# A Very Small SAT Solver

For those who don’t know, my co-supervisor (yes, that is the person who does the opposite of a supervisor) is Koen Claessen. Koen is important to me, he is the man who got me into research by picking me up in the second year of my B.Sc. at Chalmers. Actually, Ramona Enache also deserves a special thanks for her involvement in that. Anyway, that’s a story for a different time. Back to the topic of this post. For those who don’t know him, Koen likes SAT. Koen likes SAT a lot. So to honour Koen I’m going to show you how to build a truly tiny SAT solver in Haskell.

We are going to go through a few iterations of our design, gradually refining it to eventually end up with a one line 97 character marvel of an incomprehensible program. All the code for this blogpost is available here.

We begin by trying to understand the problem. In SAT we are given a propositional logic formula in Conjunctive Normal Form (CNF). CNF simply means that the formula is a conjunction of disjunctions. As an example, below is a CNF with variables x1, x2, and x3.
(x1 ∨ x2) ∧ (¬x2 ∨ x3)
We are asked to find an assignment to the variables of the formula which make the formula true. In the example above one such assignment is
x1 = ⊤, x2 = ⊥, x3 = ⊤
Where denotes `True` and `False`. If there is no such assignment, as is the case for the formula x1 ∧ ¬x1, we are asked to return `UNSAT`.

With an intuition for the problem formed we are ready to start to tackle the problem in Haskell. We start with some definitions. A Literal is either a variable, a positive Literal, or the negation of a variable, a negative Literal. One way to represent this in Haskell is with a fancy sum type:

``````type Variable = Int
data Literal  = Positive Variable | Negative Variable``````

However, we are going to go for something slightly simpler:

``type Literal = Int``

Where a positive integer `x` represents a positive Literal and `-x` the negation of the Literal. We disallow `0` as a Literal. A Clause is a disjunction of Literals, we represent this simply as a list of `Literal`s:

``type Clause = [Literal]``

Similarly, a Problem is a conjunction of Clauses, represented as a list of Clauses:

``type Problem = [Clause]``

Using this representation the Problem above can be coded as:

``````example :: Problem
example = [[1, 2], [-2, 3]]``````

With our encoding of problems complete we move on to solutions. A solution to the SAT problem is an Assignment of variables to truth values. Alternatively, we can think of an Assignment as a list of true literals, with the constraint that the same variable appears at most once in the list.

``type Assignment = [Literal]``

Our solution to SAT is going to involve a simple search procedure, we pick a literal and check if it should be true or false. We do this by propagating the choice of the value of a literal to the rest of the problem, reducing the problem to a simpler one:

``````propagate :: Literal -> Problem -> Problem
propagate l p = [ filter (/= negate l) c | c <- p, l `notElem` c ]``````

With all preliminaries in place we can finally move on to building our first SAT solver:

``````solve :: Problem -> Maybe Assignment
solve []        = Just []
solve ([]:p)    = Nothing
solve ((l:c):p) =
case solve (propagate l p) of
Just assignment -> Just (l:assignment)
Nothing         -> case solve (propagate (negate l) (c:p)) of
Just assignment -> Just (negate l:assignment)
Nothing         -> Nothing``````

The type of `solve` tells us that it will take a problem and either return `Just` a satisfying assignment, or it will return `Nothing`, indicating that the problem is `UNSAT`. The implementation is just a simple backtracking search where we propagate the choice of the literal to the rest of the problem and check if the smaller problem has a solution or not and act accordingly.

If we run our solver on the problem from above we get the following solution:

``````ghci> solve example
Just [1,-2]``````

## Multiple Solutions

With our initial solver completed we are ready to start golfing. The first thing we are going to do is to make our solver slightly more powerful. Instead of having it spit out only the first solution it finds, we are going to make it enumerate solutions. The new type of `solve` is the following:

``solve :: Problem -> [Assignment]``

Instead of returning a `Maybe Assignment` we now return a list of Assignments, where `[]` represents `UNSAT`. The first two cases are straightforward:

``````solve [] = [[]]
solve ([]:p) = []``````

We simply replace `Just []` with `[[]]` and `Nothing` with `[]`. The case which actually does the solving is also relatively straightforward:

``````solve ((l:c):p) =
[ l:assignment        | assignment <- solve (propagate l p) ]
++ [ negate l:assignment | assignment <- solve (propagate (negate l) (c:p)) ]``````

We simply concatenate all the solutions we get from considering both `l` and `negate l`.

If we run this solver on the example from above we get the following:

``````ghci> solve example
[[1,-2],[1,2,3],[-1,2,3]]``````

## Do-bious syntax

Ok, so this is ultimately about using Haskell to get as small a piece of code as possible to perform the SAT task. Since we are using Haskell we have access to a very powerful abstraction, Monads. In particular, we have access to the list monad. What does this mean? In effect this means that we have a special syntax to apply the `concatMap` function. Here is how it works, if I write the code that looks like this:

``````example :: [Int]
example = do
x <- [1..5]
replicate x x``````

The code should be read “for every number x in the range 1 to 5, produce x copies of x”. That is to say, the variable `x` is bound to the elements of `[1..5]` individually. It de-sugars into code that looks, basically, like this:

``````example :: [Int]
example = concatMap (\x -> replicate x x) [1..5]``````

Which evaluates to:

``````ghci> example
[1,2,2,3,3,3,4,4,4,4,5,5,5,5,5]``````

Having a convenient syntax for doing `concatMap` is nice, but what does it give us when we are trying to solve SAT? It turns out we can also use this syntax to do pattern-matching. The following code:

``````tail :: [Int] -> [Int]
tail xs = do
(y:ys) <- [xs]
ys``````

Which de-sugars into:

``````tail :: [Int] -> [Int]
tail xs = concatMap (\ys -> case ys of { y:ys -> ys; _ -> [] }) [xs]``````

This is to say, if the pattern on the left-hand-side of the binding (`y:ys <- ...`) doesn’t match then we default to the empty list `[]` in the function being mapped. We can use this trick to simplify our solver a little bit, getting rid of one of the pattern-matches in the definition:

``````solve :: Problem -> [Assignment]
solve []    = [[]]
solve (c:p) = do
(l:c) <- [c]
([l:as | as <- solve (propagate l p)] ++ [negate l:as | as <- solve (propagate (negate l) (c:p))])``````

This code can be further simplified by breaking the implicit backtracking from the recursive calls to solve out:

``````solve :: Problem -> [Assignment]
solve []    = [[]]
solve (c:p) = do
(l:c)  <- [c]
(l, p) <- [(l, p), (negate l, c:p)]
map (l:) solve (propagate l p)``````

We have broken the choice between and ¬ℓ out into another implicit call to `concatMap`, where the mapped function (the “continuation”) consists of the recursive call to `solve`. Finally we are ready for the 97 character version of our solver, all we need to do is to inline `propagate` and further inline some of the `do` notation.

``f=filter;s[]=[[]];s(c:p)=do(l:c)<-[c];(l,p)<-[(l,p),(-l,c:p)];(l:)<\$>s(f(/=0-l)<\$>f(notElem l)p)``

We have renamed `solve` to `s` in order to keep the number of characters to a minimum, but other than that there is no significant difference between this code and the one above, other than that we have used the inline version of `map`, `(<\$>)`, to reduce the number of necessary brackets to make everything parse correctly.