TagElimination and Jones Optimality

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Tag-Elimination and Jones Optimality

• Walid Taha (Yale)
• Henning Makholm (DIKU)
• John Hughes (Chalmers)

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Compiling by Specialising Interpreters
The Mix Equation (mix prog data) data prog
data data The First Futamura Projection (mix
interp prog) data interp prog data Optimal
Specialisation mix self_interp prog prog
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Compiling by Specialising Interpreters
Typed
The Mix Equation (mix prog data) data prog
data data The First Futamura Projection (mix
interp prog) data interp prog data Optimal
Specialisation mix self_interp prog prog
interp Prog -gt Val -gt Val
??????
Val -gt Val
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Compiling by Specialising Interpreters
Typed
The Mix Equation (mix prog data) data prog
data data The First Futamura Projection (mix
interp prog) data interp prog data Optimal
Specialisation mix self_interp prog prog
interp Prog -gt Val -gt Val
??????
Val -gt Val
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The Tagging Problem
data Val N Int F (Val -gt Val) eval (Const n)
env N n eval (Lam x e) env F (?v. eval
e (bind x v env)) eval (App e e) env unF
(eval e env) (eval e env)
?f.?x.f (f x)
mix
F (?f. F (?x. unF f (unF f x)))
Unwanted tags -- cost 2x
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This Paper Recovering Optimality
Optimal Specialisation tagElim (mix self_interp
prog) (typeOf prog) prog
????????
Val -gt Val
Eliminates tags, converts to type of prog.
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Havent we heard this before?

• Type specialisation (Hughes)
• specialised an interpreter for ?-calculus to
  terms without tags -- but not a self-interpreter!
• large step away from partial evaluation.
• Tag elimination (Taha)
• eliminated tags (provably), but for a tiny
  language -- no self-interpreter!
• Optimal typed specialisation (Makholm)
• first-order language, complex -- many stages
• experimentally optimal, but no proof!

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This Paper

• Higher-order language ?-calculus with data
  types.
• A complete self-interpreter.
• Standard partial evaluation, followed by a simple
  tag-elimination post-processor.
• Provably optimal specialisation.
• Implementation matching the theory.

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The Language
? D V ? -gt ?
S-expression data
Values in interpreted programs data V E D F
(V -gt V)
s x (s.s) u car cdr atom? o
cons equal? e x e e ??x.e fix x.e
s u e o e e if e e e E e unE e
F e unF e
No case expression!
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The Self-Interpreter
fix eval. ??env. ?e. (if (equal? (car e) quote)
(E (car (cdr e))) (if (equal? (car e) E) (E
(unE (eval env (car (cdr e)))))
??????? V
?????????? D
Run-time type check
????????????? V
e and E e evaluate to the same representation! OK,
since they have different types.
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Tag-Elimination Annotations

• Each tag/untag operation is annotated
• k (keep), or
• e (eliminate)

Erasure
Tag Elimination
Ee e E e Ek e E e unEe e unE
e unEk e unE e
Ee e e Ek e E e unEe e
e unEk e unE e
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Annotating the Self-Interpreter
fix eval. ??env. ?e. (if (equal? (car e) quote)
(Ee (car (cdr e))) (if (equal? (car e) E) (Ek
(unEe (eval env (car (cdr e)))))
mix
eval env (E (quote x))
Ek (unEe (Ee x))
_
E x
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The Analysis
How do we know e is well typed? _
changes types! E.g. unFe (Ee x) (Ee y)
x y
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The Analysis
Richer Annotated type system
How do we know e is well typed? _
changes types! E.g. unFe (Ee x) (Ee y)
x y
??? E D
F a -gt a
unFk (Ek x) (Ek y) unF (E x) (E y)
??? V
V -gt (V -gt V)
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Annotated Types
c V E D F (c -gt c) can be produced by
_ a term of type V a D c a -gt
a can be produced by _
c V D D a -gt a a -gt a
V V E D D F (c -gt c) c -gt
c D D a -gt a a -gt a
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Roadmap
self_interp
Well-typed, but not well-annotated.

prog
mix self_interp prog
Well-typed, but perhaps not well-annotated.
Analysis succeeds?
Some sensible default.
tagElim (mix )
Well-typed, prog.
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Wrapping and Unwrapping
Wltagt a -gt a
Ultagt a -gt a
WltDgt x x WltVgt x x WltE Dgt x E x WltF agt f F
(Wltagt f) Wlta-gtagt f
?x. Wltagt (f (Ultagt x))
UltDgt x x UltVgt x x UltE Dgt x unE x UltF agt f
Ultagt (unF f) Ulta-gtagt f
?x. Ultagt (f (Wltagt x))
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Correctness of Tag Elimination
Now we can define the tag elimination
transformation ? e if e a tagElim e
a ? ? Ultagt e otherwise
Correctness
? e a. tagElim e a ? Ultagt e
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Mix Self_interp A Closer Look
mix self_interp performs a simple syntax-directed
translation ?(?x. e) Fe (?x. ?e) ?(E e) Ek
(unEe (?e)) ?(e e) unFe (?e) (?e) ?(unE e)
Ee (unEk (?e)) Extends to a translation of
types ?D E D ?V V ?(?-gt?) F (??-gt??)
e ? ? ?e ??
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Typed Partial Evaluation and Optimality
The Typed Mix Equation e e a ? tmix e e a
? Ultagt (e e) Optimal Typed Specialisation e
? ? tmix self_interp e (??) ? e
Satisfied by tmix e e a tagElim (mix e e) a
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Conclusions

• The mix equation and Jones-optimality
  reformulated to make them consistent in a typed
  setting.
• Tag elimination is a simple post-processor which
  achieves optimality in theory and in practice,
  for a fairly rich language, building on existing
  PE technology.
• Future work
• Polymorphism?
• User-defined datatypes?
• Dynamic tag tests?