CSE216-Homework 1 Solved

1         Recursion and Higher-order Functions
In this section, you may not use any functions available in the OCaml library that already solves all or most of the question. For example, OCaml provides a List.rev function, but you may not use that in this section.

1. Write a recursive function pow, which takes two integer parameters x and n, and returns xn. Also write a function float pow, which does the same thing, but for x being a float. n is still a non-negative integer.
(6)
2. Write a function compress to remove consecutive duplicates from a list.

# compress ["a";"a";"b";"c";"c";"a";"a";"d";"e";"e";"e"];;

- : string list = ["a"; "b"; "c"; "a"; "d"; "e"]
(6)
3. Write a function remove if of the type 'a list -> ('a -> bool) -> 'a list, which takes a list and a predicate, and removes all the elements that satisfy the condition expressed in the predicate.

# remove_if [1;2;3;4;5] (fun x -> x mod 2 = 1);;

- : int list = [2; 4]
(6)
4. Some programming languages (like Python) allow us to quickly slice a list based on two integers i and j, to return the sublist from index i (inclusive) and j (not inclusive). We want such a slicing function in OCaml as well.

Write a function slice as follows: given a list and two indices, i and j, extract the slice of the list containing the elements from the ith (inclusive) to the jth (not inclusive) positions in the original list.

# slice ["a";"b";"c";"d";"e";"f";"g";"h"] 2 6;;

-    : string list = ["c"; "d"; "e"; "f"]

Invalid index arguments should be handled gracefully. For example,

# slice ["a";"b";"c";"d";"e";"f";"g";"h"] 3 2;;

-    : string list = []

# slice ["a";"b";"c";"d";"e";"f";"g";"h"] 3 20;

-    : string list = ["d";"e";"f";"g";"h"];

You do not, however, need to worry about handling negative indices.
(6)
5. Write a function equivs of the type ('a -> 'a -> bool) -> 'a list -> 'a list list, which partitions a list into equivalence classes according to the equivalence function.

# equivs (=) [1;2;3;4];;

-    : int list list = [[1];[2];[3];[4]]

# equivs (fun x y -> (=) (x mod 2) (y mod 2)) [1; 2; 3; 4; 5; 6; 7; 8];;

-    : int list list = [[1; 3; 5; 7]; [2; 4; 6; 8]]
(8)
6. Goldbach’s conjecture states that every positive even number greater than 2 is the sum of two prime numbers. E.g., 18 = 5+13, or 42 = 19+23. It is one of the most famous conjectures in number theory. It is unproven, but verified for all integers up to 4 × 1018. Write a function goldbachpair : int -> int * int to find two prime numbers that sum up to a given even integer. The returned pair must have a non-decreasing order.
(8)
 

# goldbachpair 10;; (* must return (3, 7) and not (7, 3) *)

- : int * int = (3, 7)

Note that the decomposition is not always unique. E.g., 10 can be written as 3+7 or as 5+5, so both (3, 7) and (5, 5) are correct answers.

7. Write a function called equiv on, which takes three inputs: two functions f and g, and a list lst. It returns true if and only if the functions f and g have identical behavior on every element of lst.

# let f i = i * i;; val f : int -> int = <fun> # let g i = 3 * i;; val g : int -> int = <fun>

# equiv_on f g [3];;

-    : bool = true

# equiv_on f g [1;2;3];;

-    : bool = false
(8)
8. Write a functions called pairwisefilter with two parameters: (i) a function cmp that compares two elements of a specific T and returns one of them, and (ii) a list lst of elements of that same type T. It returns a list that applies cmp while taking two items at a time from lst. If lst has odd size, the last element is returned “as is”.

# pairwisefilter min [14; 11; 20; 25; 10; 11];;

-    : int list = [11; 20; 10]

# (* assuming that shorter : string * string -> string = <fun> already exists *)

# pairwisefilter shorter ["and"; "this"; "makes"; "shorter"; "strings"; "always"; "win"];;

-    : string list = ["and"; "makes"; "always"; "win"]
(8)
9. Write the polynomial function, which takes a list of tuples and returns the polynomial function corresponding to that list. Each tuple in the input list consists of (i) the coefficient, and (ii) the exponent.

# (* below is the polynomial function f(x) = 3x^3 - 2x + 5 *)

# let f = polynomial [3, 3; -2, 1; 5, 0];; val f : int -> int = <fun>

# f 2;;

- : int = 25
(8)
10. The power set of a set S is the set of all subsets of S (including the empty set and the entire set). Write a function powerset of the type 'a list -> 'a list list, which treats lists as unordered sets, and returns the powerset of its input list. You may assume that the input list has no duplicates.

# powerset [3; 4; 10];;

- : int list list = [[]; [3]; [4]; [10]; [3; 4]; [3; 10]; [4; 10]; [3; 4; 10]];

2       Data Types
(8)
1. Let us define a language for expressions in Boolean logic:
(8)
type bool_expr =

| Lit of string

| Not of bool_expr

| And of bool_expr * bool_expr

| Or of bool_expr * bool_expr using which we can write expressions in prefix notation. E.g., (a∧b)∨(¬a) is Or(And(Lit("a"),Lit("b")), Not(Lit("a"))). Your task is to write a function truth table, which takes as input a logical expression in two literals and returns its truth table as a list of triples, each a tuple of the form:

(truth-value-of-first-literal, truth-value-of-second-literal, truth-value-of-expression)

For example,

# (* the outermost parentheses are needed for OCaml to parse the third argument correctly as a bool_expr *)

# truth_table "a" "b" (And(Lit("a"), Lit("b")));;

-        : (bool * bool * bool) list = [(true, true, true); (true, false, false); (false, true, false); (false, false, false)]

2

CSE 216                                                                        Homework I   Submission Deadline: Sep 25, 2020

 

2. Some data types are naturally recursive. Trees (or tree-like data structures) come to mind, since a subtree rooted under some node is a tree by itself. Here, you are being asked to define the abstract syntax tree of a simple arithmetic language. The specifications are as follows:

•   An arithmetic expression (expr) will be either a numeric constant called Const, or a variable called Var, or addition (Plus), multiplication (Mult), difference (Minus), or division (Div) of two arithmetic expressions.

•   Since we are dealing with binary arithmetic operators, the arguments are defined with the names Arg1 and Arg2.

           Using this type definition, we can represent simple arithmetic expressions in OCaml.         For example,

2x + 3(y− 1) can be represented as

# Plus {Mult {Arg1 = Const 2; Arg1 = Var "x"}; Mult {Arg1 = Const 3; Arg2 = Minus {Arg1 = Var "y"; Const 1}}};;

- : expr = Plus { Mult {Arg1 = Const 2; Arg1 = Var "x"}; Mult {Arg1 = Const 3;

Arg2 = Minus {Arg1 = Var "y"; Const 1}}}
(10)
3. Finally, write a function called evaluate, which takes as input a single arithmetic expression you defined in above, and gives as output its final value. You may assume that the input arithmetic expression to this function will only non-negative integer numeric values (i.e., no variables and no floats).
(10)
# let expr = Plus {Mult {Arg1 = Const 2; Arg1 = Const 3}; Mult {Arg1 = Const 3;

Arg2 = Minus {Arg1 = Const 4; Const 1}}} in evaluate(expr);;