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Background

Background. Verifying a compiler for a simple language with exceptions (MPC 04). Calculating an abstract machine that is correct by construction (TFP 05). This Talk. We show how to calculate a compiler for a simple language of arithmetic expressions;

francesca-roberts
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Background

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  1. Background Verifying a compiler for a simple language with exceptions (MPC 04). Calculating an abstract machine that is correct by construction (TFP 05).

  2. This Talk • We show how to calculate a compiler for a simple language of arithmetic expressions; • Our new approach builds upon and refines earlier work by Wand and Danvy. • We make essential use of dependent types, and our work is formalised in Agda.

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

  4. Step 1 - Sequencing Rewrite the semantics in combinatory form using a special purpose sequencing operator. Raising a type to a power: a3 a  a  a  a = 3 arguments

  5. Sequencing: (;) :: an a1+m an+m Example (n = 3, m = 2): g f ; g = f

  6. We can now rewrite the semantics: eval :: Expr  Int0 eval (Val n) = return n eval (Add x y) = eval x ; (eval y ; add) where Only uses the powers 0,1,2. return :: Int  Int0 return n = n add :: Int2 add = λn m  m+n

  7. Step 2 - Continuations Generalise the semantics to arbitrary powers by making the use of continuations explicit. Definition: A continuation is a function that is applied to the result of another computation.

  8. Aim: define a new semantics eval’ :: Expr  Int1+m Intm such that eval’ e c = eval e ; c and hence halt = λn  n eval e = eval’ e halt

  9. = eval (Add x y) ; c = (eval x ; (eval y ; add)) ; c = eval x ; (eval y ; (add ; c)) = eval’ x (eval y ; (add ; c)) = eval’ x (eval’ y (add ; c)) Case: e = Add x y eval’ (Add x y) c eval (Add x y) ; c (eval x ; (eval y ; add)) ; c eval x ; (eval y ; (add ; c)) eval’ x (eval y ; (add ; c))

  10. New semantics: eval :: Expr  Int0 eval e = eval’ e halt eval’ :: Expr  Int1+m Intm eval’ (Val n) c = return n ; c eval’ (Add x y) c = eval’ x (eval’ y (add ; c)) The semantics now uses arbitrary powers.

  11. Step 3 - Defunctionalize Make the semantics first-order again, by applying the defunctionalization technique. Basic idea: Represent the functions of type Intn we actually need using a datatype Term n.

  12. New semantics: eval :: Expr Term0 eval e = eval’ e Halt eval’ :: Expr Term (1+m)Term m eval’ (Val n) c = Return n c eval’ (Add x y) c = eval’ x (eval’ y (Add c)) The semantics is now first-order again.

  13. New datatype: data Term n where Halt :: Term 1 Return :: Int  Term (1+n)  Term n Add :: Term (1+n)  Term (2+n) Interpretation: exec :: Term n -> Intn exec Halt = halt exec (Return n c) = return n ; exec c exec (Add c) = add ; exec c

  14. Example Add (Val 1) (Add (Val 2) (Val 3)) eval Return 1 $ Return 2 $ Return 3 $ Add $ Add $ Halt exec 6

  15. Summary • Purely calculational development of a compiler for simple arithmetic expressions; • Use of dependent types arises naturally during the process to keep track of stack usage; • Technique has also been used to calculate a compiler for a language with exceptions.

  16. Ongoing and Further Work • Using an explicit stack type; • Other control structures; • Relationship to other approaches; • Further exploiting operads.

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