# 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 an interface to the CryptoMiniSat [CMS] SAT solver which can be used interchangably with DIMACS-based solvers. 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.

## 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()
p cnf 3 2
-2 0
1 0


## 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)                                                # optional - cryptominisat
Polynomial Sequence with 36 Polynomials in 0 Variables


REFERENCES: