Tim Sheard Oregon Graduate Institute

1. Fundamentals of Staged Computation Tim Sheard Oregon Graduate Institute Lecture 6: Monadic Staging of the RE language CSE 510 Section FSC Winter 2004

2. Assignments • Assignment 4 is now posted on webpage • Due in 1 week on Feb 1, 2005 • Reading Assignment now on webpage • Staging Algebraic Datatypes • By Tim Sheard and Iavor Diatchki • Must be read by next Tueday Feb. 1 • Volunteers for presentation? Cs510 FSC Winter 2005

3. Recall RE Language datatype 'a Maybe = Just of 'a | Nothing; datatype RE = Lit of string | Or of (RE*RE) | Concat of (RE*RE) | Star of RE; fun prefix [] xs = Just xs | prefix (y::ys) (x::xs) = if y=x then prefix ys xs else Nothing | prefix ys xs = Nothing; Cs510 FSC Winter 2005

4. One interpreter fun eval (Lit s) input = (case prefix (explode s) input of Just more => Just(s,more) | Nothing => Nothing) | eval (Or(a,b)) input = (case eval a input of Nothing => eval b input | Just pair => Just pair) | eval (Concat(a,b)) input = (case (eval a input) of Just(zs,rest) => (case eval b rest of Just (bs,more) => Just (zs ^ bs,more) | Nothing => Nothing) | Nothing => Nothing) | eval (Star x) input = let fun f input = case eval x input of Nothing => ("",input) | Just (cs,zs) => let val (bs,ws) = f zs in (cs ^ bs,ws) end in Just(f input) end; Cs510 FSC Winter 2005

5. Staging it fun Seval (Lit s) input = <case prefix (explode ~(lift s)) ~input of Just more => Just(~(lift s),more) | Nothing => Nothing> | Seval (Or(a,b)) input = <case ~(Seval a input) of Nothing => ~(Seval b input) | Just pair => Just pair> | Seval (Concat(a,b)) input = <case ~(Seval a input) of Just(zs,rest) => (case ~(Seval b <rest>) of Just (bs,more) => Just (zs ^ bs,more) | Nothing => Nothing) | Nothing => Nothing> Cs510 FSC Winter 2005

6. Staging continued Note how the helper function “f” is in the generated code. | Seval (Star x) input = <let fun f input = case ~(Seval x <input>) Nothing => ("",input) | Just (cs,zs) => let val (bs,ws) = f zs in (cs ^ bs,ws) end in Just(f ~input) end>; Cs510 FSC Winter 2005

7. What does it generate? val t1 = Concat(Or(Lit "tim",Lit "mary") ,Star (Lit "x")); fun test x = <fn s => ~(Seval x <s>)>; fun f x = (run(test t1)) (explode x); test t1; Cs510 FSC Winter 2005

8. -| test t1; val it = <(fn a => (case (case (case %prefix (%explode "tim") a of Just o => Just("tim",o) | Nothing => Nothing) of Nothing => (case %prefix (%explode "mary") a of Just n => Just("mary",n) | Nothing => Nothing) | Just m => Just m) of Just(c,b) => (case let fun f g = (case (case %prefix (%explode "x") g of Just l => Just("x",l) | Nothing => Nothing) of Nothing => ("",g) | Just(i,h) => let val (k,j) = f h in (i %^ k,j) end) in Just (f b) end of Just(e,d) => Just(c %^ e,d) | Nothing => Nothing) | Nothing => Nothing))> : <char list -> (string * char list) Maybe> Cs510 FSC Winter 2005

9. Monadic version <Do %mm { a <- %try (%prefix2 (%explode "tim")) (%prefix2 (%explode "mary")) ; b <- %star (%prefix2 (%explode "x")) ; Return %mm (a %@ %concat b) }> : <char list M> • What is the Monad used here? Cs510 FSC Winter 2005

10. The “meaning” type Eval :: RE -> char list -> (string * char list) Maybe datatype 'a M = M of (char list -> ('a* char list) Maybe); fun unM (M x) = x; • Eval2 :: RE -> char list M Cs510 FSC Winter 2005

11. Implementing the Monad val mm = let fun return x = M (fn s => Just(x,s)); fun bind (M f) g = let fun h s = case f s of Just(a,s2) => unM (g a) s2 | Nothing => Nothing in M h end in Mon(return,bind) end; Cs510 FSC Winter 2005

12. Operations on the monad fun test (M f) = let fun h s = case f s of Nothing => Just(Nothing,s) | Just(a,rest) => Just(Just a,rest) in M h end; fun prefix2 s = let fun h x = case prefix s x of Nothing => Nothing | Just m => Just(s,m) in M h end; fun try (M f) (M g) = let fun h s = case f s of Nothing => g s | Just pair => Just pair in M h end Cs510 FSC Winter 2005

13. Monadic Version fun star m = Do mm { mx <- test m ; case mx of Just x => Do mm { xs <- star m ; Return mm (x::xs) } | Nothing => Return mm []}; fun eval2 (Lit s) = prefix2 (explode s) | eval2 (Or(a,b)) = try (eval2 a) (eval2 b) | eval2 (Concat(a,b)) = Do mm { x <- eval2 a ; y <- eval2 b ; Return mm (x@y)} | eval2 (Star x) = Do mm { xss <- star (eval2 x) ; Return mm (concat xss) } Cs510 FSC Winter 2005

14. Staged Monadic Version fun Seval2 (Lit s) = <prefix2 (explode ~(lift s))> | Seval2 (Or(a,b)) = <try ~(Seval2 a) ~(Seval2 b)> | Seval2 (Concat(a,b)) = <Do mm { x <- ~(Seval2 a) ; y <- ~(Seval2 b) ; Return mm (x@y)}> | Seval2 (Star x) = <Do mm { xss <- star ~(Seval2 x) ; Return mm (concat xss) }>; Cs510 FSC Winter 2005

15. Results -| Seval2 t1; val it = <Do %mm { a <- %try (%prefix2 (%explode "tim")) (%prefix2 (%explode "mary")) ; b <- %star (%prefix2 (%explode "x")) ; Return %mm (a %@ %concat b) }> : <char list M> Note calls to the monadic operators “try”, “star”, “test” (embedded in call to “star”), and “prefix2” Cs510 FSC Winter 2005

16. Cs510 FSC Winter 2005