# Satisfiability of boolean formulas

The goal of this exercise is to solve the satisfiability problem for boolean formulas, which consists in determining whether a propositional formula is satisfiable, i.e. is true under some valuation.

Open a new file sat.ml.

# 1 Simple method

## 1.1 Formulas

We define a type

type var = int

for variables. A propositional formula is either

• a variable,
• a conjunction or a disjunction of formulas,
• a negation of a formula
• true or false.

Define a type formula for formulas (the constructors should be named Var, And, Or, Not, True and False).

## 1.2 Substitution

Define a substitution procedure subst of type var -> formula -> formula -> formula which replaces a given variable $$X$$ by a formula $$B$$ in a formula $$A$$. In mathematical notation, the resulting formula is often noted $$A[B/X]$$. In order to test whether you have a precise variable v, in your match you should not write

| Var v -> ...

(which will match any variable) but

| Var x when x = v -> ...

## 1.3 Free variables

Define a function free_var of type formula -> var which returns an arbitrary free variable of the formula (a variable which occurs in the formula). In the case where there is no such variable, the exception Not_found will be raised. [NB: if you would rather have a function of type formula -> var option instead of raising and exception, this is fine.]

## 1.4 Evaluation

Define a function eval of type formula -> bool which evaluates a closed formula (a formula without free variables).

## 1.5 Satisfiability

We admit that a formula $$A$$ with a free variable $$X$$ is satisfiable if and only if either $$A[\top/X]$$ or $$A[\bot/X]$$ is satisfiable: trying both cases is called splitting on $$X$$.

Define a function sat of type formula -> bool which determines whether a formula is satisfiable.

You can test it with

let () =
let x = Var 0 in
let x'= Not x in
let y = Var 1 in
let y'= Not y in
let a = And (And(Or (x,y), Or (x',y)), Or (x',y')) in
let b = And (And(Or (x,y), Or (x',y)), And(Or (x,y'), Or (x',y'))) in
assert (sat a);
assert (not (sat b))

and add your own tests! The assert function will raise an exception if its argument is false (try to assert false).

# 2 DPLL

In practice, the above procedure is very slow and could be improved. For instance, on the formula $X\land\lnot X\land X_1\land\ldots\land X_n$ it will try all the possible valuations for $$X_1,\ldots,X_n$$, whereas it is easy to see that it cannot be satisfied for $$X=\top$$ and $$X=\bot$$, independently of the values of the other variables.

We now implement the DPLL algorithm on which are based most modern SAT solvers. It works much faster in practice, although the problem is NP-complete. We admit that every boolean formula can be put into a conjunctive normal form (or cnf):

• a literal is either a variable or the negation of a variable,
• a clause is a disjunction of literals,
• a conjunctive normal form is a conjunction of clauses.

and we aim at determining the satisfiability of such formulas. They can be represented by the following types:

type var = int
type literal = bool * var (* false means negated *)
type clause = literal list
type cnf = clause list

Note that the cnf [] corresponds to true whereas the clause [] corresponds to false.

## 2.1 Operations on lists

We will be using standard operations on lists. They are part of the List module of the standard library, but we shall recode them here as an exercise.

Define a function list_mem of type 'a -> 'a list -> bool which determines whether an element is present in a list. Test it with

let () =
assert (list_mem 2 [1;2;3]);
assert (not (list_mem 5 [1;2;3]));
assert (not (list_mem 1 []))

Define a function list_map of type ('a -> 'b) -> 'a list -> 'b list which applies a function to every element of a list. Test it with

let () =
assert (list_map (fun x -> 2*x) [1;2;3] = [2;4;6]);
assert (list_map (fun _ -> ()) [] = [])

Define a function list_filter of type ('a -> bool) -> 'a list -> 'a list which, in a list, keeps only the elements satisfying a given predicate.

let () =
let even x = x mod 2 = 0 in
assert (list_filter even [1;2;3;4;6] = [2;4;6])

## 2.2 Substitution

Observe that, given a variable $$X$$ and a formula in cnf $A=C_1\land\ldots\land C_n$ where the $$C_i$$ are the clauses, a cnf for the formula $$A[\top/X]$$ can be obtained from $$A$$ by

• removing all the clauses $$C_i$$ containing $$X$$,
• removing $$\lnot X$$ from all the remaining clauses.

and similarly for $$A[\bot/X]$$. Can you see why?

Using the functions on lists coded above, implement a substitution function subst_cnf of type var -> bool -> cnf -> cnf which replaces a variable $$X$$ by either $$\top$$ or $$\bot$$ in a formula in cnf.

## 2.3 Simple satisfiability

Implement again the function dpll of type cnf -> bool, using splitting as in the first part, but noting that

• the cnf [] is true,
• a cnf containing the clause [] is false

(in particular, a cnf without free variables is in one of those two cases). This should allow to conclude that a formula is not satisfiable without giving a value to every variable. You can use the function List.hd of type 'a list -> 'a which returns the first element of a list (or raises Failure "hd" if the list is empty). We also recall that you can extract the first and second components of a pair with the functions fst and snd.

Test your function with the same example as previously:

let () =
let x = true, 0 in
let x'= false,0 in
let y = true, 1 in
let y'= false,1 in
let a = [[x;y]; [x';y]; [x';y']] in
let b = [[x;y]; [x';y]; [x;y']; [x';y']] in
assert (dpll a);
assert (not (dpll b))

The algorithm can be further improved by carefully choosing the variables on which we split. In particular, we should eliminate first the variables for which we will easily be able to conclude (at least one branch of the splitting).

## 2.4 Unitary clauses

A clause $$C_i$$ is unitary when it is reduced to a literal. If $$C_i=X$$, we know that $$X$$ must be true in order for the formula to be satisfied (the case $$X=\bot$$ will immediately fail), and similarly in the case where $$C_i=\lnot X$$.

Program a function unit of type cnf -> literal which finds a unitary clause in a formula (or raises Not_found if there is none).

Test it with

let () =
let x = true, 0 in
let y = true, 1 in
let y'= false,1 in
assert (unit [[x;y]; [x]; [y;y']] = x)

## 2.5 Pure literals

A literal $$X$$ (resp. $$\lnot X$$) is pure in a formula $$A$$ if $$\lnot X$$ (resp. $$X$$) does not occur in any clause of $$A$$ (i.e. the variable always occurs with the same polarity). What can we say of the satisfiability of a formula which contains a pure literal?

Program a function pure of type cnf -> literal which finds a pure literal in a formula (or raises Not_found if there is none).

## 2.6 The DPLL algorithm

Modify the function dpll so that it operates in the following way on a given cnf

• if the formula is empty then it returns true,
• if the formula contains the empty clause then it returns false,
• if the formula contains a unitary clause then it replaces the corresponding variable by the only possible value,
• if the formula contains a pure literal then it replaces the corresponding variable by the value which preserves satisfiability,
• otherwise, it splits on an arbitrary variable.

## 2.7 Optional: the MOM heuristics

In the last case, we can still (hope to) improve the algorithm by carefully choosing the variables on which we split. The MOM heuristics (for Maximum number of Occurences in the Minimum length) consists in choosing a variable which occurs most among clauses of minimal length.

Program a function mom of type cnf -> var which finds such a variable (or raises Not_found if the formula has no variable), and use it in dpll.

## 2.8 Optional: the Jeroslow-Wang heuristics

Given a clause $$C$$, we write $$|C|$$ for the number of literals in $$C$$. Given a formula $$A$$ and a literal $$L$$, we write $$C\in_L A$$ to indicate that $$C$$ is a clause of $$A$$ containing $$L$$ and define $\mathcal{J}(L)=\sum_{C\in_L A}\left(\frac12\right)^{|C|}$ The Jeroslow-Wang heuristics consists in splitting on the variable corresponding to a literal which maximizes $$\mathcal{J}$$.

Program a function jw of type cnf -> literal -> float which computes the above quantity.

Program a function jw_var of type cnf -> var which finds a variable according to the Jeroslow-Wang heuristics, and use it in dpll.

# 3 Testing

The CNF file format is a text format to encode SAT problems, of which you can find many examples on the net.

## 3.1 CNF files

A parser for the CNF file format can be implemented as follows:

(** Parse a CNF file. *)
let parse f : cnf =
let ic = open_in f in
let n = in_channel_length ic in
let s = Bytes.create n in
really_input ic s 0 n;
close_in ic;
Bytes.to_string s
in
let f = load_file f in
let f = String.map (function '\t' -> ' ' | c -> c) f in
let f = String.split_on_char '\n' f in
let f = List.map (String.split_on_char ' ') f in
let f = List.filter (function "c"::_ | "p"::_ -> false | _ -> true) f in
let f = List.flatten f in
let aux (a,c) = function
| "" -> (a,c)
| "0" -> (c::a,[])
| n ->
let n = int_of_string n in
let x = if n < 0 then (false,-n) else (true,n) in
(a,x::c)
in
fst (List.fold_left aux ([],[]) f)

You can then test the file file.cnf by something like

let () = assert (dpll (parse "file.cnf"))

It is advised that you compile your program in native mode, so that it is faster, by typing on a console

ocamlopt sat.ml -o sat

You can then run the program with

./sat

You can even pass a file name as argument by executing

./sat file.cnf

and the file name can be retrieved in OCaml with Sys.argv.(0) (see the documentation).

Here are some examples of satisfiable files:

and unsatisfiable ones:

## 3.2 A Sudoku solver

A Sudoku can be encoded as a 9×9 matrix cells of integers such that cells.(i).(j)$$=9$$ means that the cell $$(i,j)$$ is empty and $$0\leq$$cell.(i).(j)$$<9$$ means that the cell contains the number cells.(i).(j)$$+1$$. For instance, the game

 . . 8 6 5 . . 2 1 . 9 3 4 . 2 . 8 . 5 . . . . . 3 . . 3 8 . . 7 . 1 . 9 . 6 . . . . . 4 . 2 . 9 . 1 . . 7 3 . . 5 . . . . . 6 . 1 . 3 . 9 8 2 . 7 3 . . 2 6 1 . .

is encoded as

let simple_sudoku =
[|[|9;9;7;9;8;2;4;9;9|];
[|5;4;9;3;9;1;9;9;9|];
[|9;1;0;9;7;9;2;9;9|];
[|2;7;9;9;5;9;1;9;8|];
[|9;6;9;9;9;9;9;0;9|];
[|0;9;8;9;3;9;9;6;2|];
[|9;9;4;9;0;9;6;2;9|];
[|9;9;9;2;9;8;9;1;5|];
[|9;9;5;7;1;9;0;9;9|]|]

Our aim is now to encode such a game as a SAT problem. We define

let var i j n : var =
let i = i mod 9 in
let j = j mod 9 in
9*(9*i+j)+n

the function which returns a variable depending on $$0\leq i<9$$, $$0\leq j<9$$ and $$0\leq n<9$$. In a valuation, the value of var i j n is true means that the cell $$(i,j)$$ contains the value $$n$$. We now want to generate formulas which encode the fact that a valuation gives rise to a solution to a sudoku game.

Define a function sudoku of type int array array -> cnf, which takes a Sudoku game as above, and returns a cnf which is satisfiable if and only if the Sudoku game is. The function must generate formulas expressing that

• there is at least one number in each cell,
• each number occurs at most once in a row,
• each number occurs at most once in a column,
• each number occurs at most once in a square,
• the solution respect the game given in argument.

It can be shown that these conditions suffice (e.g. they imply that there is at most one number in a cell).

Test it with

let () =
assert (dpll (sudoku simple_sudoku))

Here is also an example of a game of medium difficulty:

let medium_sudoku =
[|[|9;1;9;7;4;3;9;9;9|];
[|9;9;9;5;9;9;9;9;7|];
[|9;0;9;9;9;9;9;9;8|];
[|1;9;9;9;9;9;9;8;2|];
[|9;6;4;2;9;7;5;1;9|];
[|7;8;9;9;9;9;9;9;6|];
[|3;9;9;9;9;9;9;5;9|];
[|2;9;9;9;9;1;9;9;9|];
[|9;9;9;6;5;0;9;3;9|]|]

a hard one:

let hard_sudoku =
[|[|9;9;9;9;4;9;8;9;9|];
[|8;7;9;9;9;9;9;4;9|];
[|9;3;4;6;9;9;2;9;9|];
[|9;9;3;1;9;9;7;6;9|];
[|9;9;9;0;9;3;9;9;9|];
[|9;6;5;9;9;8;0;9;9|];
[|9;9;7;9;9;0;5;2;9|];
[|9;1;9;9;9;9;9;7;0|];
[|9;9;6;9;2;9;9;9;9|]|]

and an unsolvable one:

let unsolvable_sudoku =
[|[|1;9;9;8;9;9;9;9;9|];
[|9;9;9;9;9;9;9;5;9|];
[|9;9;9;9;9;0;9;9;9|];
[|4;9;1;5;9;9;3;9;6|];
[|9;9;9;9;9;3;0;9;9|];
[|9;9;9;9;8;7;9;1;2|];
[|9;9;9;9;9;2;9;7;9|];
[|9;9;4;9;0;9;9;9;9|];
[|9;9;6;9;9;9;9;9;9|]|]

As usual, you can test those with

let () =
assert (dpll (sudoku medium_sudoku))

let () =
assert (dpll (sudoku hard_sudoku))

let () =
assert (not (dpll (sudoku unsolvable_sudoku)))

# 4 Putting in conjunctive normal form

Define a procedure cnf of type formula -> cnf which puts a formula into conjunctive normal form. It might be a good idea to first define a function of type bool -> formula -> cnf which puts a formula, or its negation (depending on the boolean argument), in cnf.