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)