1. Com2010 - Functional Programming Higher Order Functions and Computation Patterns (II) Marian Gheorghe Lecture 11 Module homepage Mole & http://www.dcs.shef.ac.uk/~marian ©University of Sheffieldcom2010

2. 18. Higher-Order Functions and Computation Patterns 18.3 … primitive recursion 18.4 Efficiency and recursion patterns 18.5 Partial functions and errors 18.6 More higher-order on lists Summary ©University of Sheffieldcom2010

3. Iterations of different functions: f_1, f_2, …,f_n Example. The function fact(n) = n! may be obtained as 1  *2  2  *3 …  *n  n! or similar to get [1, 2, …, n] Obs. At step n we multiply by n or append n to the list. fact :: Int -> Int -- construct n! fact n | n == 0 = 1 | n > 0 = fact(n-1) * n natList :: Int -> [Int] -- constructs seq of naturals natList n | n == 0 = [] | n > 0 = natList (n-1) ++ [n] Primitive recursion - example ©University of Sheffieldcom2010

4. The pattern of primitive recursion: primRec::(Int->a->a)->a->Int->a -- 1st param: iter function, depends on iteration stage -- 2nd param: initial value -- 3rd param: iteration length primRec f x n | n==0 = x | n>0 = f n (primRec f x (n-1)) Examples myFact :: Int ->Int myFact = primRec (\n x -> x*n) 1 myNatList ::Int -> [Int] myNatList = primRec (\n x -> x++[n]) [] Primitive recursion - pattern ©University of Sheffieldcom2010

5. Fibonacci sequence: fib :: Int -> Int – nice, simple, but inefficient (exp in n) fib n | n == 0 = 0 | n == 1 = 1 | n > 1 = fib(n-2) + fib(n-1) Fibonacci sequence; efficient version fibNextStep :: (Int, Int) -> (Int, Int) fibNextStep (x, y) = (y, x + y) -- uses myIter iteration function; efficient (linear in n) fastFib :: (Int -> Int) -- fast Fibonacci sequence fastFib = fst. (myIter fibNextStep (0,1)) Run fib, fastFib for 30 and watch the time !! Efficiency of recursion patterns ©University of Sheffieldcom2010

6. Iteration and primitive recursion are two general powerful elegant abstract mechanisms When fast algorithms are requested then less general solutions may be more efficient Ex: fib 20 makes 399209 reductions Whereas fastfib 20 makes 417 reduction Generality vs efficiency ©University of Sheffieldcom2010

7. Simple examples include attempts to divide by 0, to take the square root of a negative number, or the head of an empty list. applying a function defined by (primitive) recursion on natural numbers to negative numbers (fib(-2)). applying a function to an input value that is not caught by any pattern or guard in any of the function's definition equations (non- exhaustive patterns or guards). A hierarchy of` error treatment is studied Partial functions and errors ©University of Sheffieldcom2010

8. The function definition naiveFib :: Int -> In -- may loop naiveFib 0 = 0 naiveFib 1 = 1 naiveFib n = naiveFib(n-1) + naiveFib(n-2) When execute naiveFib with argument -3, naiveFib (-3) Error: control stack overflow!! fib :: Int -> Int -- may show run-time error fib 0 = 0 fib 1 = 1 fib n | n > 1 = fib(n-1) + fib(n-2) Run-time error when a value outside the range is used Errors: infinite loops; missing condition ©University of Sheffieldcom2010

9. A simple example using error built-in function newFib :: Int -> Int -- may produce error message newFib 0 = 0 newFib 1 = 1 newFib n | n > 1 = newFib(n-1) + newFib(n-2) | otherwise = error( "\nError in newFib:Fibonacci " ++"function cannot \nbe applied to" ++" negative integer “ ++show(n) ++"\n") When newFib (-3) is used then an error will be displayed Program abortion ©University of Sheffieldcom2010

10. Exceptional inputs can be covered by defining natural dummy results. Consider the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, … again. It would appear natural to extend the sequence into the negative indices by repeating 0 : …, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, … i.e. we define fib n = 0, for negative n. So, 0 would be the dummy result for negative inputs. This would give us the following implementation: extFib :: Int -> Int -- extends 0's leftwards extFib 0 = 0 extFib 1 = 1 extFib n | n > 1 = extFib(n-1) + extFib(n-2) | otherwise = 0 Dummy values (1) ©University of Sheffieldcom2010

11. A better variant using dummy values symFib :: Int -> Int -- satisfies recursion law symFib 0 = 0 symFib 1 = 1 symFib n | n > 1 = symFib(n-1) + symFib(n-2) | otherwise = symFib(n+2) - symFib(n+1) SymFib(-1)=1, symFib(-2)=-1, SymFib (-3)=2; ie … 2, -1, 1, 0, 1, 1, 2,… etc Guess what symFib is doing Dummy values (2) ©University of Sheffieldcom2010

12. To trap and process errors employ an explicit exception handling technique based on error types defined using algebraic types data Maybe a = Nothing | Just a deriving (Eq, Ord, Read, Show) The type Maybe a is simply the type a extended by an error value Nothing. Any function g that uses the result of another function may be transformed so it accepts an argument of type Maybe a rather than a. This is where the error handling occurs. We can  transmit the error through g  trap the error within g. Exception handling ©University of Sheffieldcom2010

13. It lifts the function g :: a -> b to a function mapMaybe g :: Maybe a -> Maybe b mapMaybe :: (a->b) -> Maybe a-> Maybe b mapMaybe g Nothing = Nothing mapMaybe g (Just x) = Just (g x) Example mapMaybe (*3) (my_nth 5[1,2,3]) ⇒ Nothing mapMaybe (*3) (my_nth 2[1,2,3]) ⇒ Just 9 If, however, we lift the function g::a -> b to a function of type Maybe a -> b then we are trapping the error. Transmit (map) an error ©University of Sheffieldcom2010

14. trapMaybe dummy g :: Maybe a -> b trapMaybe :: b->(a->b) -> Maybe a-> b trapMaybe dummy g Nothing = dummy trapMaybe dummy g (Just x) = g x Error handling. Steps: Error identification (use type Maybe a ) Error mapping (lift g to mapMaybe ) Error trapping (lift g to trapMaybe ) Trap an error ©University of Sheffieldcom2010

15. Nothing trapMaybe dummyInt (1+)(mapMaybe (*3)(my_nth 5[1,2,3]))  dummyInt integer value trapMaybe dummyInt (1+)(mapMaybe (*3)(my_nth 2[1,2,3]))  10 mapMaybe either sends up Nothing or Maybe value trapMaybe either produces a dummy_value or transforms Maybe value into (1+) value Example ©University of Sheffieldcom2010