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WORK IT, WRAP IT, FIX IT, FOLD IT

WORK IT, WRAP IT, FIX IT, FOLD IT. Graham Hutton and Neil Sculthorpe. This Talk. Worker/wrapper is a simple but powerful method for optimizing recursive programs; Previously, we developed theories for fix and fold , but with different correctness conditions;

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WORK IT, WRAP IT, FIX IT, FOLD IT

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  1. WORK IT, WRAP IT, FIX IT, FOLD IT Graham Hutton and Neil Sculthorpe

  2. This Talk • Worker/wrapper is a simple but powerful method for optimizing recursive programs; • Previously, we developed theories for fix and fold, but with different correctness conditions; • We combine and extend the two approaches to give a generalised worker/wrapper theory.

  3. Worker / Wrapper A technique for changing the type of a recursive program to improve its performance: wrapper program worker

  4. Fixed Points Our original formalisation was based on the use of explicit fixed points. For example: ones = 1 : ones can be rewritten as: ones = fix f f xs = 1 : xs fix f = f (fix f)

  5. The Problem Suppose we wish to change the type of a recursive program that is defined using fix. B A Type of the desired worker, fix g. Type of the original program, fix f.

  6. Assumptions We assume conversion functions rep A B abs such that: A can be faithfully represented by B. abs . rep = idA

  7. Let’s Calculate! fix f = fix (idA . f) = Rolling rule. fix (abs . rep . f) = abs (fix (rep . f . abs)) = abs (fix g) 6

  8. Summary We have derived the following factorisation: fix f abs fix g = Wrapper of type B  A. Recursive program of type A. Recursive worker of type B.

  9. Final Step We simplify the worker fix g = fix (rep . f. abs) to eliminate the overhead of repeatedly converting between the two types, by fusing together and rep abs

  10. The Worker / Wrapper Recipe • Express the original program using fix; • Choose the new type for the program; • Define appropriate conversion functions; • Apply the worker/wrapper transformation; • Simplify the resulting definitions.

  11. Generalising The technique also works for weakerassumptions: abs . rep = idA From the ‘fix’ paper. abs . rep . f = f fix (abs . rep . f) = fix f

  12. … and for otherconditions relating f and g: g = rep . f . abs From the ‘fold’ paper. rep . f = g . rep abs . g = f . abs Necessary and sufficient. fix g = rep (fix f) fix g = fix (rep . f . abs)

  13. Generalised Recipe • If the worker is already given, we aim to verify that one of the conditions is satisfied; • Otherwise, our aim is to construct the worker, using one of the conditions as a specification; • Similar assumptions and conditions also give a generalised worker/wrapper theory for fold.

  14. Example last [] = ⊥ last [x] = x last (x:xs) = last xs last [] = ⊥ last (x:xs) = work xxs work x [] = x work x (y:ys) = work yys

  15. Example fib 0 = 0 fib 1 = 1 fib (n+2) = fib n + fib (n+1) fib n = fst (work n) work 0 = (0,1) work (n+1) = (y,x+y) where (x,y) = work n

  16. Summary • Generalised technique for changing the type of a program to improve its performance; • Captures a wide variety of program optimization techniques in a single unified framework; • The paper also presents a range of new results concerning strictness side conditions.

  17. Further Work • Other recursion patterns; • Time and space analysis; • Computational effects; • Implementing the technique.

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