# Tuples. CMSC 330: Organization of Programming Languages. Examples With Tuples. Another Example

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1 CMSC 330: Organization of Programming Languages OCaml 2 Higher Order Functions Tuples Constructed using (e1,..., en) Deconstructed using pattern matching Patterns involve parens and commas, e.g., (p1,p2, ) Tuples are like C structs But without field labels Allocated on the heap Tuples can be heterogenous Unlike lists, which must be homogenous (1, ["string1"; "string2"]) is a valid tuple 1 2 Examples With Tuples let plusthree (x, y, z) = x + y + z let addone (x, y, z) = (x+1, y+1, z+1) plusthree (addone (3, 4, 5)) (* returns 15 *) let sum ((a, b), c) = (a+c, b+c) sum ((1, 2), 3) = (4, 5) let plusfirsttwo (x::y::_, a) = (x + a, y + a) plusfirsttwo ([1; 2; 3], 4) = (5, 6) let tls (_::xs, _::ys) = (xs, ys) tls ([1; 2; 3], [4; 5; 6; 7]) = ([2; 3], [5; 6; 7]) Another Example let f l = match l with x::(_::y) -> (x,y) What is f [1;2;3;4]? Possibilities: ([1],[3]) (1,3) (1,[3]) (1,4) (1,[3;4]) Remember, semicolon for lists, comma for tuples [1, 2] = [(1, 2)] = a list of size one (1; 2) = Warning: This expression should have type unit 3 4 1

2 List And Tuple Types Tuple types use * to separate components List And Tuple Types Tuple types use * to separate components Examples (1, 2) : (1, "string", 3.5) : (1, ["a"; "b"], 'c') : [(1,2)] : [(1, 2); (3, 4)] : [(1,2); (1,2,3)] : Examples (1, 2) : int * int (1, "string", 3.5) : int * string * float (1, ["a"; "b"], 'c') : int * string list * char [(1,2)] : (int * int) list [(1, 2); (3, 4)] : (int * int) list [(1,2); (1,2,3)] : error Ø Because the first list element has type int * int, but the second has type int * int * int list elements must all be of the same type 5 6 Polymorphic Types Some functions we saw require specific list types let plusfirsttwo (x::y::_, a) = (x + a, y + a) plusfirsttwo : int list * int -> (int * int) But other functions work for any list let hd (h::_) = h hd [1; 2; 3] (* returns 1 *) hd ["a"; "b"; "c"] (* returns "a" *) OCaml gives such functions polymorphic types hd : 'a list -> 'a this says the function takes a list of any element type 'a, and returns something of that type Examples Of Polymorphic Types let tl (_::t) = t tl : 'a list -> 'a list let swap (x, y) = (y, x) swap : 'a * 'b -> 'b * 'a let tls (_::xs, _::ys) = (xs, ys) tls : 'a list * 'b list -> 'a list * 'b list let eq (x,y) = x = y eq : 'a * a -> bool 7 8 2

3 Tuples Are A Fixed Size This OCaml definition # let foo x = match x with (a, b) -> a + b (a, b, c) -> a + b + c;; Would yield this error message This pattern matches values of type 'a * 'b * 'c but is here used to match values of type 'd * 'e Tuples of different size have different types Thus never more than one match case with tuples 9 Conditionals Use if...then...else like C/Java/Ruby But no parentheses, no elsif, and no end if grade >= 90 then print_string "You got an A" else if grade >= 80 then print_string "You got a B" else if grade >= 70 then print_string "You got a C" else print_string "You re not doing so well" 10 Conditionals (cont.) In OCaml, conditionals return a result The value of whichever branch is true/false Like? : in C, C++, and Java # if 7 > 42 then "hello" else goodbye";; - : string = "goodbye" # let x = if true then 3 else 4;; x : int = 3 # if false then 3 else 3.0;; This expression has type float but is here used with type int The Factorial Function Using conditionals & functions Can you write fact, the factorial function? let rec fact n = if n = 0 then 1 else n * fact (n-1);; Notice no return statements This is pretty much how it needs to be written

4 Let Rec The rec part means define a recursive function Let vs. let rec let x = e1 in e2 x in scope within e2 let rec x = e1 in e2 x in scope within e2 and e1 Why use let rec? If you used let instead of let rec to define fact let fact n = if n = 0 then 1 else n * fact (n-1) in e2 Fact is not bound here! 13 Let More Examples let f n = 10;;;; let f n = if n = 0 then 1 else n * f (n 1);;;; f 0;;;; f 1;;;; (* 1 *) (* 10 *) let f x = f in f (* Unbound value f *) let rec f x = f in f (* Bound value f *) 14 Recursion = Looping Recursion is essentially the only way to iterate (The only way we re going to talk about) Another example let rec print_up_to (n, m) = print_int n; print_string "\n"; if n < m then print_up_to (n + 1, m) Lists and Recursion Lists have a recursive structure And so most functions over lists will be recursive let rec length l = match l with [] -> 0 (_::t) -> 1 + (length t) This is just like an inductive definition Ø The length of the empty list is zero Ø The length of a nonempty list is 1 plus the length of the tail Type of length?

5 More Examples More Examples (cont.) sum l (* sum of elts in l *) let rec sum l = match l with [] -> 0 (x::xs) -> x + (sum xs) negate l (* negate elements in list *) let rec negate l = match l with (x::xs) -> (-x) :: (negate xs) (* return a list containing all the elements in the list l followed by all the elements in list m *) append (l, m) let rec append (l, m) = match l with [] -> m (x::xs) -> x::(append (xs, m)) rev l (* reverse list; hint: use append *) let rec rev l = match l with (x::xs) -> append ((rev xs), [x]) last l (* last element of l *) let rec last l = match l with [x] -> x rev takes O(n 2 ) time. Can you do better? (x::xs) -> last xs A Clever Version of Reverse let rec rev_helper (l, a) = match l with [] -> a (x::xs) -> rev_helper (xs, (x::a)) let rev l = rev_helper (l, []) Let s give it a try rev [1; 2; 3] rev_helper ([1;2;3], []) rev_helper ([2;3], [1]) rev_helper ([3], [2;1]) rev_helper ([], [3;2;1]) [3;2;1] More Examples flattenpairs l (* ('a * 'a) list -> 'a list *) let rec flattenpairs l = match l with ((a, b)::t) -> a :: b :: (flattenpairs t) take (n, l) (* return first n elts of l *) let rec take (n, l) = if n = 0 then [] else match l with (x::xs) -> x :: (take (n-1, xs))

6 Working With Lists Several of these examples have the same flavor Walk through the list and do something to every element Walk through the list and keep track of something Recall the following example code from Ruby: Here we passed a code block into the collect method Wouldn t it be nice to do the same in OCaml? a = [1,2,3,4,5] b = a.collect { x -x } 21 Anonymous Functions Recall code blocks in Ruby Here, we can think of { x print x } as a function We can do this (and more) in Ocaml where (1..10).each { x print x } range_each (1,10) (fun x -> print_int x) let rec range_each (i,j) f = if i > j then () else let _ = f i in (* ignore result *) range_each (i+1,j) f 22 Anonymous Functions Passing functions around is very common So often we don t want to bother to give them names Use fun to make a function with no name Parameter (fun x -> x + 3) 5 fun x -> x + 3 Body = 8 All Functions Are Anonymous Functions are first-class, so you can bind them to other names as you like let f x = x + 3 let g = f g 5 In fact, let for functions is just shorthand let f x = body = 8 stands for let f = fun x -> body

7 Examples Higher-Order Functions let next x = x + 1 Short for let next = fun x -> x + 1 let plus (x, y) = x + y Short for let plus = fun (x, y) -> x + y Which is short for Ø let plus = fun z -> (match z with (x, y) -> x + y) let rec fact n = if n = 0 then 1 else n * fact (n-1) Short for let rec fact = fun n -> (if n = 0 then 1 else n * fact (n-1)) In OCaml you can pass functions as arguments, and return functions as results let plus_three x = x + 3 let twice f z = f (f z) twice plus_three 5 twice : ('a->'a) -> 'a -> 'a let plus_four x = x + 4 let pick_fn n = if n > 0 then plus_three else plus_four (pick_fn 5) 0 pick_fn : int -> (int->int) Currying Curried Functions In OCaml We just saw a way for a function to take multiple arguments The function consumes one argument at a time, returning a function that takes the rest This is called currying the function Named after the logician Haskell B. Curry But Schönfinkel and Frege discovered it Ø So it should probably be called Schönfinkelizing or Fregging OCaml has a really simple syntax for currying let add x y = x + y This is identical to all of the following: let add = (fun x -> (fun y -> x + y)) let add = (fun x y -> x + y) let add x = (fun y -> x+y) Thus: add has type int -> (int -> int) add 3 has type int -> int Ø add 3 is a function that adds 3 to its argument (add 3) 4 = 7 This works for any number of arguments

8 Curried Functions In OCaml (cont.) Because currying is so common, OCaml uses the following conventions: -> associates to the right Ø Thus int -> int -> int is the same as Ø int -> (int -> int) function application associates to the left Ø Thus add 3 4 is the same as Ø (add 3) 4 Mental Shorthand You can think of curried types as defining multiargument functions Type int -> float -> float is a function that takes an int and a float and returns a float Type int -> int -> int -> int is a function that takes three ints and returns an int The bonus is that you can partially apply the function to some of its arguments And apply that to the rest of the arguments later Another Example Of Currying A curried add function with three arguments: The same as Then... let add_th x y z = x + y + z let add_th x = (fun y -> (fun z -> x+y+z)) add_th has type int -> (int -> (int -> int)) add_th 4 has type int -> (int -> int) add_th 4 5 has type int -> int add_th is 15 Implementing this is Challenging! Implementing functions that return other functions requires a clever data structure called a closure We ll see how these are implemented later In the meantime, we will explore using higher order functions, and then discuss how they are implemented

9 The Map Function Let s write the map function (like Ruby's collect) Takes a function and a list, applies the function to each element of the list, and returns a list of the results let rec map f l = match l with (h::t) -> (f h)::(map f t) let add_one x = x + 1 let negate x = -x map add_one [1; 2; 3] map negate [9; -5; 0] Type of map? The Map Function (cont.) What is the type of the map function? let rec map f l = match l with (h::t) -> (f h)::(map f t) ('a -> 'b) -> 'a list -> 'b list f l Pattern Matching With Fun match can be used within fun map (fun l -> match l with (h::_) -> h) [ [1; 2; 3]; [4; 5; 6; 7]; [8; 9] ] = [1; 4; 8] But use named functions for complicated matches May use standard pattern matching abbreviations map (fun (x, y) -> x+y) [(1,2); (3,4)] = [3; 7] 35 The Fold Function Common pattern Iterate through list and apply function to each element, keeping track of partial results computed so far let rec fold f a l = match l with [] -> a (h::t) -> fold f (f a h) t a = accumulator Usually called fold left to remind us that f takes the accumulator as its first argument What's the type of fold? = ('a -> 'b -> 'a) -> 'a -> 'b list -> 'a 36 9

10 Example Another Example let rec fold f a l = match l with [] -> a (h::t) -> fold f (f a h) t let rec fold f a l = match l with [] -> a (h::t) -> fold f (f a h) t let add a x = a + x fold add 0 [1; 2; 3; 4] fold add 1 [2; 3; 4] fold add 3 [3; 4] fold add 6 [4] fold add 10 [] 10 let next a _ = a + 1 fold next 0 [2; 3; 4; 5] fold next 1 [3; 4; 5] fold next 2 [4; 5] fold next 3 [5] fold next 4 [] 4 We just built the sum function! We just built the length function! Using Fold to Build Reverse let rec fold f a l = match l with [] -> a (h::t) -> fold f (f a h) t Can you build the reverse function with fold? let prepend a x = x::a fold prepend [] [1; 2; 3; 4] fold prepend [1] [2; 3; 4] fold prepend [2; 1] [3; 4] fold prepend [3; 2; 1] [4] fold prepend [4; 3; 2; 1] [] [4; 3; 2; 1] 39 10

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