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This project will give you practice programming in OCaml by asking you to implement code for constructing, manipulating, and solving boolean formulae. You will then use your code to model binary numbers and solve a 3x3 magic square.

Here is the code skeleton for the project.

In this part of the project, you will develop some functions for working with boolean expressions, represented using the following type:

type bexpr =
    EFalse
  | ETrue
  | EVar of string
  | EAnd of bexpr * bexpr
  | EOr of bexpr * bexpr
  | ENot of bexpr
  | EForall of string * bexpr
  | EExists of string * bexpr

Here EFalse and ETrue represent the obvious values. EVar s represents the boolean variable with name s. The constructors EAnd, EOr, and ENot represent boolean conjunction, disjunction, and negation, respectively. For example, the expression (a or b) and c could be represented as And(Or(Var "a", Var "b"), Var "c"). (We'll explain Forall and Exists in a moment.)

We will use associative lists, which are just lists of pairs, to assign truth values to variables:

type asst = (string * bool) list

Here if an asst contains the pair (s, b), then that assignment gives the variable represented by the string s the value b. When working with the type asst, you will find the functions List.assoc, List.mem_assoc, and related functions helpful. See the OCaml standard library documentation for more details.

You may assume for purposes of this project that whenever you work with an asst, variables on the left override variables on the right. E.g., the assignment ["a", true; "a", false] assigns true to Var "a".

The last two expressions, EForall and EExists, represent the similarly named quantifiers. The expression EForall(x,e) is true if e is true when x is true and when x is false. Similarly, EExists(x,e) is true if e is true either when x is true or when x is false.

You can find the definitions of the above types in boolean.mli, along with type signatures for several functions you must implement in boolean.ml:

• eval : asst -> bexpr -> bool evaluates the expression on the given assignment and returns the result as an OCaml bool. For example, given assignment ["x", true; "y", false; "z", true] and expression EAnd(EOr(EVar "x", EVar "y"), EVar "z"), the eval function should return true. Your function can assume all of the expression's free variables are specified in the assignment.
• free_vars : bexpr -> string list returns a list of the free variables of the expression. The free variables are those that are not bound by a quantifier. For example, for EExists("x", EOr(EVar "x", EVar "y")), you should return a list containing only "y". The list should contain no duplicates. The free variables may appear in any order in the list.
• sat : bexpr -> asst option should return either Some a where a is any satisfying assignment for the expression or None if the expression is not satisfiable. For example, given EOr(EVar "x", EVar "y"), the sat function could return any of the following:
  • Some ["x", true; "y", false]
  • Some ["x", false; "y", true]
  • Some ["x", true; "y", true]
  In contrast, calling sat on EAnd(EVar "x", ENot(EVar "x")) should return None. If more than one assignment is possible, you may return any assignment. You might find the free_vars function handy here. Do not worry about the efficiency of sat.

However complex a boolean expression, it represents a single bit. In the next part of the project, you will put those bits together to represent integers using the following type:

type bvec = bexpr list (* low order bit at head of list *)

A bvec is a list of boolean expressions representing bits, with the low order bit at the head of the list. For example, we could represent the number 13 as [ETrue; EFalse; ETrue; ETrue]. Zero could be represented as [] or [EFalse] or [EFalse; EFalse], etc. An unknown 4-bit integer could be represented as [EVar "a"; EVar "b"; EVar "c"; EVar "d"].

Implement the following functions in bvec.ml:

• int_of_bvec : bvec -> int returns the OCaml integer corresponding to a vector consisting solely of ETrue and/or EFalse. For example, int_of_bvec [EFalse; ETrue] returns 2. You don't have to handle the case when some element of bvec is neither ETrue nor EFalse.
• bvec_of_int : int -> bvec is the inverse of int_of_bvec. (You may want to use mod and / in writing this function.)
• subst : asst -> bvec -> bvec applies the (possibly partial) assignment to the vector. For example, subst ["x", true] [EVar "x"; ETrue; EAnd(EVar "x", EVar "y")] should return [ETrue; ETrue; EAnd(ETrue, EVar "y")].
• zero : bvec -> bexpr returns a boolean expression that is true iff the vector is zero. For example zero [EVar "x"; EVar "y"] could evaluate to EAnd(ENot(EVar "x"), ENot(EVar "y")).
• bitand : bvec -> bvec -> bvec returns a new vector representing the bitwise and of the two vectors. You can assume for this problem that the two bit vectors have the same length.
• eq : bvec -> bvec -> bexpr returns an expression representing whether the two vectors are equal. Again you can assume the bit vectors have the same length.
• add : bvec -> bvec -> bvec returns a new vector representing the sum of the two vectors. You may assume the two vectors have the same length. Since there may be a carry out of the last bit, the resulting vector will have one more element than the input vectors. You'll have to figure out the expression for addition yourself. Hint: You'll probably want to write a recursive helper function that has an extra parameter for the carry bit.

If you've gotten this far, you've now built a (likely very slow) solver for part of bitvector arithmetic! Suppose we have:

let x = [EVar "a"; EVar "b"]
let y = [EVar "c"; EVar "d"]
let z = [EFalse; EFalse; ETrue]  (* 4 *)

Then we can use our solver to answer questions like:

sat (eq (add x y) z)  (* are there two, two-bit numbers that sum to 4? *)
sat (eq (add x x) z)  (* is there a two-bit number that, doubled, is 4? *)

etc. Cool!

A magic square is an n-by-n grid of the numbers 1 through n^2 such that every number appears exactly once and the sum of each row, column, and diagonal is the same. For example, here is a 3x3 magic square:

8	1	6
3	5	7
4	9	2

In this part, you will write a function that solves the magic squares problem, using the machinery you developed in part 1. In particular, what we will do is "reduce" the problem of finding a solution to the magic squares problem to a boolean formula. We can then use the sat function to actually find a solution. Here are the functions you should write; place them in magic.ml:

• pad : bvec -> int -> bvec. The function pad v i returns a new bvec that is equal to v but whose length is the greater of i and the length of v. I.e., it pads v, if necessary, with extra Falses so that it has i bits.
• add_three : bvec -> bvec -> bvec -> bvec. This function returns a new vector that represents the sum of the three input vectors. You can assume the three bvecs have the same length.
• is_digit : bvec -> bexpr. The input is a bvec of length 4 (i.e., exactly 4 boolean formulae). This function returns a formula that is true if and only if the bvec is greater than or equal to 1 and less than or equal to 9.
• disjoint : bvec list -> bexpr. This function takes a list of bvecs and returns a formula representing whether all the bvecs are different from each other.
• is_magic : bvec list -> bexpr. This function takes a list of exactly nine bvecs and returns a formula representing whether the list is a magic square. A list [v1; v2; v3; v4; v5; v6; v7; v8; v9] represents the following square:

  v1	v2	v3
  v4	v5	v6
  v7	v8	v9

  You can assume that the bvecs in the list consist of exactly four boolean formulae each, i.e., they are four-bit numbers.

You can try out your is_magic function by seeing if it can decide that the square above is magic. You can also try replacing one or two entries by variables, and then seeing if your code correctly finds the right numbers to fill in. However, you'll have to wait a long, long time if you try making a lot of the entries in the magic square variables, since your sat function probably takes exponential time.

Your next task is to implement a few functions related to Lambda calculus. Below is an interface file lambda.mli. You must implement the last six of the listed functions in lambda.ml. (We've already implemented unparse for you.)

    type lexpr =
    | Var of string
    | Lam of string * lexpr
    | App of lexpr * lexpr

    val unparse : lexpr -> string

    val free_vars : lexpr -> string list
    val subst : string -> lexpr -> lexpr -> lexpr
    val beta : lexpr -> lexpr option
    val normal_form : lexpr -> lexpr
    val lexpr_of_int : int -> lexpr
    val add : lexpr -> lexpr -> lexpr

These functions should do the following:

• free_vars e returns a list of the free variables in e, with no duplicates.
• subst x e1 e2 returns a copy of e2 in which free occurrences of x have been replaced by e1. Be sure to implement capture-avoiding substitution, i.e., bound variables should be renamed as needed to avoid capture free variables under a lambda. For example, subst "x" (Var "y") (Lam("y", Var "x")) should return something equivalent to Lam("z", Var "y") and not Lam("y", Var "y"). (Did you have to worry about this for boolean expressions? Why not?)
• beta e applies one step of call-by-value beta reduction to e, or returns None if no reduction is possible, according to the following rules:
  • Var x amd Lam (x, e) do not reduce (they return None).
  • App (e1, e2)
    • Returns App(e1', e2) if e1 reduces to e1'.
    • Returns App(e1, e2') if e1 cannot be reduced and e2 reduces to e2'.
    • Returns subst x e2 e if e1=Lam(x,e) and e2 cannot be reduced.
    • Returns None in any other cases.
• normal_form e keeps applying beta to e until the expression cannot be reduced any more, and then returns the result.
• lexpr_of_int n returns the Church numeral (see slide 19 at the link above) encoding of a non-negative OCaml integer.
• add e1 e2 returns the sum of two Church numerals (slide 22).

Your last task is to write unit tests for parts 1 and 2 above, using OUnit. Put your test cases in the file test.ml, and be sure to add your test cases to the suites suite_boolean, suite_bvec, suit_magic, and suite_lambda, as appropriate.

We will grade this part of the project by running your test cases against correct and incorrect implementations. More specifically, we will create a correct implementation cor and a set of incorrect implementations Incor. Your test cases should detect that the incorrect implementations are buggy, but not just by trivially saying all implementations are buggy. Thus, you will receive credit for detection an incorrect implementation i∈Incor if at least one of your test cases both (1) flags i as incorrect and (2) passes cor as correct.

Put all your code in the code skeleton we have supplied, and upload your solution using Gradescope. Details will be posted here once we figure out what those details are.