• Sep 3. Updated code skeleton to use free_vars instead of vars_of.
• Sep 2. Fixed arguments of eval to be in same order as in code skeleton.
• Aug 31. Replaced "formula" by "expr." You might need to download p1.tar.gz again.

This project will help refresh your memory on programming in OCaml by asking you to write two small modules and then develop test cases for them. You may find it helpful to review the OCaml slides from CMSC 330.

Here is the code skeleton for the project.

In this part of the project, you will develop some functions for working with boolean expressions and vectors of boolean expressions. We will use the following data type to represent boolean expressions:

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

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 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.

However complex a boolean expression, it represents a single bit. We can put those bits together to represent integers using the following type:

type bvec = expr 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"].

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 -> expr -> 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 : expr -> string list returns a list of the free varialbes 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 : expr -> 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.
• 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 -> expr 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 -> expr 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!

Your next task is to implement a few functions related to Lambda calculus, which you learned about in CMSC 330. 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 expr =
    | Var of string
    | Lam of string * expr
    | App of expr * expr

    val unparse : expr -> string

    val free_vars : expr -> string list
    val subst : string -> expr -> expr -> expr
    val beta : expr -> expr option
    val normal_form : expr -> expr
    val expr_of_int : int -> expr
    val add : expr -> expr -> expr

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.
• expr_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 and suite_lambda, as appropriate. We will grade this part of the project by running your test cases against various correct and incorrect implementations. A test case will only be counted as correct if it passes a correct implementation and fails at least one incorrect implementation.

Put all your code in the code skeleton we have supplied, and upload your solution to the submit server. You can either upload your solution via the web interface, or you can run java -jar submit.jar in the project directory. Your code can use any OCaml standard library functions that you like.

Important note: On GRACE you should use the version of OCaml installed at

/afs/glue.umd.edu/class/fall2016/cmsc/430/0101/public/bin

Please post any questions you have about this on Piazza. You are welcome to install OCaml on your own machine, though we can't provide much support for that. We do recommend using opam to manage the installation process if you choose to do that. You can use opam to install both OCaml and OUnit

The Campus Senate has adopted a policy asking students to include the following statement on each assignment in every course: "I pledge on my honor that I have not given or received any unauthorized assistance on this assignment." Consequently your program is requested to contain this pledge in a comment near the top.

Please carefully read the academic honesty section of the course syllabus. Any evidence of impermissible cooperation on projects, use of disallowed materials or resources, or unauthorized use of computer accounts, will be submitted to the Student Honor Council, which could result in an XF for the course, or suspension or expulsion from the University. Be sure you understand what you are and what you are not permitted to do in regards to academic integrity when it comes to project assignments. These policies apply to all students, and the Student Honor Council does not consider lack of knowledge of the policies to be a defense for violating them. Full information is found in the course syllabus---please review it at this time.