1. Continuation-passing Style (CPS)

2. Assignment-converted/alphatized IR (.alpha)
e ::= (let ([x e] ...) e)
| (lambda (x ...) e)
| (lambda x e)
| (apply e e)
| (e e ...)
| (prim op e ...)
| (apply-prim op e)
| (if e e e)
| (call/cc e)
| x
| (quote dat)

3. Administrative normal form (ANF) (.anf)
e ::= (let ([x e]) e)
| (apply ae ae)
| (ae ae ...)
| (prim op ae ...)
| (apply-prim op ae)
| (if ae e e)
| (call/cc ae)
| ae
ae ::= (lambda (x ...) e)
| (lambda x e)
| x
| (quote dat)

4. Continuation-passing style (CPS) (.cps)
e ::= (let ([x (apply-prim op ae)]) e)
| (let ([x (prim op ae ...)]) e)
| (apply ae ae)
| (ae ae ...)
| (if ae e e)
ae ::= (lambda (x ...) e)
| (lambda x e)
| x
| (quote dat)

5. Programs in CPS require no stack and never return.
e ::= (let ([x (apply-prim op ae)]) e)
| (let ([x (prim op ae ...)]) e)
| (apply ae ae)
| (ae ae ...)
| (if ae e e)
ae ::= (lambda (x ...) e)
| (lambda x e)
| x
| (quote dat)
Programs in CPS require no stack and never return.
Instead, at each application, a continuation (a callback function) is passed forward explicitly.
Points that would otherwise have extended the stack now create a closure (where the environment saves local variables and the stack tail).
Return points become invocations of the current continuation.

6. (let ([x ((lambda (y z) z) a b)]) e)
ecps has a free var for e’s cont.
((lambda (k y z) (k 0 z)) (lambda (k v) ecps) a b)
All functions take an extra continuation parameter.
As call/cc lets us pass continuations as values, so must they (despite not using it).

7. call/cc =
(lambda (k f)
(f k k))

8. Visualizing CPS (example)

9. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))

10. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))
fn {+} (fib (- n 2)) [n = 4]

11. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))
fn {+} (fib (- n 2)) [n = 3]
fn {+} (fib (- n 2)) [n = 4]

12. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))
fn {+} (fib (- n 2)) [n = 2]
fn {+} (fib (- n 2)) [n = 3]
fn {+} (fib (- n 2)) [n = 4]

13. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))
{1}
fn {+} (fib (- n 2)) [n = 2]
fn {+} (fib (- n 2)) [n = 3]
fn {+} (fib (- n 2)) [n = 4]

14. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))
fn {+} (fib (- n 2)) [n = 3]
fn {+} (fib (- n 2)) [n = 4]

15. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))
fn {+} {1}
fn {+} (fib (- n 2)) [n = 3]
fn {+} (fib (- n 2)) [n = 4]

16. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))
fn {+} (fib (- n 2)) [n = 3]
fn {+} (fib (- n 2)) [n = 4]

17. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))
fn {+} (fib (- n 2)) [n = 4]

18. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))
fn {+} (fib (- n 2)) [n = 4]

19. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))
fn {+} (fib (- n 2)) [n = 4]

20. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))

21. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))
fn {+} (fib (- n 2)) [n = 2]
fn {+} {2}

22. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))
fn {+} (fib (- n 2)) [n = 2]
fn {+} {2}

23. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))

24. IR (define (fib n) (if (<= n 1) n (+ (fib (- n 1)) (fib (- n 2)))))
fn {+} {1}
fn {+} {2} 3

25. ANF (define (fib n) (let ([c (<= n 1)]) (if c n
0(let ([n-1 (- n 1)])
1(let ([v0 (fib n-1)])
2(let ([n-2 (- n 2)])
3(let ([v1 (fib n-2)])
4(let ([s (+ v0 v1)])
5s)))))))
ANF
(fib 4)

26. ANF (define (fib n) (let ([c (<= n 1)]) (if c n
0(let ([n-1 (- n 1)])
1(let ([v0 (fib n-1)])
2(let ([n-2 (- n 2)])
3(let ([v1 (fib n-2)])
4(let ([s (+ v0 v1)])
5s)))))))
ANF
(fib 3)
letk v0 e2 [n=4,n-1=3,…]

27. ANF (define (fib n) (let ([c (<= n 1)]) (if c n
0(let ([n-1 (- n 1)])
1(let ([v0 (fib n-1)])
2(let ([n-2 (- n 2)])
3(let ([v1 (fib n-2)])
4(let ([s (+ v0 v1)])
5s)))))))
ANF
(fib 2)
letk v0 e2 [n=3,n-1=2,…]
letk v0 e2 [n=4,n-1=3,…]

28. ANF (define (fib n) (let ([c (<= n 1)]) (if c n
0(let ([n-1 (- n 1)])
1(let ([v0 (fib n-1)])
2(let ([n-2 (- n 2)])
3(let ([v1 (fib n-2)])
4(let ([s (+ v0 v1)])
5s)))))))
ANF
(fib 1) -> 1
letk v0 e2 [n=2,n-1=1,…]
letk v0 e2 [n=3,n-1=2,…]
letk v0 e2 [n=4,n-1=3,…]

29. ANF (define (fib n) (let ([c (<= n 1)]) (if c n
0(let ([n-1 (- n 1)])
1(let ([v0 (fib n-1)])
2(let ([n-2 (- n 2)])
3(let ([v1 (fib n-2)])
4(let ([s (+ v0 v1)])
5s)))))))
ANF
(fib 0) -> 0
letk v1 e4 [v0=1,n=2,n-1=1,…]
letk v0 e2 [n=3,n-1=2,…]
letk v0 e2 [n=4,n-1=3,…]

30. ANF (define (fib n) (let ([c (<= n 1)]) (if c n
0(let ([n-1 (- n 1)])
1(let ([v0 (fib n-1)])
2(let ([n-2 (- n 2)])
3(let ([v1 (fib n-2)])
4(let ([s (+ v0 v1)])
5s)))))))
ANF
s [s=1,v0=1,v1=0,…]
letk v0 e2 [n=3,n-1=2,…]
letk v0 e2 [n=4,n-1=3,…]

31. ANF (define (fib n) (let ([c (<= n 1)]) (if c n
0(let ([n-1 (- n 1)])
1(let ([v0 (fib n-1)])
2(let ([n-2 (- n 2)])
3(let ([v1 (fib n-2)])
4(let ([s (+ v0 v1)])
5s)))))))
ANF
(fib 1) -> 1
letk v1 e4 [v0=1,n=3,n-1=2,…]
letk v0 e2 [n=4,n-1=3,…]

32. ANF (define (fib n) (let ([c (<= n 1)]) (if c n
0(let ([n-1 (- n 1)])
1(let ([v0 (fib n-1)])
2(let ([n-2 (- n 2)])
3(let ([v1 (fib n-2)])
4(let ([s (+ v0 v1)])
5s)))))))
ANF
s [s=2,v0=1,v1=1,…]
letk v0 e2 [n=4,n-1=3,…]

33. ANF (define (fib n) (let ([c (<= n 1)]) (if c n
0(let ([n-1 (- n 1)])
1(let ([v0 (fib n-1)])
2(let ([n-2 (- n 2)])
3(let ([v1 (fib n-2)])
4(let ([s (+ v0 v1)])
5s)))))))
ANF
(fib 2)
letk v1 e4 [v0=2,n=4,n-1=3,…]

34. ANF (define (fib n) (let ([c (<= n 1)]) (if c n
0(let ([n-1 (- n 1)])
1(let ([v0 (fib n-1)])
2(let ([n-2 (- n 2)])
3(let ([v1 (fib n-2)])
4(let ([s (+ v0 v1)])
5s)))))))
ANF
(fib 1) -> 1
letk v0 e2 [n=2,n-1=3,…]
letk v1 e4 [v0=2,n=4,n-1=3,…]

35. ANF (define (fib n) (let ([c (<= n 1)]) (if c n
0(let ([n-1 (- n 1)])
1(let ([v0 (fib n-1)])
2(let ([n-2 (- n 2)])
3(let ([v1 (fib n-2)])
4(let ([s (+ v0 v1)])
5s)))))))
ANF
(fib 0) -> 0
letk v1 e4 [v0=1,n=2,n-1=3,…]
letk v1 e4 [v0=2,n=4,n-1=3,…]

36. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(fib 4 print)

37. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(lambda (v0) …)
n=4 k =
(fib 3 k)
print

38. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
print
(lambda (v0) …)
n=4 k =
(lambda (v0) …)
n=3 k =
(fib 2 k)

39. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(lambda (v0) …)
n=3 k =
print
(lambda (v0) …)
n=2 k =
(fib 1 k)
(lambda (v0) …)
n=4 k =

40. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(lambda (v0) …)
n=3 k =
print
(lambda (v0) …)
n=2 k =
(k 1)
(lambda (v0) …)
n=4 k =

41. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(lambda (v0) …)
n=3 k =
print
(lambda (v1) …)
n=2 v0=1 k
(fib 0 k)
(lambda (v0) …)
n=4 k =

42. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(lambda (v0) …)
n=3 k =
print
(k 0)
(lambda (v1) …)
n=2 v0=1 k
(lambda (v0) …)
n=4 k =

43. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(lambda (v0) …)
n=3 k =
print
(k 1)
(lambda (v0) …)
n=4 k =

44. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(lambda (v1) …)
n=3 v0=1 k
print
(fib 1 k)
(lambda (v0) …)
n=4 k =

45. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(lambda (v1) …)
n=3 v0=1 k
print
(k 1)
(lambda (v0) …)
n=4 k =

46. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
print
(k 2)
(lambda (v0) …)
n=4 k =

47. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
print
(fib 2 k)
(lambda (v1) …)
n=4 v0=2 k

48. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(fib 1 k)
print
(lambda (v0) …)
n=2 k =
(lambda (v1) …)
n=4 v0=2 k

49. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(k 1)
print
(lambda (v0) …)
n=2 k =
(lambda (v1) …)
n=4 v0=2 k

50. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(fib 0 k)
print
(lambda (v1) …)
n=2 v0=1 k
(lambda (v1) …)
n=4 v0=2 k

51. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(k 0)
print
(lambda (v1) …)
n=2 v0=1 k
(lambda (v1) …)
n=4 v0=2 k

52. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
print
(k 1)
(lambda (v1) …)
n=4 v0=2 k

53. CPS (define (fib n k) (let ([c (<= n 1)]) (if c (k n)
(let ([n-1 (- n 1)])
(fib n-1
(lambda (v0)
(let ([n-2 (- n 2)])
(fib n-2
(lambda (v1)
(let ([s (+ v0 v1)])
(k s)))))))))))
CPS
(k 3)
print