CUTTING OUT CONTINUATIONS Graham Hutton and Patrick Bahr

Background In his classic 1972 paper on definitional interpreters, John Reynolds introduced two key techniques: z Continuation-passing style - Makes control flow explicit z Defunctionalisation - Makes programs first-order 1

Background These techniques are often used together: language CPS DEFUN compiler Can the twocontinuations steps be fused together !? Introduce Remove continuations 2

This Talk z A new approach to transforming high-level semantics into low-level implementations; z Only requires simple calculation techniques, and avoids the use of continuations; z Scales to exceptions, state, variable binding, loops, non-determinism, interrupts, etc. 3

Arithmetic Expressions Syntax: data Expr = Val Int | Add Expr Semantics: eval : : Expr Int eval (Val n) = n eval (Add x y) = eval x + eval y 4

Step 1 – Add 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. 5

Specification Aim: define a new semantics eval’ : : Expr Cont Int such that Cont = Int eval’ e c = c (eval e) 6

New semantics: eval’ : : Expr Cont Int eval’ (Val n) c = c n eval’ (Add x y) c = eval’ x ( n eval’ y ( m c (n + m))) Original semantics: Control flow now explicit. eval e = eval’ e ( n n) 7

Step 2 - Defunctionalise Make the semantics first-order again by applying the technique of defunctionalisation. Basic idea: Represent the continuations that we actually need using a datatype. 8

Specification Aim: define a new semantics eval’’ : : Expr CONT Int such that exec : : CONT Cont eval’’ e c = eval’ e (exec c) 9

New semantics: eval’’ : : Expr CONT Int eval’’ (Val n) c = exec c n eval’’ (Add x y) c = eval’’ x (NEXT y c) exec : : CONT Int exec (NEXT y c) n = eval’’ y (ADD n c) exec (ADD n c) m = exec c (n + m) exec HALT n=n An abstract machine for evaluating expressions. 10

Reflection We now have a two step process for calculating an abstract machine from a high-level semantics: 1 – Add a continuation 2 – Remove the continuations Can the steps be combined? 11

The Trick Start directly with the combined specification: eval’’ e c = exec c (eval e) Aim to calculate definitions for eval’’, exec and CONT that satisfy these equations. 12

In Practice z Calculating three interlinked definitions at the same time seems like an impossible task; z But. . . with experience gained from our stepwise approach, it turns out to be straightforward; z New calculation is simpler, more direct, and eliminates the use of continuations. 13

Case: e = Add x y eval’’ (Add x y) c = exec c (eval (Add x y)) = exec c (eval x + eval y) But usingto induction, Nowwe weare appear be stuck, soaswe no have definitions induction hypotheses! further can be applied. 14

To use the hypothesis for y, we need a term of form exec c’ (eval y) for some c’, i. e. we must solve: = exec c’ (eval y) exec c (eval x + eval y) We start by generalising to: exec c’ m = exec c (n + m) But we can’t simply use this as a definition for exec, because c and n are unbound. 15

Solution Package the free variables up in the argument c’ by adding a new constructor to the CONT type, ADD : : Int CONT and a new equation for the exec function: exec (ADD n c) m = exec c (n + m) 16

Now we can continue: exec c (eval x + eval y) = exec (ADD (eval x) c) (eval y) = eval’’ y (ADD (eval x) c) = . . . 17

Summary z The rest of the calculation of the abstract machine proceeds in a similar manner; z The driving force for the calculation is the desire to apply induction hypotheses; z We solve the resulting equations by adding new constructors to CONT on demand. 18

Final Result data CONT where NEXT : : Expr CONT ADD : : Int CONT HALT : : CONT eval’’ (Val n) c = exec c n eval’’ (Add x y) c = eval’’ x (NEXT y c) exec (NEXT y c) n = eval’’ y (ADD n c) exec (ADD n c) m = exec c (n + m) exec HALT n=n 19

Conclusion z Purely calculational approach to cutting out the intermediate use of continuations; z Only requires simple techniques, and scales to a wide variety of language features; z Can also be used to calculate compilers, and everything has been formalised in Coq. 20

CUTTING OUT CONTINUATIONS Graham Hutton and Patrick Bahr