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Objectives Accumulating Recursion Activity References Tail Recursion Dr. Mattox Beckman University of Illinois at Urbana-Champaign Department of Computer Science Objectives Accumulating Recursion Activity References Objectives

  1. Objectives Accumulating Recursion Activity References Tail Recursion Dr. Mattox Beckman University of Illinois at Urbana-Champaign Department of Computer Science

  2. Objectives Accumulating Recursion Activity References Objectives ◮ Identify expressions that have subexpressions in tail position. ◮ Explain the tail call optimization. ◮ Convert a direct style recursive function into an equivalent tail recursive function.

  3. Objectives Accumulating Recursion Activity References Tail Calls Tail Position A subexpression s of expressions e , if it is evaluated, will be taken as the value of e . Consider this code: ◮ if x > 3 then x + 2 else x - 4 ◮ f (x * 3) – no (proper) tail position here Tail Call A function call that occurs in tail position ◮ if h x then h x else x + g x

  4. aux n a = aux (n - 1) (a * n) where aux 0 a = a Objectives Accumulating Recursion Activity References Your Turn Find the tail calls! Example Code 1 fact1 0 = 1 2 fact1 n = n * fact1 (n - 1) 3 4 fact2 n = aux n 1 5 6 7 8 fib 0 = 0 9 fib 1 = 1 10 fib n = fib (n - 1) + fib (n - 2)

  5. Objectives Accumulating Recursion Activity References Tail Call Example ◮ If one function calls another in tail position, we get a special behavior. Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ What happens when we call foo 1 ?

  6. 1 x ret Objectives Accumulating Recursion Activity References Tail Call Example ◮ If one function calls another in tail position, we get a special behavior. Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ What happens when we call foo 1 ?

  7. ret x 2 y ret 1 Objectives Accumulating Recursion Activity References Tail Call Example ◮ If one function calls another in tail position, we get a special behavior. Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ What happens when we call foo 1 ?

  8. ret x 3 z ret 2 y ret 1 Objectives Accumulating Recursion Activity References Tail Call Example ◮ If one function calls another in tail position, we get a special behavior. Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ What happens when we call foo 1 ?

  9. 30 x ret 3 z ret 2 y ret 1 Objectives Accumulating Recursion Activity References Tail Call Example ◮ If one function calls another in tail position, we get a special behavior. Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ What happens when we call foo 1 ?

  10. 30 1 ret 3 z ret 30 2 y ret x Objectives Accumulating Recursion Activity References Tail Call Example ◮ If one function calls another in tail position, we get a special behavior. Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ What happens when we call foo 1 ?

  11. 30 1 ret 3 z 30 ret 30 2 y ret x Objectives Accumulating Recursion Activity References Tail Call Example ◮ If one function calls another in tail position, we get a special behavior. Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ What happens when we call foo 1 ?

  12. 30 30 ret 3 z 30 ret 30 2 y ret 1 x Objectives Accumulating Recursion Activity References Tail Call Example ◮ If one function calls another in tail position, we get a special behavior. Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ What happens when we call foo 1 ?

  13. 30 30 ret 3 z 30 ret 30 2 y 30 ret 1 x Objectives Accumulating Recursion Activity References Tail Call Example ◮ If one function calls another in tail position, we get a special behavior. Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ What happens when we call foo 1 ?

  14. Objectives Accumulating Recursion Activity References The Tail Call Optimization Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ If that’s the case, we can cut out the middle man …

  15. x 1 ret Objectives Accumulating Recursion Activity References The Tail Call Optimization Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ If that’s the case, we can cut out the middle man …

  16. x 1 ret y 2 ret Objectives Accumulating Recursion Activity References The Tail Call Optimization Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ If that’s the case, we can cut out the middle man …

  17. ret x 3 z ret 2 y ret 1 Objectives Accumulating Recursion Activity References The Tail Call Optimization Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ If that’s the case, we can cut out the middle man …

  18. 30 1 ret 3 z ret 2 y ret x Objectives Accumulating Recursion Activity References The Tail Call Optimization Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ If that’s the case, we can cut out the middle man …

  19. 30 30 ret 3 z ret 2 y 30 ret 1 x Objectives Accumulating Recursion Activity References The Tail Call Optimization Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ If that’s the case, we can cut out the middle man …

  20. Objectives Accumulating Recursion Activity References The Tail Call Optimization Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ If that’s the case, we can cut out the middle man … ◮ Actually, we can do even better than that.

  21. Objectives Accumulating Recursion Activity References The Optimization ◮ When a function is in tail position, the compiler will recycle the activation record ! Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10

  22. x 1 ret Objectives Accumulating Recursion Activity References The Optimization ◮ When a function is in tail position, the compiler will recycle the activation record ! Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10

  23. y 2 ret Objectives Accumulating Recursion Activity References The Optimization ◮ When a function is in tail position, the compiler will recycle the activation record ! Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10

  24. z 3 ret Objectives Accumulating Recursion Activity References The Optimization ◮ When a function is in tail position, the compiler will recycle the activation record ! Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10

  25. z 3 ret 30 Objectives Accumulating Recursion Activity References The Optimization ◮ When a function is in tail position, the compiler will recycle the activation record ! Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10

  26. z 3 ret 30 Objectives Accumulating Recursion Activity References The Optimization ◮ When a function is in tail position, the compiler will recycle the activation record ! Example 1 foo x = bar (x + 1) 2 bar y = baz (y + 1) 3 baz z = z * 10 ◮ This allows recursive functions to be written as loops internally.

  27. Objectives Accumulating Recursion Activity References Direct-Style Recursion ◮ In recursion, you split the input into the “fjrst piece” and the “rest of the input.” ◮ In direct-style recursion: the recursive call computes the result for the rest of the input, and then the function combines the result with the fjrst piece. ◮ In other words, you wait until the recursive call is done to generate your result. Direct Style Summation 1 sum [] = 0 2 sum (x : xs) = x + sum xs

  28. where aux [] a = a aux (x : xs) a = aux xs (a + x) Objectives Accumulating Recursion Activity References Accumulating Recursion ◮ In accumulating recursion: generate an intermediate result now , and give that to the recursive call. ◮ Usually this requires an auxiliary function. Tail Recursive Summation 1 sum xx = aux xx 0 2 3

  29. | odd x = fun1 xs + 1 Objectives Accumulating Recursion Activity References Convert These Functions! ◮ Here are three functions. Try converting them to tail recursion. 1 fun1 [] = 0 2 fun1 (x : xs) | even x = fun1 xs - 1 3 4 5 fun2 1 = 0 6 fun2 n = 1 + fun2 (n `div` 2) 7 8 fun3 1 = 1 9 fun3 2 = 1 10 fun3 n = fun3 (n - 1) + fun3 (n - 2)

  30. aux n a = aux (n `div` 2) (a + 1) where aux [] a = a where aux 1 a = a | odd x = aux xs (a + 1) aux (x : xs) | even x = aux xs (a - 1) Objectives Accumulating Recursion Activity References Solution for fun1 and fun2 ◮ Usually it’s best to create a local auxiliary function. 1 fun1 xx = aux xx 0 2 3 4 5 6 fun2 n = aux n 1 7 8

  31. aux n f1 f2 = aux (n - 1) f2 (f1 + f2) where aux 0 f1 f2 = f1 Objectives Accumulating Recursion Activity References Solution for fun3 ◮ Because the recursion calls itself twice, we need two accumulators. 1 fun3 n = aux n 1 1 2 3

  32. Objectives Accumulating Recursion Activity References References [DG05] Olivier Danvy and Mayer Goldberg. “There and Back Again”. In: Fundamenta Informaticae 66.4 (Jan. 2005), pp. 397–413. ISSN: 0169-2968. URL: http://dl.acm.org/citation.cfm?id=1227189.1227194 . [Ste77] Guy Lewis Steele Jr. “Debunking the ”Expensive Procedure Call” Myth or, Procedure Call Implementations Considered Harmful or, LAMBDA: The Ultimate GOTO”. In: Proceedings of the 1977 Annual Conference . ACM ’77. Seattle, Washington: ACM, 1977, pp. 153–162. URL: http://doi.acm.org/10.1145/800179.810196 .

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