# Sat¶

Sage supports solving clauses in Conjunctive Normal Form (see Wikipedia article Conjunctive_normal_form), i.e., SAT solving, via an interface inspired by the usual DIMACS format used in SAT solving [SG09]. For example, to express that:

```
x1 OR x2 OR (NOT x3)
```

should be true, we write:

```
(1, 2, -3)
```

Warning

Variable indices **must** start at one.

## Solvers¶

By default, Sage solves SAT instances as an Integer Linear Program (see
`sage.numerical.mip`

), but any SAT solver supporting the DIMACS input
format is easily interfaced using the `sage.sat.solvers.dimacs.DIMACS`

blueprint. Sage ships with pre-written interfaces for *RSat* [RS] and *Glucose*
[GL]. Furthermore, Sage provides a C++ interface to the *CryptoMiniSat* [CMS]
SAT solver which can be used interchangably with DIMACS-based solvers, but also
provides advanced features. For this last solver, the optional CryptoMiniSat
package must be installed, this can be accomplished by typing the following in the
shell:

```
sage -i cryptominisat sagelib
```

We now show how to solve a simple SAT problem.

```
(x1 OR x2 OR x3) AND (x1 OR x2 OR (NOT x3))
```

In Sage’s notation:

```
sage: solver = SAT()
sage: solver.add_clause( ( 1, 2, 3) )
sage: solver.add_clause( ( 1, 2, -3) )
sage: solver() # random
(None, True, True, False)
```

Note

`add_clause()`

creates new variables
when necessary. When using CryptoMiniSat, it creates *all* variables up to
the given index. Hence, adding a literal involving the variable 1000 creates
up to 1000 internal variables.

DIMACS-base solvers can also be used to write DIMACS files:

```
sage: from sage.sat.solvers.dimacs import DIMACS
sage: fn = tmp_filename()
sage: solver = DIMACS(filename=fn)
sage: solver.add_clause( ( 1, 2, 3) )
sage: solver.add_clause( ( 1, 2, -3) )
sage: _ = solver.write()
sage: for line in open(fn).readlines():
....: print(line)
p cnf 3 2
1 2 3 0
1 2 -3 0
```

Alternatively, there is `sage.sat.solvers.dimacs.DIMACS.clauses()`

:

```
sage: from sage.sat.solvers.dimacs import DIMACS
sage: fn = tmp_filename()
sage: solver = DIMACS()
sage: solver.add_clause( ( 1, 2, 3) )
sage: solver.add_clause( ( 1, 2, -3) )
sage: solver.clauses(fn)
sage: for line in open(fn).readlines():
....: print(line)
p cnf 3 2
1 2 3 0
1 2 -3 0
```

These files can then be passed external SAT solvers.

### Details on Specific Solvers¶

## Converters¶

Sage supports conversion from Boolean polynomials (also known as Algebraic Normal Form) to Conjunctive Normal Form:

```
sage: B.<a,b,c> = BooleanPolynomialRing()
sage: from sage.sat.converters.polybori import CNFEncoder
sage: from sage.sat.solvers.dimacs import DIMACS
sage: fn = tmp_filename()
sage: solver = DIMACS(filename=fn)
sage: e = CNFEncoder(solver, B)
sage: e.clauses_sparse(a*b + a + 1)
sage: _ = solver.write()
sage: print(open(fn).read())
p cnf 3 2
1 0
-2 0
```

### Details on Specific Converterts¶

## Highlevel Interfaces¶

Sage provides various highlevel functions which make working with Boolean polynomials easier. We construct a very small-scale AES system of equations and pass it to a SAT solver:

```
sage: sr = mq.SR(1,1,1,4,gf2=True,polybori=True)
sage: F,s = sr.polynomial_system()
sage: from sage.sat.boolean_polynomials import solve as solve_sat # optional - cryptominisat
sage: s = solve_sat(F) # optional - cryptominisat
sage: F.subs(s[0]) # optional - cryptominisat
Polynomial Sequence with 36 Polynomials in 0 Variables
```

### Details on Specific Highlevel Interfaces¶

REFERENCES:

[RS] | http://reasoning.cs.ucla.edu/rsat/ |

[GL] | http://www.lri.fr/~simon/?page=glucose |

[CMS] | http://www.msoos.org/cryptominisat2/ |

[SG09] | http://www.satcompetition.org/2009/format-benchmarks2009.html |