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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.
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, the optional CryptoMiniSat package must be installed, this can be accomplished by typing:
sage: install_package('cryptominisat') # not tested
and by running sage -b from the shell afterwards to build Sage’s CryptoMiniSat extension module.
Since by default Sage does not include any SAT solver, we demonstrate key features by instantiating a fake DIMACS-based solver. We start with a trivial example:
(x1 OR x2 OR x3) AND (x1 OR x2 OR (NOT x3))
In Sage’s notation:
sage: from sage.sat.solvers.dimacs import DIMACS
sage: solver = DIMACS(command="sat-solver")
sage: solver.add_clause( ( 1, 2, 3) )
sage: solver.add_clause( ( 1, 2, -3) )
Note
sage.sat.solvers.dimacs.DIMACS.add_clause() creates new variables when necessary. In particular, 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.
We demonstrate solving using CryptoMiniSat:
sage: from sage.sat.solvers import CryptoMiniSat # optional - cryptominisat
sage: cms = CryptoMiniSat() # optional - cryptominisat
sage: cms.add_clause((1,2,-3)) # optional - cryptominisat
sage: cms() # optional - cryptominisat
(None, True, True, False)
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
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
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 |
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