## problem definition in a nutshell

SAT is a NP-complete problem also called "satisfiability problem". It is most often presented in the form of a set of boolean clauses, each of those being a disjunction of literals. A literal is either a proposition or its negation. The problem goal is to find whether it exists a valuation for those boolean literals such that all clauses are true, and, in this case, to give such a valuation. In other words, can we give each proposition a truth value such that all clauses have at least one true literal?

A simple example is:

```a or (not b)
b or c
(not a) or (not c)
(not c)
```

which is satisfiable by the valuation `a=1, b=1, c=0`

## Dimacs syntax

There exists a simple, straighfoward standard format used to facilitate reading of a problem by a solver.

It is composed of:

• comments (lines beginning with `'c'`)
• a line defining the problem, with format `p <n> <n>`. The two numbers define the number of variables (propositions) and the number of clauses, respectively.
• a sequence of positive or negative numbers (not 0) that represent the clauses themselves. A negative number `-9` represents the literal `(not x9)` where `x9` is the ninth proposition. A positive number just represents a proposition. Each clause (a list of literals) is ended by a 0, which cannot be a proposition since there would be no way to tell `x0` from `(not x0)`.

Another example:

```c a very personal comment on this problem
c talking about clauses and literals
p 3 2
1 2 3 0
2 -3 0
```

It represents the logical problem

```x1 or x2 or x3
x2 or (not x3)
```

## problems

A list of a few problem (quite) simple on which to test a SAT-solver, in Dimacs format.

## 2018-2020: batsat

• I adapted some code from ratsat, itself a port of minisat.

The result is a SAT solver in rust, batsat with the ability to provide callbacks. I think these should be enough to be the basis for a CDCL(T) SMT solver.

## 2011: SAT solver in java

I have been working on a (relatively) efficient SAT-solver written in Java. It implements the DPLL algorithm, with the following features:

• two-watched literals for fast boolean propagation
• backjumping and clause learning with 1-UIP
• restarts
• research guided by literals activity (VSIDS)

Some lacking features are:

• better heuristics (notably for restarts)
• clause forgetting (garbage collecting learnt clauses)