CSE-321 Programming Languages Introduction to Functional Programming (Part II)

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CSE-321 Programming LanguagesIntroduction to
Functional Programming(Part II)
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• POSTECH
• March 13, 2006

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Outline

• Expressions and values V
• Variables V
• Functions V
• Types
• Polymorphism
• Recursion
• Datatypes
• Pattern matching
• Higher-order functions
• Exceptions
• Modules

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What is the Type of ?
All we know about f is that it takes booleans
as arguments.
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f bool -gt ?
fn f gt (f true, f false) (bool -gt ?) -gt ? ?
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f bool -gt 'a
fn f gt (f true, f false) (bool -gt 'a) -gt 'a
'a

• 'a
• type variable
• usually read as alpha
• means 'for any type alpha'

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Polymorphic Types

• Types involving type variables 'a, 'b, 'c, ...
• E.g.
• fn x gt x
• 'a -gt 'a
• fn x gt fn y gt (x, y)
• 'a -gt 'b -gt ('a 'b)
• fn (x 'a) gt fn (y 'a) gt x y
• 'a -gt 'a -gt bool
• ( actually does not typecheck! )

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Equality Types

• Motivation
• Equality () is not defined on every type.
• E.g.
• comparing two functions for equality?
• Type variables with equality
• ''a, ''b, ''c, ...
• ''a means 'for any type alpha for which
  equality is defined'
• fn (x ''a) gt fn (y ''a) gt x y
• ''a -gt ''a -gt bool

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Outline

• Expressions and values V
• Variables V
• Functions V
• Types V
• Recursion
• Datatypes
• Pattern matching
• Higher-order functions
• Exceptions
• Modules

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Recursion vs. Iteration

• Recursion in SML
• fun sum n if n 0 then 0else sum (n - 1) n

• Iteration in C
• int i, sum
• for (i 0, sum 0 i lt n
• i)sum n

• Recursion is not an awkward tool if you are used
  to functional programming.
• Recursion seems elegant but inefficient!

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Recursion in Action

• fun sum n if n 0 then 0else sum (n - 1) n

call stack
0
further computation
...
...
f 10
evaluation
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Funny Recursion

• fun zero n if n 0 then 0else zero (n - 1)

call stack
no further computation
evaluation
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Funny Recursion Optimized

• fun zero n if n 0 then 0else zero (n - 1)

call stack
0
0
...
...
f 10
evaluation
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Funny Recursion Further Optimized

• fun zero n if n 0 then 0else zero (n - 1)

call stack
f 10
...
0
evaluation
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Tail Recursive Function

• A tail recursive function f
• A recursive call to f is the last step in
  evaluating the function body.
• That is, no more computation remains after
  calling f itself.
• A tail recursive call needs no stack!
• A tail recursive call is as efficient as
  iteration!

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Example

• Tail recursive sum
• fun sum' accum k if k 0 then accumelse
• sum' (accum k) (k - 1)
• fun sum n sum' 0 n

• Non-tail recursive sum
• fun sum n if n 0 then 0else sum (n - 1) n

• Think about the invariant of sum
• given
• sum' accum k
• invariant
• accum (k 1) ... n

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Mathematics vs. Computer Science

• CS ½ Math?

• CS ? Math?

Math
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Factorial Function

• fac 1 1
• fac n n fac (n - 1)
• Makes sense in
• Math
• CS

• fac 1 1
• fac n fac (n 1) / n
• Makes sense in math.
• But not in CS
• it always diverges.

• "Then what kind of recursive functions are
  computationally meaningful?"
• ) domain theory in computer science
• Other examples
• real numbers vs. computable real numbers
• logic in math vs. logic in computer science

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Outline

• Expressions and values V
• Variables V
• Functions V
• Types V
• Recursion V
• Datatypes
• Pattern matching
• Higher-order functions
• Exceptions
• Modules

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Enumeration Types in C

• enum shape Circle, Rectangle, Triangle
• Great flexibility
• e.g.
• Circle 1 Rectangle
• Triangle - 1 Rectangle
• (Circle Triangle) / 2 Rectangle
• But is this good or bad?

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Datatypes in SML

• datatype shape Circle Rectangle Triangle
• No flexibility
• e.g.
• Circle 1 (x)
• Triangle - 1 (x)
• Circle Triangle (x)
• But high safety.

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Primitive Set

• datatype set
• Empty
• Singleton
• Pair
• Many
• - Empty
• val it Empty set
• - Many
• val it Many set

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Primitive Set with Arguments

• datatype set
• Empty
• Singleton
• Pair
• Many of int
• - Many 5
• val it Many 5 set

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Primitive Set with Type Parameters

• datatype 'a set
• Empty
• Singleton of 'a
• Pair of 'a 'a
• Many of int
• - Singleton 0
• val it Singleton 0 int set
• - Pair (0, 1)
• val it Pair (0, 1) int set

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Primitive Set with Type Parameters

• datatype 'a set
• Empty
• Singleton of 'a
• Pair of 'a 'a
• Many of int
• - Pair (Singleton 0, Pair (0, 1))
• val it Pair (Singleton 0, Pair (0, 1))
  int set set

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Primitive Set with Type Parameters

• datatype 'a set
• Empty
• Singleton of 'a
• Pair of 'a 'a
• Many of int
• - Empty
• val it Empty 'a set
• - Many 5
• val it Many 5 'a set

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Primitive Set with Type Parameters

• datatype 'a set
• Empty
• Singleton of 'a
• Pair of 'a 'a
• Many of int
• - Pair (0, true)
• stdIn27.1-27.15 Error operator and operand
  don't agree literal
• operator domain int int
• operand int bool
• in expression
• Pair (0,true)

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Recursive Set with Type Parameters

• datatype 'a set
• Empty
• NonEmpty of 'a 'a set
• This is essentially the definition of datatype
  list.

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Datatype list

• datatype 'a list
• nil
• of 'a 'a list
• - nil
• val it 'a list
• - 2 nil
• val it 2 int list
• - 1 (2 nil)
• val it 1,2 int list
• - 1 2 nil
• val it 1,2 int list

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Datatype list

• datatype 'a list
• nil
• of 'a 'a list
• - 1 2 nil
• - 1 2 nil
• - 1 2 nil
• val it 1,2 int list list
• - 1 2
• val it 1,2 int list list

(X)
(X)
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Using Datatypes

• We know how to create values of various
  datatypes
• shape
• set
• int set
• int list
• ...
• But how do we use them in programming?
• What is the point of creating datatype values
  that are never used?

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Outline

• Expressions and values V
• Variables V
• Functions V
• Types V
• Recursion V
• Datatypes V
• Pattern matching
• Higher-order functions
• Exceptions
• Modules

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Simple Pattern

• datatype shape Circle Rectangle Triangle
• ( convertToEnum shape -gt int )
• fun convertToEnum (x shape) int
• case x of
• Circle gt 0
• Rectangle gt 1
• Triangle gt 2

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Pattern with Arguments

• datatype 'a set
• Empty
• Singleton of 'a
• Pair of 'a 'a
• Many of int
• fun size (x 'a set) int
• case x of
• Empty gt 0
• Singleton e gt 1
• Pair (e1, e2) gt 2
• Many n gt n

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Wildcard Pattern _ "don't care"

• datatype 'a set
• Empty
• Singleton of 'a
• Pair of 'a 'a
• Many of int
• fun isEmpty (x 'a set) bool
• case x of
• Empty gt true
• _ gt false

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Pattern with Type Annotation

• datatype 'a list
• nil
• of 'a 'a list
• fun length (x 'a list) int
• case x of
• (nil 'a list) gt 0
• (_ 'a) (tail 'a list) gt
• 1 length tail

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Outline

• Expressions and values V
• Variables V
• Functions V
• Types V
• Recursion V
• Datatypes V
• Pattern matching V
• Higher-order functions
• Exceptions
• Modules

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Higher-order Functions

• Take functions as arguments.
• Return functions as the result.

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Why "Higher-order"?

• T0 int bool real unit ...
• T1 T0 -gt T0 T0 ( 1st order )
• T2 T1 -gt T1 T1 ( 2nd order )
• T3 T2 -gt T2 T 2 ( higher order )
• ...

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Higher-order Functions in List

• val exists ('a -gt bool) -gt 'a list -gt bool
• val all ('a -gt bool) -gt 'a list -gt bool
• val map ('a -gt 'b) -gt 'a list -gt 'b list
• val filter ('a -gt bool) -gt 'a list -gt 'a list
• val app ('a -gt unit) -gt 'a list -gt unit
• ( printInt int -gt unit )
• fun printInt i
• TextIO.print ((Int.toString i) "\n" )
• List.app printInt 1, 2, 3

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List Fold Function

• foldl ('a 'b -gt 'b) -gt 'b -gt 'a list -gt 'b
• foldr ('a 'b -gt 'b) -gt 'b -gt 'a list -gt 'b
• foldl f b0 a0, a1, a2, ..., an-1

a2
a0
a1
an-1
an-2
...
b0
bn-2
...
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Examples

• List.exists f l
• List.foldl (fn (a, b) gt b orelse f a) false l
• List.all f l
• List.foldl (fn (a, b) gt b andalso f a) true l
• List.app f l
• List.foldl (fn (a, _) gt f a) () l
• List.map f l
• List.foldr (fn (a, b) gt f a b) nil l
• List.filter f l
• List.foldr (fn (a, b) gt if f a then a b
• else b) nil l

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More Examples

• fun sum (l int list)
• List.foldl (fn (a, accum) gt a accum) 0 l
• fun reverse (l 'a list)
• List.foldl (fn (a, rev) gt a rev) nil l
• Whenever you need iterations over
  lists, first consider foldl and foldr.

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Outline

• Expressions and values V
• Variables V
• Functions V
• Types V
• Recursion V
• Datatypes V
• Pattern matching V
• Higher-order functions V
• Exceptions
• exception please see the course notes.
• Modules

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Outline

• Expressions and values V
• Variables V
• Functions V
• Types V
• Recursion V
• Datatypes V
• Pattern matching V
• Higher-order functions V
• Exceptions V
• Modules

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Structures and Signatures

• Structure
• collection of type declarations, exceptions,
  values, and so on.
• structure Set
• struct
• type 'a set 'a list
• val emptySet nil
• fun singleton x x
• fun union s1 s2 s1 _at_ s2
• end

• Signature
• conceptually type of structures.
• signature SET
• sig
• type 'a set
• val emptySet 'a set
• val singleton 'a -gt 'a set
• val union
• 'a set -gt 'a set -gt 'a set
• end

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Structures Signatures

• Transparent constraint
• Type definition is exported to the outside.
• structure Set SET
• struct
• type 'a set 'a list
• val emptySet nil
• fun singleton x x
• fun union s1 s2 s1 _at_ s2
• end
• - Set.singleton 1 1
• val it true bool

• Opaque constraint
• No type definition is exported.
• structure Set gt SET
• struct
• type 'a set 'a list
• val emptySet nil
• fun singleton x x
• fun union s1 s2 s1 _at_ s2
• end
• - Set.singleton 1 1
• ( Error! )

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I need to test your 57 structures!

• structure BetOne gt HW_ONE ...
• structure Bigh2000One gt HW_ONE ...
• structure BoongOne gt HW_ONE
• structure BrandonOne gt HW_ONE ...
• ...
• structure ZiedrichOne gt HW_ONE ...
• How can I test 57 structures?
• They all conform to the same signature.

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Simple (but Ugly) Solution

• fun sumTest f f nil 0 andalso f 1, 2, 3 6
• fun facTest f f 1 1 andalso f 3 6 andalso f
  7 5040
• ...
• fun allTest (sum, fac, ...)
• let
• val sumScore if sumTest sum then 10 else 0
• val facScore if facTest fac then 4 else 0
• ...
• in
• sumScore facScore ...
• end
• allTest (BetOne.sum, BetOne.fac, ...)
• allTest (Bigh2000One.sum, Bigh2000One.fac, ...)
• allTest (BoongOne.sum, BoongOne.fac, ...)
• allTest (BrandonOne.sum, BrandonOne.fac, ...)
• ...
• allTest (ZiedrichOne.sum, ZiedrichOne.fac, ...)

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Functors

• Functions on structures
• takes a structure as an argument
• returns a structure as the result

structure
structure
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HW1 Test Functor

• signature HW_TEST
• sig
• val score int
• end
• functor Hw1TestFn (P HW_ONE) HW_TEST
• struct
• structure S Hw1Solution
• val sumScore
• if P.sum nil S.sum nil andalso
• P.sum 1, 2, 3 S.sum 1, 2, 3
• then 10 else 0
• val facScore
• if P.fac 1 S.fac 1 andalso
• P.fac 3 S.fac 3 andalso
• P.fac 7 S.fac 7
• then 4 else 0
• ...
• val score sumScore facScore ...

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HW1 Test Functor in Action

• structure BetOneTest Hw1TestFn (BetOne)
• structure Bigh2000OneTest Hw1TestFn
  (Bigh2000One)
• structure BoongOneTest Hw1TestFn (BoongOne)
• structure BrandonOneTest Hw1TestFn (BrandonOne)
• ...
• structure ZiedrichOneTest Hw1TestFn
  (ZiedrichOne)
• BetOneTest.score
• Bigh2000OneTest.score
• BoongOneTest.score
• BrandonOneTest.score
• ...
• ZiedrichOneTest.score

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• So it is YOU who will grade your homework!