Type Specialisation

John Hughes

Chalmers University

Type Specialisation(With thanks to Neil Jones)

John Hughes

Chalmers University

1985: A Weird Program Called MIX!

The Mix Equation

(mix prog x) y = prog x y

The Futamura Projections

Specialised program

mix interp prog = codebecause

code data = interp prog data

1985: A Weird Program Called MIX!

The Mix Equation

(mix prog x) y = prog x y

The Futamura Projections

Specialised program

mix mix interp = compilerbecause

compiler prog = mix interp prog = code

1985: A Weird Program Called MIX!

The Mix Equation

(mix prog x) y = prog x y

The Futamura Projections

Specialised program

mix mix mix = compiler-generatorbecause

compiler-generator interp = mix mix interp = compiler

Why Did He Use Scheme?

Why not:

•A language with lazy evaluation?

•A language with types?

1987: Partial Evaluation and Mixed Computation

1987: Partial Evaluation and Mixed Computation

A ”Hughes”

•a measure of the irrelevance of a talk to the workshop topic.

1987: Partial Evaluation and Mixed Computation

Challenging Problems

•eliminating tags from typed interpreters

data V = N Int | F (V -> V)

eval (Const n) = N neval (Lam x e) = F (v. eval (bind x v ) e)…

Lam ”x” (Const 3)

F (v. N 3)

mix

1987: Partial Evaluation and Mixed Computation

Challenging Problems

•eliminating tags from typed interpreters

•type specialisation

data V = N Int | F (V -> V)…

Int

Int->Int

(Int->Int)->Int

1987: Partial Evaluation and Mixed Computation

Challenging Problems

•eliminating tags from typed interpreters

•type specialisation

•optimal specialisation of typed interpreters

mix self-interp prog prog x. 3

F (x. N 3)

1995: Neil’s Seminar @ Chalmers

•How can we recognise an interpreter for a strongly typed language?

•How can we derive a type-checker from an interpreter?

•An approach for a first-order language.

1995: Neil’s Seminar @ Chalmers

•How can we recognise an interpreter for a strongly typed language?

•How can we derive a type-checker from an interpreter?

•An approach for a first-order language.

Surely one can dobetter?

1995: Neil’s Seminar @ Chalmers

•How can we recognise an interpreter for a strongly typed language?

•How can we derive a type-checker from an interpreter?

•An approach for a first-order language.

Invitation to Dagstuhl Seminar on Partial Evaluation!

Key Idea: Offline Partial Evaluation

Annotations in the source code classify operations as

•static -- performed by mix

•dynamic -- built into residual program

eval (Const n) = N (lift n)eval (Lam x e) = F (v. eval (bind x v ) e)… bind x v = y. if x=y then v else y

Underlinedoperations are

dynamic

Key Idea: Offline Partial Evaluation

Annotations in the source code classify operations as

•static -- performed by mix

•dynamic -- built into residual program

eval (Const n) = N (lift n)eval (Lam x e) = F (v. eval (bind x v ) e)… bind x v = y. if x=y then v else y

Lam ”x” (Const 3)

F (v. N 3)

mix

Key Idea: Offline Partial Evaluation

Annotations in the source code classify operations as

•static -- performed by mix

•dynamic -- built into residual program

eval (Const n) = N (lift n)eval (Lam x e) = F (v. eval (bind x v ) e)… bind x v = y. if x=y then v else y

Don’t underlinethe tags!

But we have to!

Interesting Properties

N (z+lift 1) N Int

x. case x of N n -> N (n+lift 1) F f -> …

N Int -> N Int

f @ N (z+lift 1) N Int

Keep Track of Tags by Type Inference!

e e´ : ´ ´ ::= Int | ´-> ´ | C ´

Residual types

|- e e´ : ´

|- C e e´ : C ´

|- e e´ : Ck 1´

, xk e´ : 1´ |- ek ek´ : 2´

|- case e of {Cj xj -> ej} ek´ : 2´

Reformulate Partial Evaluation in Terms of Type Inference

e e´ : ´ ´ ::= Int | ´-> ´ | C ´ | n | …

Residual types

x. x+1 x. : 2 -> 3

x. x+z <z> : close <z : Int> in x. x+z

Real Increase in Power

Type specialisation=

Type-based analysis + type-directed transformation

Examples

•Monomorphisation

•Typed defunctionalisation

•Constructor specialisation for -calculus

By type-specialising a suitable interpreter.

Continuing Work

•Simple type specialiser as a back-end for PE to achieve optimality (PADO II).

•Principal type specialisations.

•Strategies for controlling polyvariance (type-based cloning)

•…

Continuing Work

•Simple type specialiser as a back-end for PE to achieve optimality (PADO II).

•Principal type specialisations & polymorphism.

•Strategies for controlling polyvariance (type-based cloning)

•…

Thank you Neil!