Description

This homework document consists of 3 pages. Carefully read the entire document before you start coding.
Note: All functions, unless otherwise specified, should be polymorphic (i.e., they should work with any data type). For example, if you are writing a method that should work for lists, the type must be ‘a list, and not int list.
1 Recursion and Higher-order Functions
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)
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# 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)
# 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)]
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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)
# let expr = Plus {Mult {Arg1 = Const 2; Arg1 = Const 3}; Mult {Arg1 = Const 3;
Arg2 = Minus {Arg1 = Const 4; Const 1}}} in evaluate(expr);;
– : int = 15
NOTES:
• Please remember to verify what you are submitting. Make sure you are, indeed, submitting what you think you are submitting!
• What to submit? A single .zip file comprising three .ml files. The first file must be named hw1.ml, and should contain your code for the ten questions in section 1 of this assignment. The second file must be named hw1bool.ml, and this should contain your code for question 2.1 (Boolean logic). Finally, the third file must be named hw1math.ml, with your code for question 2.2 and 2.3 (the arithmetic expression definition and evaluation).
This assignment will be graded by a script, so be absolutely sure that you follow this structure.
You don’t want to be in a situation where hw1.ml doesn’t compile because of that one function you copy-pasted from your terminal REPL and made a careless mistake (for example, forgot to include the “;;” at the end of it)!
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