CALCULATINGCORRECT COMPILERS

Graham Hutton and Patrick Bahr

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How To Make Sausages

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How To Make Compilers?

language compiler

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This Talk

A new approach to the problem of calculating compilers from high-level semantics;

Only requires simple techniques, and all the calculations have been formalised in Coq;

Scales to exceptions, state, variable binding, loops, non-determinism, interrupts, etc.

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Arithmetic Expressions

data Expr = Val Int | Add Expr Expr

eval :: Expr Int

eval (Val n) = n

eval (Add x y) = eval x + eval y

Syntax:

Semantics:

Example

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1 + (2 + 3)

Add (Val 1) (Add (Val 2) (Val 3))

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Step 1 – Stacks

Aim: define a new semantics

evalS :: Expr Stack Stack

evalS e s = eval e : s

such that

Make the manipulation of arguments explicit by transforming the semantics to use a stack.

Stack = [Int]

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evalS (Add x y) s

Case for addition:

eval (Add x y) : s

=

(eval x + eval y) : s=

add (evalS y (evalS x s))=

add (eval y : evalS x s)=

add (eval y : eval x : s)=

add (n:m:s) = m+n : s

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New semantics:

evalS :: Expr Stack Stack

evalS (Val n) s = push n s

evalS (Add x y) s = add (evalS y (evalS x s))

Stack operations:

push n s = n : s

add (n:m:s) = m+n : s

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Step 2 – Continuations

Make the flow of control explicit by transforming the semantics into continuation-passing style.

Definition:

A continuation is a function that is applied to the result of another

computation.

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Aim: define a new semantics

evalC e c s = c (evalS e s)

such that

evalC :: Expr Cont Cont

Cont = Stack Stack

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New semantics:

evalC :: Expr Cont Cont

evalC (Val n) c s = c (push n s)

evalC (Add x y) c s = evalC x (evalC y (c . add)) s

Previous semantics:

evalS :: Expr Stack Stack

evalS e = evalC e (λs s)

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Step 3 - Defunctionalise

Basic idea:

Represent the continuations that we actually need using a

datatype.

Make the semantics first-order again by applying the technique of defunctionalisation.

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comp :: Expr Code

comp e = comp’ e HALT

New semantics:

comp’ :: Expr Code Code

comp’ (Val n) c = PUSH n c

comp’ (Add x y) c = comp’ x (comp’ y (ADD c))

A compiler for arithmetic expressions!

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1 + (2 + 3)

Example

PUSH 1PUSH 2PUSH 3ADD ADDHALT

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data Code = PUSH Int Code | ADD Code | HALT

New datatype and its interpretation:

exec :: Code Stack Stack

exec (PUSH n c) s = exec c (n:s)

exec (ADD c) (n:m:s) = exec c (m+n : s)

exec HALT s = s

A virtual machine for arithmetic expressions!

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1 + (2 + 3)

Example

PUSH 1PUSH 2PUSH 3ADD ADDHALT

Stack

Code

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1 + (2 + 3)

Example

PUSH 2PUSH 3ADD ADDHALT

Stack

Code

1

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1 + (2 + 3)

Example

PUSH 3ADD ADDHALT

Stack

Code

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1 + (2 + 3)

Example

ADD ADDHALT

Stack

Code

321

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1 + (2 + 3)

Example

ADDHALT

Stack

Code

51

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1 + (2 + 3)

Example

HALT

Stack

Code

6

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1 + (2 + 3)

Example

Stack

Code

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Compiler Correctness

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Is captured by the following two equations:

These follow from defunctionalisation, or can be

verified by simple inductive proofs.

exec c (eval e : s)

exec (comp e) s eval e : s=

exec (comp’ e c) s =

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Reflection

We now have a three step process for calculating a correct compiler from a high-level semantics:

Can the steps be combined?

1 - Add a stack2 - Add a continuation3 - Remove the continuations

The Trick

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Start directly with the correctness equations:

Aim to calculate definitions for comp, comp’, exec and Code that satisfy

these equations.

exec c (eval e : s)

exec (comp e) s eval e : s=

exec (comp’ e c) s =

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In Practice

Calculating four interlinked definitions at the same time seems like an impossible task;

But… with experience gained from our stepwise approach, it turns out to be straightforward;

New calculation is simpler, more direct, and has the same structure as our stepwise version.

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Summary

Purely calculational approach to developing compilers that are correct by construction;

Only requires simple techniques, and scales to a wide variety of language features;

More sophisticated languages also introduce the idea of using partial specifications.

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Further Work

Register-based machines;

Real source/target languages;

Mechanical assistance;

EPSRC application.

CALCULATINGCORRECT COMPILERS

Graham Hutton and Patrick Bahr