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Type Analysis andTyped Compilation

• Stephanie Weirich
• Cornell University

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Outline

• Typed Compilation background
• Type Analysis background
• Initial framework - Type Passing
• Problems with Type Passing
• Type Erasure framework
• Closure Conversion comparison
• Related Work

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Traditional Compilation
Compilation is a series of translations between
several languages
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Typed Compilation
Most of those languages are typed, and the types
are translated with the terms
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Typed Compilation

• Safety
• well-typed programs cant go wrong
• Types describe invariants maintained by the
  compiler
• Performance
• Types provide information which may be used by
  the compiler for optimization
• Tag-free garbage collection
• Data layout control

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Type Analysis

• Create functions from types to values
• Code may branch on an unknown type

fun toString xa gt (typecase a of int gt
Int.toString char gt Char.toString
gt fn (fst,snd) gt ( (toString
fst) , (toString snd) ) )
x
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Data Layout Optimization

• Parametric polymorphism requires uniformity of
  representation for all arguments, regardless of
  their types.

kinds k T4 T types t B4 t array a
t -gtt terms e,f x lx.e e f Lak.e
e t ld(e,f) i e ? f if0 then
else... malloc(e,e ...) st(e,e,e) ld (t
array B4) -gt t st (t array B4 t ) -gt t
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Polymorphic Subscript
Because any array may be passed to a polymorphic
function, all arrays must look the same, no
matter the type of their elements.
Aint Array Bbool Array
ICFP '98
sub fn (Aa array,iint) gt wordsub(A,i)
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Monomorphic subscript
In languages such as C, the type of an array is
always known at its use.
int A4
bool B4
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Type Analysis to the Rescue
Type analysis allows us to determine the type at
run time.
Aint Array
Bbool Array
sub fn (Aa array,iint) gt typecase a
of int gt wordsub(A,i) bool gt
(wordsub(A,i div 32) (1ltlt(i mod 32)))
ltgt 0
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ICFP '98
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Initial Framework - ??iML

• Make type abstraction/application explicit (as in
  System F)
• (? a. e) int
• Add typecase operator
• typecase t of
• int gt ...
• b g gt ...
• Code execution (operational semantics) now relies
  on the type system

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Type Passing Semantics
subint(A,3)
(?a.?(Aa array,iint). typecase a of
int gt wordsub(A,i) bool gt (wordsub(A,i div
32) (1ltlt(i mod 32))) ltgt 0) int
(A,3)
typecase int of int gt wordsub(A,3)
bool gt (wordsub(A,3 div 32) (1ltlt(3 mod
32))) ltgt 0
wordsub(A,3)
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Typechecking
To typecheck a typecase term we annotate the
term with its return type tostring????a.
?xa. (typecase d.d -gt string a of int
gt Int.toString char gt Char.toString
b g gt fn (fst,snd)bg gt (
(toStringb fst) ,
(toStringg snd) ) ) x Then we check that
each branch satisfies that type with the
appropriate type substitution. The return type of
the entire term is a substituted for d in the
annotated type.
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Problems

• Issues about expressiveness
• Complicates low-level constructs
• Cant express some optimizations
• Cant express abstraction boundaries

ICFP '98
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Complexity
Complexity

• Polymorphic closure conversion
• Minamide et al. 1996
• Morrisett et al. 1998
• Duplication of effort
• Optimization of runtime behavior
• Explicit modeling of low level computation
• Allocation Semantics
• Typed Assembly Languages

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Inefficiency
Inefficiency

• Must pass all types even if some are never
  examined
• TIL -- eliminates unexamined run-time types in ad
  hoc manner in translation to untyped calculus

ICFP '98
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Loss of Abstraction

• No way to hold types abstract if they can always
  be examined
• Clients allowed to break abstraction barriers and
  infer more information than desired

Host
Client
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bool ref
capability
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Solution

• Pass terms that represent types

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Type Erasure Semantics
sub int Rint (A,3)
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Type Erasure Semantics
sub int Rint (A,3)
(?a.?xR(a). ?(Aa array,iint). typecase x
of Rint gt wordsub(A,i) Rbool gt
(wordsub(A,i div 32) (1ltlt(i mod 32)))
ltgt 0) int Rint (A,3)
typecase Rint of Rint gt wordsub(A,3)
Rbool gt (wordsub(A,3 div 32) (1ltlt(3
mod 32))) ltgt 0
wordsub(A,3)
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Type Erasure Semantics
sub int Rint (A,3)
(??.?xR(?). ?(A? array,iint). typecase x
of Rint gt wordsub(A,i) Rbool gt
(wordsub(A,i div 32) (1ltlt(i mod 32)))
ltgt 0) int Rint (A,3)
typecase Rint of Rint gt wordsub(A,3)
Rbool gt (wordsub(A,3 div 32) (1ltlt(3
mod 32))) ltgt 0
wordsub(A,3)
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Formalization

• Special representation terms
• Rint
• R? (x,y)
• A term e which represents a type ? has the
  special type R(?).
• R?(Rint,Rint) R(int ? int)
• Instead of a type, the argument to typecase is a
  term of type R(?)
• The type system tracks the correspondence between
  a type and its representation

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Typechecking
To typecheck a typecase term we still annotate
the term with its return type tostring????a.
?xa. ?yR(a). (typecase d.d -gt string y
of Rint gt Int.toString Rchar gt
Char.toString R(r1,r2) as bg gt fn
(fst,snd)bg gt ( (toStringb fst
r1) , (toStringg snd r2)
) ) x We require that the argument to the
typecase be of type R(t) for the typecase term to
typecheck. We can then substitute a for d in the
annotated type as before.
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Solution
Solution

• Everything that happens at run-time is described
  by the terms
• Can go to a type erasure semantics
• Optimization
• Traditional code optimizers can optimize type
  representations
• Sophisticated techniques still possible
• If a representation is not provided a type may
  not be analyzed

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Typed Closure Conversion

• At run-time, a first-class function is
  represented by a code pointer
• But now x is unbound -- so we change the type of
  l2 to take another argument
• A closure is just a code pointer paired with the
  values of its free variables -- its environment

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Typed Closure Conversion

• Application just extracts the environment and
  applies the function to it

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Existential Types

• Unfortunately, the type of the closure now
  depends on its free variables
• Existential types hold the types of free
  variables abstract

clos (intint -gt int) int
l1 ?xint. pack (l2,x) as ?env.((intenv
-gt int)env) hiding int
unpack (env,clos) (f 3) in (1 clos)(4,2 clos)
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Polymorphic Closure Conversion

• In a type passing semantics a function may also
  have free type variables at run time
• They also must be part of the closure, via
  translucent types
• The type environment is then abstracted using
  existential kinds

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Closure Conversion

• In a type erasure semantics, only the type
  representation remains at run time.
• At the time of closure creation, the type
  arguments may be given to the function.

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Multi-stage Type Analysis
Final Step of typed compilation compile to a
typed assembly language
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Multi-stage Type Analysis

• How do we preserve the meaning of typecase when
  the types themselves change?
• in TALx86 both int and float are compiled into
  B4, the type of 4-byte values
• int ? int may be compiled into a variety of
  types depending on the calling convention

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Related Work

• Harper, Morrisett - POPL 95
• Minamide, Morrisett, Harper - POPL 96
• Minamide - 2nd Fuji Intl Workshop on Functional
  and Logic Programming 96
• Morrisett, Walker, Crary, Glew - POPL 98
• Crary, Weirich, Morrisett - ICFP 98
• Crary, Weirich - ICFP 99

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Low-level Type Analysis

• How do we analyze types with quantifiers?
• In TALx86 every function (polymorphic or not) is
  compiled into a polymorphic code pointer

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Areas for Future Work
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Outline

• Introduction
• Typed Compilation
• Type Analysis in general
• toString example
• Type Analysis in compilation
• bit array example
• Initial framework
• syntax -- from examples
• semantics
• type passing
• problems
• complication of theory
• cant express efficient code
• loss of abstraction
• Type erasure semantics
• syntax
• dynamic semantics
• static semantics
• Closure Conversion example

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A note about typechecking

• In this example wordsub has a strange type
• a array int -gt int
• It would better if it were of type
• int array int -gt int
• Then argument to subscript must allways be an int
  array. But that forgets its actual type of bool
  array. So we create a special type, called packed
  array, with a type-level type analysis operator.

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Type Passing Semantics
subint(A,3)
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ICFP '98
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Type Passing Semantics
subint(A,3)
(??.?(A? array,iint). typecase ? of
int gt wordsub(A,3) bool gt (wordsub(A,3 div
32) (1ltlt(3 mod 32))) ltgt 0) int
(A,3)
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Type Passing Semantics
subint(A,3)
(??.?(A? array,iint). typecase ? of
int gt wordsub(A,3) bool gt (wordsub(A,3 div
32) (1ltlt(3 mod 32))) ltgt 0) int
(A,3)
typecase int of int gt wordsub(A,3)
bool gt (wordsub(A,3 div 32) (1ltlt(3 mod
32))) ltgt 0
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Formalization

• Special representation terms
• Rint
• R? (x,y)

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Formalization

• Special representation terms
• Rint
• R? (x,y)
• A term e which represents a type ? has the
  special type R(?).
• R?(Rint,Rint) R(int ? int)

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Type Erasure Semantics
sub int Rint (A,3)
(??.?xR(?). ?(A? array,iint). typecase x
of Rint gt wordsub(A,3) Rbool gt
(wordsub(A,3 div 32) (1ltlt(3 mod 32)))
ltgt 0) int Rint (A,3)
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Type Erasure Semantics
sub int Rint (A,3)
(??.?xR(?). ?(A? array,iint). typecase x
of Rint gt wordsub(A,3) Rbool gt
(wordsub(A,3 div 32) (1ltlt(3 mod 32)))
ltgt 0) int Rint (A,3)
typecase Rint of Rint gt wordsub(A,3)
Rbool gt (wordsub(A,3 div 32) (1ltlt(3
mod 32))) ltgt 0
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Type based compilation
Terms
Types
Source Language
Intermediate Language
Machine Language
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ICFP '98
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Multi-stage Type Analysis
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Type Level Type Analysis
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Type Safe Language

• Give us guarentees about the run-time behavior of
  programs
• Types abstractly describe the run-time flow of
  values

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Traditional Compilation
( fn x gt x1 ) 3
Source File
Type Inference Checking
( fn x int gt x 1 ) 3
( ? x . x 1 ) 3
Untyped IL
l1 push 3 call l2 retn l2 mov
eax,esp4 add eax,eax mov
esp4, eax retn
Machine Code
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Type Based Compilation
( fn x gt x1 ) 3
Source File
Type Inference Checking
( fn x int gt x 1 ) 3
( ? x int . x 1 ) 3
Typed IL
l1 push 3 call l2 retn l2 mov
eax,esp4 add eax,eax mov
esp4, eax retn
Machine Code
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Why Typed Compilation

• Safety -- assurances about compiler correctness
• Type based optimizations
• For example ...

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But we dont always know the types, what then ?

• For example -- Parametric polymorphism
• Introduce an operator into the language that can
  distinguish types
• Typecase !

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Need a language with this operator - lmli
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Second example

• print

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Performance and Safety
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ICFP '98
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Safety
Terms
Types
Source
IL
Machine
TAL
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ICFP '98
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Type Passing Semantics

• Used by the language??iML , the Intermediate
  language of TIL/ML and FLINT compilers
• Unlike most calculi where types may be erased
  prior to run-time, types do have an operational
  significance -- they are arguments to typecase
  terms.

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ICFP '98
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Intensional Type Analysis

• Valuable element of type-directed compilers
• Allows otherwise untypeable optimizations
• Specialized data layout
• Tag-free Garbage Collection
• Polymorphic marshalling
• ...

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ICFP '98
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Solution

• More efficient
• Only pass type representations when necessary
• Traditional code optimizers can help
• Sophisticated techniques still possible
• Recovers abstraction
• Can withhold representation from clients

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