Functional Programming Lecture 11 - higher order functions

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Functional ProgrammingLecture 11 - higher
order functions
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Rule of Cancellation

• If f t1 -gt t2 -gt -gt tn -gt t
• and f is applied to arguments
• e1 t1, ek tk, where kltn
• then the resulting type is given by cancelling
  out the types t1 to tk
• t1 -gt t2 -gt -gt tk -gt tk1 -gt tn-gt t
• i.e.
• (f e1 ek ) tk1 -gt tk2 -gt tn
• Example max Int -gt Int -gt Int
• So, max 2 Int -gt Int

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• Example max Int -gt Int -gt Int
• So, max 2 Int -gt Int
• (max 2) is just a function.
• It is the function that takes the maximum of 2
  and another number.
• given
• max x y if xgty then x else y
• then
• (max 2) y if 2gty then 2 else y
• This is an example of partial evaluation. (The
  function max has been partially evaluated, to
  give another function, max 2.)

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Function Application and -gt

• Associativity
• Function application is left associative.
• i.e. f x y (f x) y
• f x y f (x y) !!
• E.g.
• max Int -gt Int -gt Int
• max 2 3 (max 2) 3
• max (2 3) (doesnt make sense)

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Function Application and -gt

• Associativity
• Function space symbol -gt is right associative.
• i.e. a -gt b -gt c means a -gt (b -gt c)
• not (a -gt b) -gt
  c
• f a -gt (b -gt c)
• g (a -gt b) -gt c
• a
• b f c (a-gtb)
  g c
• E.g.
• max Int -gt (Int -gt Int)
• max 2 Int -gt Int

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Partial evaluation

• When you partially evaluate a function, the new,
  resulting function must be enclosed in ( ..
  ).
• The new function is also called an operator
  section.
• Examples
• (2) the function which adds 2 to its argument.
• (2) the function which adds 2 to its argument.
• (gt2) the function which compares its argument
  with 2.
• (3) the function the puts 3 at the front of a
  list.
• (\n) the function which puts a newline at the
  end of a string.

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An aside

• Displaying strings with newline characters.
• \n is the newline character.
• If you type in
• gt show test\ntest
• the result is test\ntest
• To display a string, and interpret the special
  symbols, use putStr String -gt IO String.
• If you type
• gt putStr test\ntest
• the result is
• test
• test IO ()

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Anonymous functions

• Functions do not have to have names.
• E.g.
• \m -gt nm
• This defines a function with one argument that
  returns the sum of n and that argument.
• The arguments - given between \ and -gt
• The result - given after the -gt.
• \ stands for the Greek letter l.

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Using a Lambda-defined function

• Define a function comp2 which applies a function
  f to each of its 2 arguments before applying
  function g to the results.
• f
• 
  g
• f
• comp2 (a -gt b) -gt (b -gt b -gt c) -gt (a -gt a -gt
  c)
• f g
  resulting function
• comp2 f g \x y -gt g (f x) (f y)
• e.g.
• comp2 sq add 3 4 gt add (sq 3) (sq 4) gt 916 gt
  25

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Curry and Uncurry

• A function which takes its arguments one at a
  time is said to be a curried function, or in
  curried form.
• (Named after Haskell Curry, a logician who worked
  on combinatory logic and showed relationships to
  l-calculus.)
• E.g. max Int -gt Int -gt Int
• is in curried form.
• x
• y
• Most of the functions in this course so far have
  been curried.
• A function which takes all its arguments
  bundled together as a tuple is said to an
  uncurried function, or in uncurried form.
• E.g. maxUC (Int,Int) -gt Int
• is in uncurried form.
• (x,y)

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• There is an obvious relationship between the
  curried and uncurried versions.
• Namely, max x y maxUC (x,y)
• We can turn an uncurried function into a curried
  one, and vice-versa.
• Suppose g is an uncurried function, g (a,b)
  -gtc.
• Then (curry g) a -gt b -gt c.
• curry ((a,b) -gtc) -gt (a -gt b -gt c)
• curry g x y g (x,y)
• Recall function application is left assoc.
• i.e. (curry g) x y g (x,y)

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• Similarly,
• suppose f is an curried function, f a -gt b
  -gtc.
• Then (uncurry f) (a,b) -gt c.
• uncurry (a -gt b -gt c) -gt ((a,b) -gtc)
• uncurry f (x,y) f x y
• So, (uncurry f) (x,y) f x y
• curry and uncurry are defined in the standard
  prelude.
• E.g.

Preludegt uncurry () (2,1) 3
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• Finally,
• we would expect that curry and uncurry are
  inverses
• E.g.
• curry (uncurry f) f
• uncurry (curry g) g
• It works for specific cases
• uncurry max maxUC
• curry maxUC max
• So, curry (uncurry max) curry maxUC max
• uncurry (curry maxUC) uncurry max maxUC
• But to prove it for an arbitrary function,
  requires induction.