# Exercises

##### Exercise: spec game [✭✭✭]

Pair up with another programmer and play the specification game with them. Take turns being the specifier and the devious programmer. Here are some suggested functions you could use:

• num_vowels : string -> int
• is_sorted : 'a list -> bool
• sort : 'a list -> 'a list
• max : 'a list -> 'a
• is_prime : int -> bool
• is_palindrome : string -> bool
• second_largest : int list -> int
• depth : 'a tree -> int

##### Exercise: poly spec [✭✭✭]

Let's create a data abstraction (a module that represents some kind of data) for single-variable integer polynomials of the form $c_n x^n + \ldots + c_1 x + c_0.$ Let's assume that the polynomials are dense, meaning that they contain very few coefficients that are zero. Here is an incomplete interface for polynomials:

(** [Poly] represents immutable polynomials with integer coefficients. *)
module type Poly = sig
(** [t] is the type of polynomials *)
type t

(** [eval x p] is [p] evaluated at [x].
Example:  if [p] represents $3x^3 + x^2 + x$, then
[eval 10 p] is . *)
val eval : int -> t -> int
end


(The use of \$ above comes from LaTeX, in which mathematical formulas are surrounded by dollar signs. Similarly, ^ represents exponentiation in LaTeX.)

Finish the design of Poly by adding more operations to the interface. Consider what operations would be useful to a client of the abstraction:

• How would they create polynomials?
• How would they combine polynomials to get new polynomials?
• How would they query a polynomial to find out what it represents?

Write specification comments for the operations that you invent. Keep in mind the spec game as you write them: could a devious programmer subvert your intentions?

##### Exercise: poly impl [✭✭✭]

Implement your specification of Poly. As part of your implementation, you will need to choose a representation type t. Hint: recalling that our polynomials are dense might guide you in choosing a representation type that makes for an easier implementation.

##### Exercise: int set rep [✭✭✭]

Consider this interface for integer sets. Suppose that you wanted the to_list implementation to run in constant time, perhaps at the expense of other operations being less efficient. Implement the interface in a file named intset.ml. First choose a representation type, then document its abstraction function and representation invariant. Inside the implementation, define a rep_ok function. Insert applications of it in the appropriate places of your implementation to guarantee that all input and output values satisfy the representation invariant.

##### Exercise: interval arithmetic [✭✭✭✭]

Specify and implement a data abstraction for interval arithmetic. Be sure to include the abstraction function, representation invariant, and rep_ok. Also implement a to_string function, or a format function as seen in the notes on functors.

##### Exercise: association list maps [✭✭✭]

Consider the MyMap signature in maps.ml. Create two implementations of it, both with the representation type ('k * 'v) list. The functions in the interface should mostly be trivially implementable with the association list functions in the standard library List module. Your first implementation should prohibit any key from appearing twice in the list; and your second should allow it. Start each implementation by documenting the AF and RI for t, and only after you do that, implement the functions.

##### Exercise: function maps [✭✭✭✭]

Implement the MyMap signature using the representation type 'k -> 'v. That is, a map is represented as an OCaml function from keys to values. Your solution will make heavy use of higher-order functions.

## A Buggy Queue

Download buggy_queues.ml, which efficiently implements queues with two lists—but with a couple bugs deliberately injected.

##### Exercise: AF and RI [✭]

The TwoListQueue module documents an abstraction function and a representation invariant, but they are not clearly identified as such. Modify the comments to explicitly identify the abstraction function and representation invariant.

##### Exercise: rep ok [✭✭]

Write a rep_ok function for TwoListQueue. Its type should be t -> t. It should raise an exception whenever the representation invariant does not hold. Modify the other functions exposed by the Queue signature to (i) check that rep_ok holds for any queues passed in, and (ii) check that rep_ok also holds for any queues passed out. Hint: you will need to add nine applications of rep_ok.

##### Exercise: test with rep ok [✭✭✭]

There are two bugs we deliberately injected into TwoListQueue. Both are places where we failed to apply norm to ensure that a queue is in normal form. Figure out where those are by testing each operation of TwoListQueue in the toplevel to see where your rep_ok raises an exception. Fix each bug by adding an application of norm.

Hint: to find one of the bugs, you will need to build a queue of length at least 2.