INF 231 - Functional Algorithmic and Programming¶

Lecture 4: Recursion¶

Below are the commands demonstrated during the lecture session (plus some additional commands).

Remember:

Try them yourself.
Tweak them: modify them and try variants of those commands.
Your curiosity is your best pedagogical tool.

Disclaimer: the functions in the code below are defined most of the times without SPECIFICATION. Remember that defining a function always consists in giving its specification and then implementation.

Defining some recursive functions¶

(* factorial *)
let rec fact (n:int):int=
  if (n=0) then 1
  else n*fact (n-1) ;;

(* sum of integers *)
let rec sum (n:int):int=
  if (n=0) then 0
  else n + sum (n-1) ;;

(* Quotient of the Euclidean division *)
let rec quotient (a:int) (b:int):int =
  if (a<b) then 0
  else 1 + quotient (a-b) b ;; 


(* Remainder of the Euclidian division *)
let rec remainder (a:int) (b:int):int=
  if (a<b) then a
  else remainder (a-b) b ;;

remainder 7 3 ;;

(* Addition with only incrementation and substraction *)
let incr (n:int):int = n+1 ;;

let rec add (n:int) (m:int):int=
  match n with
    | 0 -> m
    | n -> add (n-1) (incr m) ;;

(* Multiplication with only addition and substraction *)
let rec mult (n:int) (m:int):int=
  match n with
    | 0 -> m
    | n -> m + mult (n-1) m ;;

(* Calling a recursive function *)
fact 3;;
fact (4);;
quotient 10 5 ;;
quotient 22 7 ;;

(* Tracing a function *)
#trace fact;;

fact 3;;

val fact : int -> int = <fun>

val sum : int -> int = <fun>

val quotient : int -> int -> int = <fun>

val remainder : int -> int -> int = <fun>

- : int = 1

val incr : int -> int = <fun>

val add : int -> int -> int = <fun>

val mult : int -> int -> int = <fun>

- : int = 6

- : int = 24

Let's practice !

(* Remainder of the Euclidian division *)
let rec remainder (a:int) (b:int):int =
    if (a<b) then a
    else remainder (a-b) b ;;

remainder 16 7 ;;

(* The Fibonacci series *)
let rec fib (n:int):int=
  if (n<=1) then 1
  else fib(n-1) + fib(n-2) ;;

(* Power *)
let rec power1 (a:float) (n:int):float=
  if (n=0) then 1.
  else (if n>0 then a *. (power1 a (n-1))
        else 1. /. (power1 a (n-1))
       ) ;;

let rec power2 (a:float) (n:int):float=
  match n with
    | 0 -> 1.
    | n when (n mod 2=0) -> power2 (a*.a) (n / 2)
    | _ -> a *. power2 (a*.a) ((n-1) / 2) ;; (* n when (n mod 2=1) *)

power1 8. 10 ;;

power2 4. 6 ;;

#trace power1;;
#trace power2;;

power1 3. 20;;
power2 3. 20;;

val remainder : int -> int -> int = <fun>

- : int = 2

val fib : int -> int = <fun>

val power1 : float -> int -> float = <fun>

val power2 : float -> int -> float = <fun>

- : float = 1073741824.

- : float = 4096.

power1 is now traced.
power2 is now traced.
power1 <-- 3.
power1 --> <fun>
power1* <-- 20
power1 <-- 3.
power1 --> <fun>
power1* <-- 19
power1 <-- 3.
power1 --> <fun>
power1* <-- 18
power1 <-- 3.
power1 --> <fun>
power1* <-- 17
power1 <-- 3.
power1 --> <fun>
power1* <-- 16
power1 <-- 3.
power1 --> <fun>
power1* <-- 15
power1 <-- 3.
power1 --> <fun>
power1* <-- 14
power1 <-- 3.
power1 --> <fun>
power1* <-- 13
power1 <-- 3.
power1 --> <fun>
power1* <-- 12
power1 <-- 3.
power1 --> <fun>
power1* <-- 11
power1 <-- 3.
power1 --> <fun>
power1* <-- 10
power1 <-- 3.
power1 --> <fun>
power1* <-- 9
power1 <-- 3.
power1 --> <fun>
power1* <-- 8
power1 <-- 3.

- : float = 3486784401.

power1 --> <fun>
power1* <-- 7
power1 <-- 3.
power1 --> <fun>
power1* <-- 6
power1 <-- 3.
power1 --> <fun>
power1* <-- 5
power1 <-- 3.
power1 --> <fun>
power1* <-- 4
power1 <-- 3.
power1 --> <fun>
power1* <-- 3
power1 <-- 3.
power1 --> <fun>
power1* <-- 2
power1 <-- 3.
power1 --> <fun>
power1* <-- 1
power1 <-- 3.
power1 --> <fun>
power1* <-- 0
power1* --> 1.
power1* --> 3.
power1* --> 9.
power1* --> 27.
power1* --> 81.
power1* --> 243.
power1* --> 729.
power1* --> 2187.
power1* --> 6561.
power1* --> 19683.
power1* --> 59049.
power1* --> 177147.
power1* --> 531441.
power1* --> 1594323.
power1* --> 4782969.
power1* --> 14348907.
power1* --> 43046721.
power1* --> 129140163.
power1* --> 387420489.
power1* --> 1162261467.
power1* --> 3486784401.
power2 <-- 3.
power2 --> <fun>
power2* <-- 20
power2 <-- 9.
power2 --> <fun>
power2* <-- 10
power2 <-- 81.
power2 --> <fun>
power2* <-- 5
power2 <-- 6561.
power2 --> <fun>
power2* <-- 2
power2 <-- 43046721.
power2 --> <fun>
power2* <-- 1
power2 <-- 1853020188851841.
power2 --> <fun>
power2* <-- 0
power2* --> 1.
power2* --> 43046721.
power2* --> 43046721.
power2* --> 3486784401.
power2* --> 3486784401.
power2* --> 3486784401.

Mutually recursive functions, on an example¶

let rec even (n:int):bool = if n=0 then true else odd (n-1)
   and odd (n:int):bool = if n=0 then false else even (n-1) ;;

even 4;;
even 5;;

#trace even;;
#trace odd;;

even 3;;

val even : int -> bool = <fun>
val odd : int -> bool = <fun>

- : bool = true

- : bool = false

even is now traced.
odd is now traced.
even <-- 3
odd <-- 2
even <-- 1
odd <-- 0
odd --> false
even --> false
odd --> false
even --> false

- : bool = false

Termination¶

let rec mc n = if (n> 100) then (n-10) else mc (mc (n+11)) ;;

#trace mc;;

mc 89;;
mc 10;;


let rec weird_fact (n:int):int =
  if n=0 then 1
  else (weird_fact (n+1))/(n+1);;

weird_fact 1;;

let rec sum (n:int):int = if (n=0) then 0 else n + sum (n-1);;

#trace sum;;

sum 10 ;;

val mc : int -> int = <fun>

mc is now traced.
mc <-- 89
mc <-- 100
mc <-- 111
mc --> 101
mc <-- 101
mc --> 91
mc --> 91
mc <-- 91
mc <-- 102
mc --> 92
mc <-- 92
mc <-- 103
mc --> 93
mc <-- 93
mc <-- 104
mc --> 94
mc <-- 94
mc <-- 105
mc --> 95
mc <-- 95
mc <-- 106
mc --> 96
mc <-- 96
mc <-- 107
mc --> 97
mc <-- 97
mc <-- 108
mc --> 98
mc <-- 98
mc <-- 109
mc --> 99
mc <-- 99
mc <-- 110
mc --> 100
mc <-- 100
mc <-- 111
mc --> 101
mc <-- 101
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc <-- 10
mc <-- 21
mc <-- 32
mc <-- 43
mc <-- 54
mc <-- 65
mc <-- 76
mc <-- 87
mc <-- 98
mc <-- 109
mc --> 99
mc <-- 99
mc <-- 110
mc --> 100
mc <-- 100
mc <-- 111
mc --> 101
mc <-- 101
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc <-- 91
mc <-- 102
mc --> 92
mc <-- 92
mc <-- 103
mc --> 93
mc <-- 93
mc <-- 104
mc --> 94
mc <-- 94
mc <-- 105
mc --> 95
mc <-- 95
mc <-- 106
mc --> 96
mc <-- 96
mc <-- 107
mc --> 97
mc <-- 97
mc <-- 108
mc --> 98
mc <-- 98
mc <-- 109
mc --> 99
mc <-- 99
mc <-- 110
mc --> 100
mc <-- 100
mc <-- 111
mc --> 101
mc <-- 101
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc <-- 91
mc <-- 102
mc --> 92
mc <-- 92
mc <-- 103
mc --> 93
mc <-- 93
mc <-- 104
mc --> 94
mc <-- 94
mc <-- 105
mc --> 95
mc <-- 95
mc <-- 106
mc --> 96
mc <-- 96
mc <-- 107
mc --> 97
mc <-- 97
mc <-- 108
mc --> 98
mc <-- 98
mc <-- 109
mc --> 99
mc <-- 99
mc <-- 110
mc --> 100
mc <-- 100
mc <-- 111
mc --> 101
mc <-- 101
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc <-- 91
mc <-- 102
mc --> 92

- : int = 91

- : int = 91

val weird_fact : int -> int = <fun>

mc <-- 92
mc <-- 103
mc --> 93
mc <-- 93
mc <-- 104
mc --> 94
mc <-- 94
mc <-- 105
mc --> 95
mc <-- 95
mc <-- 106
mc --> 96
mc <-- 96
mc <-- 107
mc --> 97
mc <-- 97
mc <-- 108
mc --> 98
mc <-- 98
mc <-- 109
mc --> 99
mc <-- 99
mc <-- 110
mc --> 100
mc <-- 100
mc <-- 111
mc --> 101
mc <-- 101
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc <-- 91
mc <-- 102
mc --> 92
mc <-- 92
mc <-- 103
mc --> 93
mc <-- 93
mc <-- 104
mc --> 94
mc <-- 94
mc <-- 105
mc --> 95
mc <-- 95
mc <-- 106
mc --> 96
mc <-- 96
mc <-- 107
mc --> 97
mc <-- 97
mc <-- 108
mc --> 98
mc <-- 98
mc <-- 109
mc --> 99
mc <-- 99
mc <-- 110
mc --> 100
mc <-- 100
mc <-- 111
mc --> 101
mc <-- 101
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc <-- 91
mc <-- 102
mc --> 92
mc <-- 92
mc <-- 103
mc --> 93
mc <-- 93
mc <-- 104
mc --> 94
mc <-- 94
mc <-- 105
mc --> 95
mc <-- 95
mc <-- 106
mc --> 96
mc <-- 96
mc <-- 107
mc --> 97
mc <-- 97
mc <-- 108
mc --> 98
mc <-- 98
mc <-- 109
mc --> 99
mc <-- 99
mc <-- 110
mc --> 100
mc <-- 100
mc <-- 111
mc --> 101
mc <-- 101
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc <-- 91
mc <-- 102
mc --> 92
mc <-- 92
mc <-- 103
mc --> 93
mc <-- 93
mc <-- 104
mc --> 94
mc <-- 94
mc <-- 105
mc --> 95
mc <-- 95
mc <-- 106
mc --> 96
mc <-- 96
mc <-- 107
mc --> 97
mc <-- 97
mc <-- 108
mc --> 98
mc <-- 98
mc <-- 109
mc --> 99
mc <-- 99
mc <-- 110
mc --> 100
mc <-- 100
mc <-- 111
mc --> 101
mc <-- 101
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc <-- 91
mc <-- 102
mc --> 92
mc <-- 92
mc <-- 103
mc --> 93
mc <-- 93
mc <-- 104
mc --> 94
mc <-- 94
mc <-- 105
mc --> 95
mc <-- 95
mc <-- 106
mc --> 96
mc <-- 96
mc <-- 107
mc --> 97
mc <-- 97
mc <-- 108
mc --> 98
mc <-- 98
mc <-- 109
mc --> 99
mc <-- 99
mc <-- 110
mc --> 100
mc <-- 100
mc <-- 111
mc --> 101
mc <-- 101
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc <-- 91
mc <-- 102
mc --> 92
mc <-- 92
mc <-- 103
mc --> 93
mc <-- 93
mc <-- 104
mc --> 94
mc <-- 94
mc <-- 105
mc --> 95
mc <-- 95
mc <-- 106
mc --> 96
mc <-- 96
mc <-- 107
mc --> 97
mc <-- 97
mc <-- 108
mc --> 98
mc <-- 98
mc <-- 109
mc --> 99
mc <-- 99
mc <-- 110
mc --> 100
mc <-- 100
mc <-- 111
mc --> 101
mc <-- 101
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91
mc --> 91

Stack overflow during evaluation (looping recursion?).
Raised by primitive operation at file "[14]", line 11, characters 7-25
Called from file "[14]", line 11, characters 7-25
Called from file "[14]", line 11, characters 7-25
Called from file "[14]", line 11, characters 7-25
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Recursive types¶

type weird_road =
   LEFT of weird_road
  | RIGHT of weird_road ;;

type safer_road =
   LEFT of safer_road
  | RIGHT of safer_road
  | Stop ;;

let r1 = LEFT (RIGHT Stop) ;;

type meal = KFC of chicken | HOME of egg and chicken = COMEFROM of egg and egg = COMEFROM of chicken ;;

type weird_road = LEFT of weird_road | RIGHT of weird_road

type safer_road = LEFT of safer_road | RIGHT of safer_road | Stop

val r1 : safer_road = LEFT (RIGHT Stop)

File "[12]", line 12, characters 81-100:
Warning 30: the constructor COMEFROM is defined in both types chicken and egg.

type meal = KFC of chicken | HOME of egg
and chicken = COMEFROM of egg
and egg = COMEFROM of chicken

Peano natural numbers¶

type natPeano = Zero | Suc of natPeano;;

Zero ;;

(* 
Suc Suc Zero ;; (* parenthesis needed *)
*)

Suc (Suc Zero) ;; 

let three = Suc (Suc (Suc Zero));;

let rec natPeano2int (n:natPeano):int =
  match n with
      Zero -> 0
    | Suc nprime -> 1 + natPeano2int nprime ;;

natPeano2int three;;

let rec int2natPeano (n:int):natPeano =
  match n with
      0 -> Zero
    | nprime -> Suc (int2natPeano (n-1)) ;; 

int2natPeano 5;;

type natPeano = Zero | Suc of natPeano

- : natPeano = Zero

- : natPeano = Suc (Suc Zero)

val three : natPeano = Suc (Suc (Suc Zero))

val natPeano2int : natPeano -> int = <fun>

- : int = 3

val int2natPeano : int -> natPeano = <fun>

- : natPeano = Suc (Suc (Suc (Suc (Suc Zero))))

Some functions around Peano numbers¶

let rec sumPeano (n1:natPeano) (n2:natPeano):natPeano =
  match n1 with
      Zero -> n2
    | Suc n1prime -> sumPeano n1prime (Suc n2) ;;

let two = Suc (Suc Zero);;

sumPeano two three ;;

sumPeano Zero three;;

let rec prodPeano (n1:natPeano) (n2:natPeano):natPeano =
  match n1 with
      Zero -> Zero
    | Suc Zero -> n2
    | Suc nprime1 -> sumPeano (prodPeano nprime1 n2) n2 ;;

let six = prodPeano three two ;;

prodPeano three Zero ;;

prodPeano Zero three ;;

let rec factPeano (n:natPeano):natPeano = 
  match n with 
      Zero -> Suc Zero
    | Suc Zero -> Suc Zero
    | Suc nprime -> prodPeano n (factPeano nprime) ;;

factPeano (Suc (Suc Zero)) ;;

factPeano six ;;

natPeano2int (factPeano six) ;;

let rec fact (n:int):int = 
  match n with 
      0 | 1 -> 1
    | n -> n * (fact (n-1)) ;;

fact 6 ;;

val sumPeano : natPeano -> natPeano -> natPeano = <fun>

val two : natPeano = Suc (Suc Zero)

- : natPeano = Suc (Suc (Suc (Suc (Suc Zero))))

- : natPeano = Suc (Suc (Suc Zero))

val prodPeano : natPeano -> natPeano -> natPeano = <fun>

val six : natPeano = Suc (Suc (Suc (Suc (Suc (Suc Zero)))))

- : natPeano = Zero

- : natPeano = Zero

val factPeano : natPeano -> natPeano = <fun>

- : natPeano = Suc (Suc Zero)

Polynomials of one variable¶

type coef = int ;;
type degree = int ;;
type monomial = coef * degree ;;

type polynomial = Mn of monomial | Plus of monomial * polynomial ;;

let x4:coef*degree = (1,4);;

let two_x:coef*degree = (2,1);;
let three:monomial = (3,0);;

let poly:polynomial = Plus (x4, Plus (two_x, Mn three)) ;;
let polyprime:polynomial = Plus (three, Plus (two_x, Mn x4)) ;;

let rec dg_max (p:polynomial):int =
  match p with
      Mn (c,d) -> d
    | Plus ((c,d), pprime) -> max d (dg_max pprime) ;; 
    
dg_max poly ;;
dg_max polyprime ;;

type coef = int

type degree = int

type monomial = coef * degree

type polynomial = Mn of monomial | Plus of monomial * polynomial

val x4 : coef * degree = (1, 4)

val two_x : coef * degree = (2, 1)

val three : monomial = (3, 0)

val poly : polynomial = Plus ((1, 4), Plus ((2, 1), Mn (3, 0)))

val polyprime : polynomial = Plus ((3, 0), Plus ((2, 1), Mn (1, 4)))

val dg_max : polynomial -> int = <fun>

Disadvantages of this modelisation

poly = polyprime ;;

let poly1 = Plus ((0,5), poly) and poly2 = Plus ((-1,4),poly) in (dg_max poly1, dg_max poly2) ;;

- : bool = false

- : int * int = (5, 4)

Model 2¶

type polynomial = Zero | Plus of monomial * polynomial ;;

let poly = Plus (x4, Plus (two_x, Plus (three, Zero))) ;;

let polyprime = Plus (three, Plus (x4, Plus (two_x, Zero)));;

let rec well_formed_acc (p:polynomial) (n:int):bool=
  match p with 
    | Zero -> true
    | Plus ((c,d),pprime) -> c <> 0 && d<n && well_formed_acc pprime d ;;

let well_formed (p:polynomial):bool=
  match p with
    | Zero -> true
    | Plus ((c,d),pprime) -> c <> 0 && well_formed_acc pprime d ;;

well_formed poly ;;
well_formed polyprime ;;

let dg_max (p:polynomial):int=
  if well_formed p then
    match p with 
    Zero -> 0
    | Plus ((c,d),pprime) -> d
  else
    failwith "Polynomial is ill-formed" ;;

dg_max poly ;;

let rec add_mono_poly (m:monomial) (p:polynomial):polynomial=
  let (c,d)=m in
    if (c=0 || not (well_formed p)) then failwith "Ill-formed arugments"
    else (
    match p with 
      | Zero -> Plus (m,Zero)
      | Plus ((cc,dd), pp) -> 
      if (d> dd) then Plus (m, p)
      else (
        if (d=dd) then 
          let coef = c+cc in if coef=0 then pp else Plus ((coef,dd),pp)
        else (* dd > d*)
        Plus  ((cc,dd), add_mono_poly m pp)
      )
    )
;;

let rec add_polynomials (p1:polynomial) (p2:polynomial):polynomial=
  match p1,p2 with
      Zero,p -> p
    | p,Zero -> p
    | Plus (m,p), _ -> add_polynomials p (add_mono_poly m p2)
;;

poly;;
add_polynomials poly poly;;

type polynomial = Zero | Plus of monomial * polynomial

val poly : polynomial = Plus ((1, 4), Plus ((2, 1), Plus ((3, 0), Zero)))

val polyprime : polynomial =
  Plus ((3, 0), Plus ((1, 4), Plus ((2, 1), Zero)))

val well_formed_acc : polynomial -> int -> bool = <fun>

val well_formed : polynomial -> bool = <fun>

- : bool = true

- : bool = false

val dg_max : polynomial -> int = <fun>

- : int = 4

val add_mono_poly : monomial -> polynomial -> polynomial = <fun>