Functional Programming Lecture 2 - Functions

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Functional ProgrammingLecture 2 - Functions

• Muffy Calder

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Function Definitions

• f Int -gt Int
• -- multiply argument by 100
• f x x100
• f takes an Int and returns an Int the value of
  f x is x100
• The mantra comment, type, equation

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Defining Equation of Function

• Fn_name arg1 arg2 argn
• expression that can use
• arg1argn
• E.g. g x y x y gt 10

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Evaluation or Simplification

• Given
• f x y xy
• Evaluate f 3 4
• f 3 4 gt 3 4 12
• Now evaluate f 3 (f 4 5)
• f 3 (f 4 5)gt 3 f 4 5gt 3 (45)gt 3
  20gt 60
• Note there is an alternative evaluation!
• Does it matter which one we pick? (Stay tuned)

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Programs (Scripts)

• What is meant by given?
• A set of definitions is saved in a file.
• E.g.
• x 2
• y 4
• z 100x -30y
• f Int -gt Int -gt Int
• f x y x 2y
• You use a program by asking it to perform a
  calculation, e.g. calculate f 2 3.
• gt f 2 3

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Equations vs. Assignments

• Assignment - command to update a variable
• e.g. x x1
• x ------? ----? 3
• Equation - permanent assertion that two values
  are equal e.g. x3 or x y1
• x -----? 3
• You obey assignments and solve equations.
• Equations in a script can appear in any order!

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and

• Ada x x1
• take the old value of x and increment it
• This is a common and useful operation.
• Haskell x x1
• Assert that x and x1 are the same value.
• This equation has no solution.
• It is a syntactically-valid statement, but it is
  not solvable. There is no solution for x. So
  this equation cannot be true!

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Side Effects andEquational Reasoning

• Equations are timeless -there is no notion of
  changingthe value of a variable
• Mathematics (algebra) is all about reasoning with
  equations, not with the contents of computer
  locations -- that keep changing.

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Local functions where

• Sometimes we want to make a function definition
  local (as opposed to global)
• For example
• f x y (g x) y
• where
• g z z 32
• The function g is local to the function f.
• Alternative
• f x y (g x) y
• g z z 32
• h x (g x) y
• In this case, the function g is available to
  other functions.

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Local names let expressions

• let x 1
• y 2
• z 3
• in xyz
• The variables x, y, z are local
• A script is just one big let expression.

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A let expression is an expression!

• E.g.
• 2 let a xy in (a1)(a-1)
• 2 (let a 5 in (a1)(a-1) ) 3
• - allows you to make somelocal definitions
• - it is not a block of statements to be executed.

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Conditional expressions

• if boolean_exp
• then exp1 else exp2
• Haskell evaluates boolean_exp and returns the
  appropriate expression.
• exp1 and exp2 must have the same type
• E.g.
• f x if xgt3 then x2 else x-2
• g x if xgt3 then x2 else False (wrong)
• What are the types of f and g?

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Guarded expressions

• f x
• xgt0 True
• otherwise False
• In general
• Name x1 x2 .. xn
• guard1 e1
• guard2 e2
• .
• otherwise e
• Haskell evaluates the guards, from the top, and
  returns the appropriate expression.
• guard1, ... must have the Bool type
• exp1, ... must have the same type
• Note
• f x
• xgt0 True
• otherwise 2 (wrong)

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Examples

• positive Int -gt Bool
• positive x
• (xgt0) True
• otherwise False
• or
• positive x if (xgt0) then True else False
• max Int -gt Int -gt Int
• max x y
• (xgty) x
• otherwise y
• or
• max x y if (xgty) then x else y
• maxthree Int -gt Int -gt Int -gt Int
• maxthree x y z
• ((xgty) (xgtz) x
• (ygtz) y
• otherwise z

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Examples

• or
• maxthree x y z max x (max y z)
• mystery Int -gt Int -gt Int -gt Bool
• mystery m n p not ((mn) (np))
• Evaluate
• mystery 0 1 2
• mystery 0 1 1
• mystery 1 1 1

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Recursion

• Functions can be recursive (i.e. the function can
  occur on the rhs of the defining equation).
• This is fundamental to functional programming.
• Example
• fact Int -gt Int
• fact 1 1
  -- fact.0
• fact n n (fact n-1)
  -- fact.1
• So, fact 4 gt 4 (fact 3)
• gt 4 3 (fact 2)
• gt 4 3 2 (fact 1)
• gt 4 3 2 1
• gt 24
• fact.0 is called the base case.

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Off-side Rule

• mystery Int -gt Int -gt Int -gt Bool
• mystery m n p not ((mn) (np))
• the box
• Whatever is typed in the box is part of the
  definition.
• So,
• f Int -gt Int
• f x 42 --okay
• g Int -gt Int --(start new
  definition)
• g x --okay
• 36
• h Int -gt Int
• h y 42 --okay
• 1
• hy

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Functional Data Structures

• Allow you to build aggregate objects from smaller
  values
• Built-in data structures
• Lists
• Tuples
• Arrays
• User defined data structures