PS7: Set Yourself Free

• Dueish: Wed. Apr. 12.
• Notes:
  • This pset has 100 total points.
  • This pset contains one solo problems worth 15 points.
  • It is strongly recommended that you do the non-solo SML problems (Problems 4 and 5) first, to get more practice with SML and modules/abstract datatypes that will be valuable going forward.
  • This second problem involves reading parts of a long paper. It is best to spread this problem out over multiple sittings.
• Submission:
  • In your yourFullName CS251 Spring 2017 Folder, create a Google Doc named yourFullName CS251 PS7.
  • At the top of your yourFullName CS251 PS7 doc, include your name, problem set number, date of submission, and an approximation of how long each problem part took.
  • For all parts of all problems, include all answers (including SML and Racket code) in your PS7 google doc. Format code using a fixed-width font, like Consolas or Courier New. You can use a small font size if that helps.
  • In Problems 1, 4, and 5, you will create the following code files from starter files populating your ~wx/cs251/sml/ps7 directory on your wx Virtual Machine:
    • Problem 1: yourAccountName-marbles.sml
    • Problem 4: yourAccountName-FunSet.sml
    • Problem 5: yourAccountName-OperationTreeSet.sml

    For these problems:

    • include all functions from the four files named yourAccountName... in your Google Doc (so that I can comment on them).
    • For Problem 4b, don’t forget to include your English answers in your Google Doc.

    • Drop a copy of your ~wx/cs251/sml/ps7 folder in your ~/cs251/drop/ps07 drop folder on cs.wellesley.edu by executing the following (replacing gdome by your cs server username):

      scp -r ~wx/cs251/sml/ps7 gdome@cs.wellesley.edu:/students/gdome/cs251/drop/ps07

  • For Problem 2 (Backus paper): Include English answers to all parts in your Google Doc.
  • For Problem 3:
    • Be sure that your deep-reverse definition in yourAccountName-ps6-deep-reverse.rkt also appears in your Google Doc.
    • Drop a copy of your yourAccountName-ps6-deep-reverse.rkt in your ~/cs251/drop/ps06 drop folder on cs.wellesley.edu.

Starting this Problem Set

Problems 1, 4, and 5 involve starter files in the ~wx/cs251/sml/ps7 directory in your wx Virtual Machine.

To create this directory, execute the following two commands in a wx VM shell:

cd ~/cs251/sml
git pull origin master

1. Solo Problem: Losing Your Marbles (15 points)

This is a solo problem. This means you must complete it entirely on your own without help from any other person and without consulting resources other than course materials or online documentation. You may ask Lyn for clarification, but not for help.

Put all your definitions for this problem in the starter file ~/cs251/sml/ps7/marbles.sml. When you finish this problem, rename the file to yourAccountName-ps7-marbles.sml before you submit it.

In this problem, you will define an SML function satisfying the following specification:

val marbles: int -> int -> int list list

Assume that m is a non-negative integer and that c is a positive integer. Given m marbles and a row of c cups, marbles m c returns a sorted list of all configurations whereby all m marbles are distributed among the c cups. Each configuration should be a list of length c whose elements are integers between 0 and m and the sum of whose elements is m. The returned list should be ordered lexicographically (i.e., in dictionary order).

At the end of this problem description are numerous sample invocations of the marbles function.

Your task is to define the marbles function in SML so that it satisfies the above specification and has the same behavior as the sample invocations below.

Notes:

• As usual, you should use divide/conquer/glue as your problem-solving strategy. Strive to make your solution as simple as possible. For example, do not use more base cases than are strictly necessary.

• Don’t forget the very powerful notion of “wishful” thinking, in which you blindly apply a recursive function to smaller versions of the same problem and combine their results. Study the examples carefully to see how the result of a call to marbles is stitched together from the results of calls to “smaller” versions of marbles.

• Your marbles function should generate the elements in sorted order without calling any kind of sorting function.

• You are expected to use higher-order list functions where they can simplify your definition.

• The stater file begins with the following helper functions, which you might find useful:

  fun cons x ys = x :: ys (* curried list consing operator *)
                        
  fun range lo hi = (* list all ints from lo up to (but not including) hi *)
    List.tabulate(hi - lo, fn i => i + lo)

  Reminder: these and other helper functions must be defined above your definition of marbles, because definition order matters in SML.

• Feel free to define any additional auxiliary functions you find helpful.

• To display elements beyond the default cutoff length for lists, use Control.Print.printLength := 100;

  - Control.Print.printLength := 100;
  val it = () : unit
  
  - marbles 0 1;
  val it = [[0]] : int list list
  
  - marbles 0 2;
  val it = [[0,0]] : int list list
  
  - marbles 0 3;
  val it = [[0,0,0]] : int list list
  
  - marbles 0 4;
  val it = [[0,0,0,0]] : int list list
  
  - marbles 1 1;
  val it = [[1]] : int list list
  
  - marbles 1 2;
  val it = [[0,1],[1,0]] : int list list
  
  - marbles 1 3;
  val it = [[0,0,1],[0,1,0],[1,0,0]] : int list list
  
  - marbles 1 4;
  val it = [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] : int list list
  
  - marbles 2 1;
  val it = [[2]] : int list list
  
  - marbles 2 2;
  val it = [[0,2],[1,1],[2,0]] : int list list
  
  - marbles 2 3;
  val it = [[0,0,2],[0,1,1],[0,2,0],[1,0,1],[1,1,0],[2,0,0]] : int list list
  
  - marbles 2 4;
  val it =
    [[0,0,0,2],[0,0,1,1],[0,0,2,0],[0,1,0,1],[0,1,1,0],[0,2,0,0],[1,0,0,1],
     [1,0,1,0],[1,1,0,0],[2,0,0,0]] : int list list
  
  - marbles 3 1;
  val it = [[3]] : int list list
  
  - marbles 3 2;
  val it = [[0,3],[1,2],[2,1],[3,0]] : int list list
  
  - marbles 3 3;
  val it =
    [[0,0,3],[0,1,2],[0,2,1],[0,3,0],[1,0,2],[1,1,1],[1,2,0],[2,0,1],[2,1,0],
     [3,0,0]] : int list list
  
  - marbles 3 4;
  val it =
    [[0,0,0,3],[0,0,1,2],[0,0,2,1],[0,0,3,0],[0,1,0,2],[0,1,1,1],[0,1,2,0],
     [0,2,0,1],[0,2,1,0],[0,3,0,0],[1,0,0,2],[1,0,1,1],[1,0,2,0],[1,1,0,1],
     [1,1,1,0],[1,2,0,0],[2,0,0,1],[2,0,1,0],[2,1,0,0],[3,0,0,0]]
    : int list list
  
  - marbles 4 1;
  val it = [[4]] : int list list
  
  - marbles 4 2;
  val it = [[0,4],[1,3],[2,2],[3,1],[4,0]] : int list list
  
  - marbles 4 3;
  val it =
    [[0,0,4],[0,1,3],[0,2,2],[0,3,1],[0,4,0],[1,0,3],[1,1,2],[1,2,1],[1,3,0],
     [2,0,2],[2,1,1],[2,2,0],[3,0,1],[3,1,0],[4,0,0]] : int list list
  
  - marbles 4 4;
  val it =
    [[0,0,0,4],[0,0,1,3],[0,0,2,2],[0,0,3,1],[0,0,4,0],[0,1,0,3],[0,1,1,2],
     [0,1,2,1],[0,1,3,0],[0,2,0,2],[0,2,1,1],[0,2,2,0],[0,3,0,1],[0,3,1,0],
     [0,4,0,0],[1,0,0,3],[1,0,1,2],[1,0,2,1],[1,0,3,0],[1,1,0,2],[1,1,1,1],
     [1,1,2,0],[1,2,0,1],[1,2,1,0],[1,3,0,0],[2,0,0,2],[2,0,1,1],[2,0,2,0],
     [2,1,0,1],[2,1,1,0],[2,2,0,0],[3,0,0,1],[3,0,1,0],[3,1,0,0],[4,0,0,0]]
    : int list list
  
  - marbles 5 1;
  val it = [[5]] : int list list
  
  - marbles 5 2;
  val it = [[0,5],[1,4],[2,3],[3,2],[4,1],[5,0]] : int list list
  
  - marbles 5 3;
  val it =
    [[0,0,5],[0,1,4],[0,2,3],[0,3,2],[0,4,1],[0,5,0],[1,0,4],[1,1,3],[1,2,2],
     [1,3,1],[1,4,0],[2,0,3],[2,1,2],[2,2,1],[2,3,0],[3,0,2],[3,1,1],[3,2,0],
     [4,0,1],[4,1,0],[5,0,0]] : int list list
  
  - marbles 5 4;
  val it =
    [[0,0,0,5],[0,0,1,4],[0,0,2,3],[0,0,3,2],[0,0,4,1],[0,0,5,0],[0,1,0,4],
     [0,1,1,3],[0,1,2,2],[0,1,3,1],[0,1,4,0],[0,2,0,3],[0,2,1,2],[0,2,2,1],
     [0,2,3,0],[0,3,0,2],[0,3,1,1],[0,3,2,0],[0,4,0,1],[0,4,1,0],[0,5,0,0],
     [1,0,0,4],[1,0,1,3],[1,0,2,2],[1,0,3,1],[1,0,4,0],[1,1,0,3],[1,1,1,2],
     [1,1,2,1],[1,1,3,0],[1,2,0,2],[1,2,1,1],[1,2,2,0],[1,3,0,1],[1,3,1,0],
     [1,4,0,0],[2,0,0,3],[2,0,1,2],[2,0,2,1],[2,0,3,0],[2,1,0,2],[2,1,1,1],
     [2,1,2,0],[2,2,0,1],[2,2,1,0],[2,3,0,0],[3,0,0,2],[3,0,1,1],[3,0,2,0],
     [3,1,0,1],[3,1,1,0],[3,2,0,0],[4,0,0,1],[4,0,1,0],[4,1,0,0],[5,0,0,0]]
    : int list list

2. Backus’s Paper (25 points)

This problem is about John Backus’s 1977 Turing Award Lecture: Can Programming be Liberated from the von Neumann Style? A Functional Style and its Algebra of Programs. His paper can be found here.

You should begin this problem by reading Sections 1–11 and 15–16 of this paper. (Although Sections 12–14 are very interesting, they require more time than I want you to spend on this problem.)

Section 11.2 introduces the details of the FP language. Backus uses many notations that may be unfamiliar to you. For example:

• p₁ → e₁; … ; p_n → e_n; e_n+1 is similar to the Racket expression (ifp₁e₁… (ifp_ne_ne_n+1) … ).

• ⟨e₁, …, e_n⟩ denotes the sequence of the n values of the expressions e₁, … e_n. φ denotes the empty sequence. Because FP is dynamically typed, such sequences can represent both tuples and lists from Python and OCaml.

• The symbol ⊥ (pronounced “bottom”) denotes the value of an expression that doesn’t terminate (i.e., it loops infinitely) or terminates with an error.

• If f is a function and x is an object (atom or sequence of objects), then f : x denotes the result of applying f to x.

• [f₁, …, f_n] is a functional form denoting a sequence of n functions, f₁ through f_n. The application rule for this functional form is [f₁, …, f_n] : x = ⟨f₁ : x, … , f_n : x⟩ — i.e., the result of applying a sequence of n functions to an object x is an n-element sequence consisting of the results of applying each of the functions in the function sequence to x.

Consult Lyn if you have trouble understanding Backus’s notation.

Answer the following questions about Backus’s paper. Your answers should be concise but informative.

1. (2 points) One of the reasons this paper is well-known is that in it Backus coined the term “von Neumann bottleneck”. Describe what this is and its relevance to the paper.

2. (2 points) Many programming languages have at least two syntactic categories: expressions and statements. Backus claims that expressions are good but statements are bad. Explain his claim.

3. (3 points) In Sections 6, 7, and 9 of the paper, Backus discusses three problems/defects with von Neumann languages. Summarize them.

4. (3 points) What are applicative languages and how do they address the three problems/defects mentioned by Backus for von Neumann languages?

5. (2 points) The FP language Backus introduces in Section 11 does not support abstraction expressions like Racket’s lambda. Why did Backus make this decision in FP?

6. (3 points) Backus wrote this paper long before the development of Java and Python. Based on his paper, how do you think he would evaluate these two languages?

7. (10 points) Consider the following FP definition:

   Def F ≡ α/+ ◦ αα× ◦ αdistl ◦ distr ◦ [id, id]

   What is the value of F⟨2, 3, 5⟩? Show the evaluation of this expression in a sequence of small-step algebra-like steps.

3. Deep Reverse (10 points)

We saw in lecture that tree recursion on trees represented as s-expressions could be expressed rather elegantly. For example:

(define (atom? x)
  (or (number? x) (boolean? x) (string? x) (symbol? x)))

(define (num-atoms sexp)
  (if (atom? sexp)
      1
      (foldr + 0 (map num-atoms sexp))))

> (num-atoms '((a (b c) d) e (((f) g h) i j k)))
11
> (num-atoms '(a b c d))
4
> (num-atoms 'a)
1
> (num-atoms '())
0

(define (list-atoms sexp)
  (if (atom? sexp)
      (list sexp)
      (foldr append null (map list-atoms sexp))))

> (list-atoms '((a (b c) d) e (((f) g h) i j k)))
'(a b c d e f g h i j k)
> (list-atoms '(a b c d))
'(a b c d)
> (list-atoms 'a)
'(a)
> (list-atoms '())
'()

In this problem, you will define a function (deep-reverse sexp) that returns a new s-expression in which the order of the children at every node of the s-expression tree sexp is reversed.

> (deep-reverse '((a (b c) d) e (((f) g h) i j k)))
'((k j i (h g (f))) e (d (c b) a))
> (deep-reverse '(a b c d))
'(d c b a)
> (deep-reverse 'a)
'a
> (deep-reverse '())
'()

Notes:

• Begin with this starter file ps7-deep-reverse-starter.rkt, which you should rename to yourAccountName-ps7-deep-reverse.rkt. Add your definition of deep-reverse to this file.

• Your definition should have form similar to the defintions for num-atoms and list-atoms, but you’ll want to use something other than foldr.

4. Fun Sets (22 points)

In SML, we can implement abstract data types in terms of familiar structures like lists and trees. But we can also use functions to implement data types. In this problem, you will investigate a compelling example of using functions to implement sets.

Your ~wx/cs251/sml/ps7 folder contains the starter file FunSet.sml. This includes the same SET signature we studied in class:

signature SET =
sig
    (* The type of sets *)
    type ''a t 

    (* An empty set *)
    val empty : ''a t

    (* Construct a single-element set from that element. *)
    val singleton : ''a -> ''a t

    (* Check if a set is empty. *)
    val isEmpty : ''a t -> bool

    (* Return the number of elements in the set. *)
    val size : ''a t -> int

    (* Check if a given element is a member of the given set. *)
    val member : ''a -> ''a t -> bool

    (* Construct a set containing the given element and all elements
       of the given set. *)
    val insert : ''a -> ''a t -> ''a t

    (* Construct a set containing all elements of the given set
       except for the given element. *)
    val delete : ''a -> ''a t -> ''a t

    (* Construct the union of two sets. *)
    val union : ''a t -> ''a t -> ''a t

    (* Construct the intersection of two sets. *)
    val intersection : ''a t -> ''a t -> ''a t

    (* Construct the difference of two sets
       (all elements in the first set but not in the second.) *)
    val difference : ''a t -> ''a t -> ''a t

    (* Construct a set from a list of elements.
       Do not assume the list elements are unique. *)
    val fromList : ''a list -> ''a t 

    (* Convert a set to a list without duplicates. 
       The elements in the resulting list may be in any order. *)
    val toList : ''a t -> ''a list

    (* Construct a set from a predicate function:
       the resulting set should contain all elements for which
       this predicate function returns true.

       This acts like math notation for sets.  For example:
         { x | x mod 3 = 0 }
       would be written:
         fromPred (fn x => x mod 3 = 0)
    *)
    val fromPred : (''a -> bool) -> ''a t

    (* Convert a set to a predicate function. *)
    val toPred : ''a t -> ''a -> bool

    (* Convert a set to a string representation, given a function
       that converts a set element into a string representation. *)
    val toString : (''a -> string) -> ''a t -> string

    (* Convert a set to a list without duplicates. 
       The elements in the resulting list may be in any order. *)
    val toList : ''a t -> ''a list

    (* Construct a set from a predicate function:
       the resulting set should contain all elements for which
       this predicate function returns true.

       This acts like math notation for sets.  For example:
         { x | x mod 3 = 0 }
       would be written:
         fromPred (fn x => x mod 3 = 0)
    *)
    val fromPred : (''a -> bool) -> ''a t

    (* Convert a set to a predicate function. *)
    val toPred : ''a t -> ''a -> bool

    (* Convert a set to a string representation, given a function
       that converts a set element into a string representation. *)
    val toString : (''a -> string) -> ''a t -> string

end

In this problem, you will flesh out the skeleton of the FunSet structure below. When you finish this probelm, you should rename this file to yourAccountName-FunSet.sml before you submit it.

exception Unimplemented (* Placeholder during development. *)
exception Unimplementable (* Impossible to implement *)

(* Implement a SET ADT using predicates to represent sets. *)
structure FunSet :> SET = struct

    (* Sets are represented by predicates. *)
    type ''a t = ''a -> bool 

    (* The empty set is a predicate that always returns false. *)
    val empty = fn _ => false

    (* The singleton set is a predicate that returns true for exactly one element *)
    fun singleton x = fn y => x=y

    (* Determining membership is trivial with a predicate *)
    fun member x pred = pred x

    (* complete this structure by replacing "raise Unimplemented"
       with implementations of each function. Many of the functions
       *cannot* be implemented; for those, use raise Unimplementable
       as there implementation *)
    fun isEmpty _ = raise Unimplemented
    fun size _ = raise Unimplemented
    fun member _ = raise Unimplemented
    fun insert _ = raise Unimplemented
    fun delete _ = raise Unimplemented
    fun union _ = raise Unimplemented
    fun intersection _ = raise Unimplemented
    fun difference _ = raise Unimplemented
    fun fromList _ = raise Unimplemented
    fun toList _ = raise Unimplemented
    fun fromPred _ = raise Unimplemented
    fun toPred _ = raise Unimplemented
    fun toString _ = raise Unimplemented

end

The fromPred and toPred operations are based on the observation that a membership predicate describes exactly which elements are in the set and which are not. Consider the following example:

- val s235 = fromPred (fn x => (x = 2) orelse (x = 3) orelse (x = 5));
val s235 = - : int t
- member 2 s235;
val it = true : bool
- member 3 s235;
val it = true : bool
- member 5 s235;
val it = true : bool
- member 4 s235;
val it = false : bool
- member 100 s235;
val it = false : bool

The set s235 consists of exactly those elements satisfying the predicate passed to fromPred – in this case, the integers 2, 3, and 5.

Defining sets in terms of predicates has many benefits. Most important, it is easy to specify sets that have infinite numbers of elements! For example, the set of all even integers can be expressed as:

fromPred (fn x => (x mod 2) = 0)

This predicate is true of even integers, but is false for all other integers. The set of all values of a given type is expressed as fromPred (fn x => true). Many large finite sets are also easy to specify. For example, the set of all integers between 251 and 6821 (inclusive) can be expressed as

fromPred (fn x => (x >= 251) && (x <= 6821))

Although defining sets in terms of membership predicates is elegant and has many benefits, it has some big downsides. The biggest one is that several functions in the SET signature simply cannot be implemented. You will explore this downside in this problem.

1. (18 points) Flesh out the code skeleton for the FunSet structure in FunSet.sml. Some value bindings cannot be implement; for these, use raise Unimplementable as the implementation.

   At the end of the starter file are the following commented out test cases. Uncomment these test cases to test your implementation. Feel free to add additional tests.

   open FunSet
   
   (* range helper function *)                           
   fun range lo hi = if lo >= hi then [] else lo::(range (lo + 1) hi)
   
   (* Test an int pred set on numbers from 0 to 100, inclusive *)
   fun intPredSetToList predSet = List.filter (toPred predSet) (range 0 101)
   
   val mod2Set = fromPred (fn x => x mod 2 = 0)
   val mod3Set = fromPred (fn x => x mod 3 = 0)
   val lowSet = fromList (range 0 61)
   val highSet = fromList (range 40 101)
   val smallSet = insert 17 (insert 19 (insert 23 (singleton 42)))
   val smallerSet = delete 23 (delete 19 (delete 57 smallSet))
   
   (* Be sure to print all details *)                      
   val _ = Control.Print.printLength := 101;                
   
   val smallSetTest = intPredSetToList(smallSet)
   val smallerSetTest = intPredSetToList(smallerSet)
   val mod3SetTest = intPredSetToList(mod3Set)
   val mod2SetUnionMod3SetTest = intPredSetToList(union mod2Set mod3Set)
   val mod2SetIntersectionMod3SetTest = intPredSetToList(intersection mod2Set mod3Set)
   val mod2SetDifferenceMod3SetTest = intPredSetToList(difference mod2Set mod3Set)
   val mod3SetDifferenceMod2SetTest = intPredSetToList(difference mod3Set mod2Set)
   val bigIntersection = intPredSetToList(intersection (intersection lowSet highSet)
                                                       (intersection mod2Set mod3Set))
   val bigDifference = intPredSetToList(difference (difference lowSet highSet)
                                                   (difference mod2Set mod3Set))

   For the test expressions generating lists, the results are as follows:

   val smallSetTest = [17,19,23,42] : int list
   val smallerSetTest = [17,42] : int list
   val mod3SetTest =
     [0,3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,
      78,81,84,87,90,93,96,99] : int list
   val mod2SetUnionMod3SetTest =
     [0,2,3,4,6,8,9,10,12,14,15,16,18,20,21,22,24,26,27,28,30,32,33,34,36,38,39,
      40,42,44,45,46,48,50,51,52,54,56,57,58,60,62,63,64,66,68,69,70,72,74,75,76,
      78,80,81,82,84,86,87,88,90,92,93,94,96,98,99,100] : int list
   val mod2SetIntersectionMod3SetTest =
     [0,6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96] : int list
   val mod2SetDifferenceMod3SetTest =
     [2,4,8,10,14,16,20,22,26,28,32,34,38,40,44,46,50,52,56,58,62,64,68,70,74,76,
      80,82,86,88,92,94,98,100] : int list
   val mod3SetDifferenceMod2SetTest =
     [3,9,15,21,27,33,39,45,51,57,63,69,75,81,87,93,99] : int list
   val bigIntersection = [42,48,54,60] : int list
   val bigDifference =
     [0,1,3,5,6,7,9,11,12,13,15,17,18,19,21,23,24,25,27,29,30,31,33,35,36,37,39]
     : int list

2. (4 points) For each function that you declared being unimplementable, explain in English why it is unimplementable. Give concrete examples where appropriate.

5. Operation Tree Sets (28 points)

A very different way of representing a set as a tree is to remember the structure of the set operations empty, insert, delete, union, intersection, and difference used to create the set. For example, consider the set t created as follows:

val s = (delete 4 (difference (union (union (insert 1 empty)
                                            (insert 4 empty))
                              (union (insert 7 empty)
                                     (insert 4 empty)))
                  (intersection (insert 1 empty)
                                (union (insert 1 empty)
                                       (insert 6 empty)))))

Abstractly, s is the singleton set {7}. But one concrete representation of s is the following operation tree:

One advantage of using such operation trees to represent sets is that the empty, insert, delete, union, difference, and intersection operations are extremely cheap — they just create a new tree node with the operands as subtrees, and thus take constant time and space! But other operations, such as member and toList, can be more expensive than in other implementations.

In this problem, you are asked to flesh out the missing operations in the skeleton of the OperationTreeSet structure shown below. Your ~wx/cs251/sml/ps7 folder contains the starter file OperationTreeSet.sml. When you finish this problem, rename the file to yourAccountName-OperationTreeSet.sml before you submit it.

structure OperationTreeSet :> SET = struct

    datatype ''a t = Empty
                   | Insert of ''a * ''a t
                   | Delete of ''a * ''a t
                   | Union of ''a t * ''a t
                   | Intersection of ''a t * ''a t
                   | Difference of ''a t * ''a t
                                                               
    val empty = Empty
    fun insert x s = Insert(x,s)
    fun singleton x = Insert(x, Empty)
    fun delete x s = Delete(x, s)
    fun union s1 s2 = Union(s1,s2)
    fun intersection s1 s2 = Intersection(s1,s2)
    fun difference s1 s2 = Difference(s1,s2)

    fun member _ _ = raise Unimplemented  
    fun toList _ = raise Unimplemented
    fun isEmpty _ = raise Unimplemented
    fun size _  = raise Unimplemented
    fun toPred _ = raise Unimplemented
    fun toString eltToString _ = raise Unimplemented
    fun fromList _ = raise Unimplemented

    fun fromPred _ = raise Unimplementable

end

In the OperationTreeSet structure, the set datatype t is create by constructors Empty, Insert, Delete, Union, Intersection, and Difference. The empty, singleton, insert, delete, union, intersection, difference operations are easy and have been implemented for you. You are responsible for fleshing out the definitions of the member, toList, size, isEmpty, toPred, toString, and fromList operations.

Notes:

• Your implementation of member should not use the toList function. Instead, it should be defined by case analysis on the structure of the operation tree.

• Your toList function should also be defined by case analysis on the structure of the operation tree. The member function should not be used in the implementation of toList (because it can be very inefficient). Keep in mind that sets are unordered, and your toList function may return lists of elements in any order, but it should not have any duplicates.

• In your toList definition, be very careful with including toList inside functional arguments, because this can often cause the same list to be calculated a tremendous number of times, leading to tests that take a very long time for large sets. If some of the test cases for large sets don’t return in a reasonable time, this is probably the cause.

• Your implementation of size and isEmpty should use the toList function. Indeed, it is difficult to implement these functions by a direct case analysis on the operation tree. Note that size will only work correcty if the output of toList does not contain duplicates.

• In the implementation of toString, the function String.concatWith is particularly useful.

• In the implementation of fromList, for lists with >= 2 elements, you should first split the list into two (nearly) equal-length sublists using List.take and List.drop and union the results of turning the sublists into sets. This yields a height-balanced operation tree.

• OperationTreeSet.sml ends with two collections of test cases. The first collection involves “small” sets that are defined without using fromList. The second collection involves “big” sets that are defined using fromList. The expected outputs to these tests can be found in this transcript file. Because of the way the testing functions are written, the integer elements in the outputs for member and toPred will be in sorted order from low to high, but the integers in the outputs for toList and toString may be in any order.

• The testing code for most functions (all except for member and assumes that toList works correctly. So you must implement toList for for most of the tests to work.