An ANF to CNF Converter using a Dense/Sparse Strategy¶

This converter is based on two converters. The first one, by Martin Albrecht, was based on [CB2007], this is the basis of the “dense” part of the converter. It was later improved by Mate Soos. The second one, by Michael Brickenstein, uses a reduced truth table based approach and forms the “sparse” part of the converter.

AUTHORS:

Martin Albrecht - (2008-09) initial version of ‘anf2cnf.py’
Michael Brickenstein - (2009) ‘cnf.py’ for PolyBoRi
Mate Soos - (2010) improved version of ‘anf2cnf.py’
Martin Albrecht - (2012) unified and added to Sage

Classes and Methods¶

class sage.sat.converters.polybori.CNFEncoder(solver, ring, max_vars_sparse=6, use_xor_clauses=None, cutting_number=6, random_seed=16)¶

Bases: sage.sat.converters.anf2cnf.ANF2CNFConverter

ANF to CNF Converter using a Dense/Sparse Strategy. This converter distinguishes two classes of polynomials.

1. Sparse polynomials are those with at most max_vars_sparse variables. Those are converted using reduced truth-tables based on PolyBoRi’s internal representation.

2. Polynomials with more variables are converted by introducing new variables for monomials and by converting these linearised polynomials.

Linearised polynomials are converted either by splitting XOR chains – into chunks of length cutting_number – or by constructing XOR clauses if the underlying solver supports it. This behaviour is disabled by passing use_xor_clauses=False.

__init__(solver, ring, max_vars_sparse=6, use_xor_clauses=None, cutting_number=6, random_seed=16)¶

Construct ANF to CNF converter over ring passing clauses to solver.

INPUT:

solver - a SAT-solver instance
ring - a sage.rings.polynomial.pbori.BooleanPolynomialRing
max_vars_sparse - maximum number of variables for direct conversion
use_xor_clauses - use XOR clauses; if None use if solver supports it. (default: None)
cutting_number - maximum length of XOR chains after splitting if XOR clauses are not supported (default: 6)
random_seed - the direct conversion method uses randomness, this sets the seed (default: 16)

EXAMPLES:

We compare the sparse and the dense strategies, sparse first:

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
-2 0
1 0
sage: e.phi
[None, a, b, c]

Now, we convert using the dense strategy:

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_dense(a*b + a + 1)
sage: _ = solver.write()
sage: print(open(fn).read())
p cnf 4 5
1 -4 0
2 -4 0
4 -1 -2 0
-4 -1 0
4 1 0
sage: e.phi
[None, a, b, c, a*b]

Note

This constructor generates SAT variables for each Boolean polynomial variable.

__call__(F)¶

Encode the boolean polynomials in F .

INPUT:

F - an iterable of sage.rings.polynomial.pbori.BooleanPolynomial

OUTPUT: An inverse map int -> variable

EXAMPLES:

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, max_vars_sparse=2)
sage: e([a*b + a + 1, a*b+ a + c])
[None, a, b, c, a*b]
sage: _ = solver.write()
sage: print(open(fn).read())
p cnf 4 9
-2 0
1 0
1 -4 0
2 -4 0
4 -1 -2 0
-4 -1 -3 0
4 1 -3 0
4 -1 3 0
-4 1 3 0

sage: e.phi
[None, a, b, c, a*b]

clauses(f)¶

Convert f using the sparse strategy if f.nvariables() is at most max_vars_sparse and the dense strategy otherwise.

INPUT:

f - a sage.rings.polynomial.pbori.BooleanPolynomial

EXAMPLES:

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, max_vars_sparse=2)
sage: e.clauses(a*b + a + 1)
sage: _ = solver.write()
sage: print(open(fn).read())
p cnf 3 2
-2 0
1 0
sage: e.phi
[None, a, b, c]

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, max_vars_sparse=2)
sage: e.clauses(a*b + a + c)
sage: _ = solver.write()
sage: print(open(fn).read())
p cnf 4 7
1 -4 0
2 -4 0
4 -1 -2 0
-4 -1 -3 0
4 1 -3 0
4 -1 3 0
-4 1 3 0

sage: e.phi
[None, a, b, c, a*b]

clauses_dense(f)¶

Convert f using the dense strategy.

INPUT:

f - a sage.rings.polynomial.pbori.BooleanPolynomial

EXAMPLES:

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_dense(a*b + a + 1)
sage: _ = solver.write()
sage: print(open(fn).read())
p cnf 4 5
1 -4 0
2 -4 0
4 -1 -2 0
-4 -1 0
4 1 0
sage: e.phi
[None, a, b, c, a*b]

clauses_sparse(f)¶

Convert f using the sparse strategy.

INPUT:

f - a sage.rings.polynomial.pbori.BooleanPolynomial

EXAMPLES:

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
-2 0
1 0
sage: e.phi
[None, a, b, c]

monomial(m)¶

Return SAT variable for m

INPUT:

m - a monomial.

OUTPUT: An index for a SAT variable corresponding to m.

EXAMPLES:

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_dense(a*b + a + 1)
sage: e.phi
[None, a, b, c, a*b]

If monomial is called on a new monomial, a new variable is created:

sage: e.monomial(a*b*c)
5
sage: e.phi
[None, a, b, c, a*b, a*b*c]

If monomial is called on a monomial that was queried before, the index of the old variable is returned and no new variable is created:

    sage: e.monomial(a*b)
    4
    sage: e.phi
    [None, a, b, c, a*b, a*b*c]

.. note::

    For correctness, this function is cached.

permutations(length, equal_zero)¶

Return permutations of length length which are equal to zero if equal_zero and equal to one otherwise.

A variable is false if the integer in its position is smaller than zero and true otherwise.

INPUT:

length – the number of variables
equal_zero – should the sum be equal to zero?

EXAMPLES:

sage: from sage.sat.converters.polybori import CNFEncoder
sage: from sage.sat.solvers.dimacs import DIMACS
sage: B.<a,b,c> = BooleanPolynomialRing()
sage: ce = CNFEncoder(DIMACS(), B)
sage: ce.permutations(3, True)
[[-1, -1, -1], [1, 1, -1], [1, -1, 1], [-1, 1, 1]]

sage: ce.permutations(3, False)
[[1, -1, -1], [-1, 1, -1], [-1, -1, 1], [1, 1, 1]]

phi¶

Map SAT variables to polynomial variables.

EXAMPLES:

sage: from sage.sat.converters.polybori import CNFEncoder
sage: from sage.sat.solvers.dimacs import DIMACS
sage: B.<a,b,c> = BooleanPolynomialRing()
sage: ce = CNFEncoder(DIMACS(), B)
sage: ce.var()
4
sage: ce.phi
[None, a, b, c, None]

split_xor(monomial_list, equal_zero)¶

Split XOR chains into subchains.

INPUT:

monomial_list - a list of monomials
equal_zero - is the constant coefficient zero?

EXAMPLES:

sage: from sage.sat.converters.polybori import CNFEncoder
sage: from sage.sat.solvers.dimacs import DIMACS
sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: ce = CNFEncoder(DIMACS(), B, cutting_number=3)
sage: ce.split_xor([1,2,3,4,5,6], False)
[[[1, 7], False], [[7, 2, 8], True], [[8, 3, 9], True], [[9, 4, 10], True], [[10, 5, 11], True], [[11, 6], True]]

sage: ce = CNFEncoder(DIMACS(), B, cutting_number=4)
sage: ce.split_xor([1,2,3,4,5,6], False)
[[[1, 2, 7], False], [[7, 3, 4, 8], True], [[8, 5, 6], True]]

sage: ce = CNFEncoder(DIMACS(), B, cutting_number=5)
sage: ce.split_xor([1,2,3,4,5,6], False)
[[[1, 2, 3, 7], False], [[7, 4, 5, 6], True]]

to_polynomial(c)¶

Convert clause to sage.rings.polynomial.pbori.BooleanPolynomial

INPUT:

c - a clause

EXAMPLES:

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, max_vars_sparse=2)
sage: _ = e([a*b + a + 1, a*b+ a + c])
sage: e.to_polynomial( (1,-2,3) )
a*b*c + a*b + b*c + b

var(m=None, decision=None)¶

Return a new variable.

This is a thin wrapper around the SAT-solvers function where we keep track of which SAT variable corresponds to which monomial.

INPUT:

m - something the new variables maps to, usually a monomial
decision - is this variable a decision variable?

EXAMPLES:

sage: from sage.sat.converters.polybori import CNFEncoder
sage: from sage.sat.solvers.dimacs import DIMACS
sage: B.<a,b,c> = BooleanPolynomialRing()
sage: ce = CNFEncoder(DIMACS(), B)
sage: ce.var()
4

zero_blocks(f)¶

Divide the zero set of f into blocks.

EXAMPLES:

sage: B.<a,b,c> = BooleanPolynomialRing()
sage: from sage.sat.converters.polybori import CNFEncoder
sage: from sage.sat.solvers.dimacs import DIMACS
sage: e = CNFEncoder(DIMACS(), B)
sage: sorted(sorted(d.items()) for d in e.zero_blocks(a*b*c))
[[(c, 0)], [(b, 0)], [(a, 0)]]

Note

This function is randomised.

An ANF to CNF Converter using a Dense/Sparse Strategy¶

Classes and Methods¶

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