Exercises

Exercise: complex synonym [✭]

Here is a signature for complex numbers, which have a real and imaginary component:

module type ComplexSig = sig
  val zero : float * float
  val add : float * float -> float * float -> float * float
end

Improve that code by adding type t = float * float to the signature. Show how the signature can be written more tersely because of the type synonym.

Exercise: complex encapsulation [✭✭]

Here is a structure that matches the signature from the previous exercise:

module Complex : ComplexSig = struct
  type t = float * float
  let zero = (0., 0.)
  let add (r1,i1) (r2,i2) = r1 +. r2, i1 +. i2
end

Investigate what happens if you make the following changes (each independently), and explain why any errors arise:

  • remove zero from the structure
  • remove add from the signature
  • change zero in the structure to let zero = 0, 0
Exercise: big list queue [✭✭]

Use the following code to create ListQueue's of exponentially increasing length: 10, 100, 1000, etc. How big of a queue can you create before there is a noticeable delay? How big until there's a delay of at least 10 seconds? (Note: you can abort utop computations with Ctrl-C.)

(* Creates a ListQueue filled with [n] elements. *)
let fill_listqueue n =
  let rec loop n q =
    if n=0 then q
    else loop (n-1) (ListQueue.enqueue n q) in
  loop n ListQueue.empty

Exercise: big two-list queue [✭✭]

Use the following function to create TwoListQueue's of exponentially increasing length:

let fill_twolistqueue n =
  let rec loop n q =
    if n=0 then q
    else loop (n-1) (TwoListQueue.enqueue n q) in
  loop n TwoListQueue.empty

Now how big of a queue can you create before there's a delay of at least 10 seconds?

Exercise: queue efficiency [✭✭✭]

Compare the implementations of enqueue in ListQueue vs. TwoListQueue. Explain in your own words why the efficiency of ListQueue.enqueue is linear time in the length of the queue. Hint: consider the @ operator. Then explain why adding \(n\) elements to the queue takes time that is quadratic in \(n\).

Now consider TwoListQueue.enqueue. Suppose that the queue is in a state where it has never had any elements dequeued. Explain in your own words why TwoListQueue.enqueue is constant time. Then explain why adding \(n\) elements to the queue takes time that is linear in \(n\).

(Note: the enqueue and dequeue operations for TwoListQueue remain constant time even after interleaving them, but showing why that is so require the study of amortized analysis, which we will not cover here.)

Exercise: binary search tree dictionary [✭✭✭]

Write a module BstDict that implements the Dictionary module type using the tree type.

Exercise: fraction [✭✭✭]

Write a module that implements the Fraction module type below:

module type Fraction = sig
  (* A fraction is a rational number p/q, where q != 0.*)
  type t

  (* [make n d] is n/d. Requires d != 0. *)
  val make : int -> int -> t

  val numerator : t -> int
  val denominator : t -> int
  val to_string : t -> string
  val to_float : t -> float

  val add : t -> t -> t
  val mul : t -> t -> t
end
Exercise: fraction reduced [✭✭✭]

Modify your implementation of Fraction to ensure these invariants hold of every value v of type t that is returned from make, add, and mul:

  1. v is in reduced form

  2. the denominator of v is positive

For the first invariant, you might find this implementation of Euclid's algorithm to be helpful:

(* [gcd x y] is the greatest common divisor of [x] and [y].
 * requires: [x] and [y] are positive.
 *)
let rec gcd (x:int) (y:int) : int =
  if x = 0 then y
  else if (x < y) then gcd (y - x) x
  else gcd y (x - y)

Map

The next few exercises explore the Map module in the OCaml standard library. It is an implementation of a dictionary data structure. Recall that dictionaries map keys to values. If a key \(k\) maps to a value \(v\), we say that \(v\) is bound to \(k\).

Exercise: make char map [✭]

To create a map, we first have to use the Map.Make functor to produce a module that is specialized for the type of keys we want. Type the following in utop:

# module CharMap = Map.Make(Char);;

The output tells you that a new module named CharMap has been defined, and it gives you a signature for it. Find the values empty, add, and remove in that signature. Explain their types in your own words.

Exercise: char ordered [✭]

The Map.Make functor requires its input module to match the Map.OrderedType signature. Look at that signature as well as the signature for the Char module. Explain in your own words why we are allowed to pass Char as an argument to Map.Make.

Exercise: use char map [✭✭]

Using the CharMap you just made, create a map that contains the following bindings:

  • 'A' maps to "Alpha"
  • 'E' maps to "Echo"
  • 'S' maps to "Sierra"
  • 'V' maps to "Victor"

Use CharMap.find to find the binding for 'E'.

Now remove the binding for 'A'. Use CharMap.mem to find whether 'A' is still bound.

Use the function CharMap.bindings to convert your map into an association list. Are the correct three bindings active in it?

Exercise: bindings [✭✭]

Investigate the documentation of the Map.S signature to find the specification of bindings. Which of these expressions will return the same association list?

  1. CharMap.(empty |> add 'x' 0 |> add 'y' 1 |> bindings)

  2. CharMap.(empty |> add 'y' 1 |> add 'x' 0 |> bindings)

  3. CharMap.(empty |> add 'x' 2 |> add 'y' 1 |> remove 'x' |> add 'x' 0 |> bindings)

Check your answer in utop.

Exercise: date order [✭✭]

Here is a type for dates:

type date = { month:int; day:int }

For example, March 31st would be represented as {month=3; day=31}. Our goal in the next few exercises is to implement a map whose keys have type date.

Obviously it's possible to represent invalid dates with type date—for example, { month=6; day=50 } would be June 50th, which is not a real date. The behavior of your code in the exercises below is unspecified for invalid dates.

To create a map over dates, we need a module that we can pass as input to Map.Make. That module will need to match the Map.OrderedType signature. Create such a module. Here is some code to get you started:

module Date = struct
  type t = date
  let compare ...
end

Recall the specification of compare in Map.OrderedType as you write your Date.compare function.

Exercise: calendar [✭✭]

Use the Map.Make functor with your Date module to create a DateMap module. Then define a calendar type as follows:

type calendar = string DateMap.t

The idea is that calendar maps a date to the name of an event occurring on that date.

Using the functions in the DateMap module, create a calendar with a few entries in it, such as birthdays or anniversaries.

Exercise: print calendar [✭✭]

Write a function print_calendar : calendar -> unit that prints each entry in a calendar in a format similar the inspiring examples in the previous exercise. Hint: use DateMap.iter, which is documented in the Map.S signature.

Exercise: is for [✭✭✭]

Write a function is_for : string CharMap.t -> string CharMap.t that given an input map with bindings from \(k_1\) to \(v_1\), ..., \(k_n\) to \(v_n\), produces an output map with the same keys, but where each key \(k_i\) is now bound to the string "\(k_i\) is for \(v_i\)". For example, if m maps 'a' to "apple", then is_for m would map 'a' to "a is for apple". Hint: there is a one-line solution that uses a function from the Map.S signature. To convert a character to a string, you could use String.make. An even fancier way would be to use Printf.sprintf.

Exercise: first after [✭✭✭]

Write a function first_after : calendar -> Date.t -> string that returns the name of the first event that occurs strictly after the given date. If there is no such event, the function should raise Not_found, which is an exception already defined in the standard library. Hint: there is a one-line solution that uses two functions from the Map.S signature.

Exercise: sets [✭✭✭]

The standard library Set module is quite similar to the Map module. Use it to create a module that represents sets of case-insensitive strings. Strings that differ only in their case should be considered equal by the set. For example, the sets {"grr", "argh"} and {"aRgh", "GRR"} should be considered the same, and adding "gRr" to either set should not change the set. Assuming your module is named CisSet, here is some test code:

# CisSet.(equal (of_list ["grr"; "argh"]) (of_list ["GRR"; "aRgh"]))
- : bool = true

Writing functors

Our goal in the next series of exercises is to write a functor that, given a module supporting a to_string function, returns a module supporting a print function that prints that string.

Exercise: ToString [✭✭]

Write a module type ToString that specifies a signature with an abstract type t and a function to_string : t -> string.

Exercise: Print [✭✭]

Write a functor Print that takes as input a module named M of type ToString. The structure returned by your functor should have exactly one value in it, print, which is a function that takes a value of type M.t and prints a string representation of that value.

Exercise: Print Int [✭✭]

Create a module named PrintInt that is the result of applying the functor Print to a new module Int. You will need to write Int yourself. The type Int.t should be int. Hint: do not seal Int.

Experiment with PrintInt in utop. Use it to print the value of an integer.

Exercise: Print String [✭✭]

Create a module named PrintString that is the result of applying the functor Print to a new module MyString. You will need to write MyString yourself. Hint: do not seal MyString.

Experiment with PrintString in utop. Use it to print the value of a string.

Exercise: Print reuse [✭]

Explain in your own words how Print has achieved code reuse, albeit a very small amount.

Exercise: Print String reuse revisited [✭✭]

The PrintString module you created above supports just one operation: print. It would be great to have a module that supports all the String module functions in addition to that print operation, and it would be super great to derive such a module without having to copy any code.

Define a module StringWithPrint. It should have all the values of the built-in String module. It should also have the print operation, which should be derived from the Print functor rather than being copied code.

Hint: use two include statements.

Compilation Units

The next couple exercises play with compilation units.

Exercise: implementation without interface [✭]

Create a file named date.ml. In it put exactly the following code:

type date = { month:int; day:int }
let make_date month day = {month; day}
let get_month d = d.month
let get_day d = d.day
let to_string d = (string_of_int d.month) ^ "/" ^ (string_of_int d.day)

Compile that file to bytecode:

$ ocamlbuild date.cmo

Now start utop and type the following to use the module you've just created:

# #directory "_build";;
# #load "date.cmo";;

# let j1 = Date.make_date 1 1;;
val j1 : Date.date = {Date.month = 1; day = 1}    

# j1.day;;
- : int = 1

# Date.to_string j1;;
- : string = "1/1"

Exercise: implementation with interface [✭]

After doing the previous exercise, also create a file named date.mli. In it put exactly the following code:

type date = { month:int; day:int; }
val make_date : int -> int -> date
val get_month : date -> int
val get_day : date -> int
val to_string : date -> string

Recompile date.ml to bytecode:

$ ocamlbuild date.cmo

Restart utop and re-issue the same phrases as before:

# #directory "_build";;
# #load "date.cmo";;

# let j1 = Date.make_date 1 1;;
val j1 : Date.date = {Date.month = 1; day = 1}    

# j1.day;;
- : int = 1

# Date.to_string j1;;
- : string = "1/1"

Exercise: implementation with abstracted interface [✭]

After doing the previous two exercises, edit date.mli and change the first declaration in it to be exactly the following:

type date

The type date is now abstract. Recompile date.ml to bytecode:

$ ocamlbuild date.cmo

Restart utop and re-issue the same phrases as before. The responses to two of them will change. Explain in your own words those changes.

# #directory "_build";;
# #load "date.cmo";;
# let j1 = Date.make_date 1 1;;
# j1.day;;
# Date.to_string j1;;

Add a declaration to date.mli:

val format : Format.formatter -> date -> unit

And add a definition of format to date.ml. Hint: use Format.fprintf and Date.to_string.

Now recompile, load utop, and install the printer by issuing the directive

#install_printer Date.format;;

after loading date.cmo. Reissue the other phrases to utop as you did in the exercises above. The response from one phrase will change in a helpful way. Explain why.

Challenge exercise: Algebra

Download this file: algebra.ml. It contains two signatures and four structures:

  • Ring is signature that describes the algebraic structure called a ring, which is an abstraction of the addition and multiplication operators.

  • Field is a signature that describes the algebraic structure called a field, which is like a ring but also has an abstraction of the division operation.

  • IntRing and FloatRing are structures that implement rings in terms of int and float.

  • IntField and FloatField are structures that implement fields in terms of int and float.

  • IntRational and FloatRational are structures that implement fields in terms of ratios (aka fractions)—that is, pairs of int and pairs of float.

(For afficionados of abstract algebra: of course these representations don't necessarily obey all the axioms of rings and fields because of the limitations of machine arithmetic. Also, the division operation in IntField is ill-defined on zero. Try not to worry about that.)

Using this code, you can write expressions like the following:

# FloatField.(of_int 9 + of_int 3 / of_int 4 |> to_string);;
- : string = "9.75"

# IntRational.(
    let half = one / (one+one) in 
    let quarter = half*half in 
    let three = one+one+one in 
    let nine = three*three in 
    to_string (nine + (three*quarter))
  );;
- : string = "39/4"
Exercise: refactor arith [✭✭✭✭]

The file algebra.ml contains a great deal of duplicated code. Refactor the code to improve the amount of code reuse it exhibits. To do that, use include, functors, and introduce additional structures and signatures as needed.

There isn't necessarily a right answer here, but it is possible to eliminate all the duplicated code. Here's some advice to guide you toward a good solution:

  • No name should be directly declared in more than one signature. For example, (+) should not be directly declared in Field; it should be reused from an earlier signature. By "directly declared" we mean a declaration of the form val name : .... An indirect declaration would be one that results from an include.

  • You need only three direct definitions of the algebraic operations and numbers (plus, minus, times, divide, zero, one): once for int, once for float, and once for ratios. For example, IntField.(+) should not be directly defined as Stdlib.(+); rather, it should be reused from elsewhere. By "directly defined" we mean a definition of the form let name = .... An indirect definition would be one that results from an include or a functor application.

  • The rational structures can both be produced by a single functor that is applied once to IntField and once to FloatField.

  • It's possible to eliminate all duplication of of_int, such that it is directly defined exactly once, and all structures reuse that definition; and such that it is directly declared in only one signature. This will require the use of functors. It will also require inventing an algorithm that can convert an integer to an arbitrary Ring representation, regardless of what the representation type of that Ring is.

When you're done, the types of all the modules should remain unchanged. You can easily see those types by running ocamlc -i algebra.ml, which will originally output the following:

module type Ring =
  sig
    type t
    val zero : t
    val one : t
    val ( + ) : t -> t -> t
    val ( ~- ) : t -> t
    val ( * ) : t -> t -> t
    val to_string : t -> string
    val of_int : int -> t
  end
module type Field =
  sig
    type t
    val zero : t
    val one : t
    val ( + ) : t -> t -> t
    val ( ~- ) : t -> t
    val ( * ) : t -> t -> t
    val ( / ) : t -> t -> t
    val to_string : t -> string
    val of_int : int -> t
  end
module IntRing : Ring
module IntField : Field
module FloatRing : Ring
module FloatField : Field
module IntRational : Field
module FloatRational : Field

The final output of that command on your solution might define additional types, but the ones above should remain literally identical.

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