def buchberger_improved(F):
"""
Compute a Groebner basis using an improved version of Buchberger's
algorithm as presented in [BW1993]_, page 232.
This variant uses the Gebauer-Moeller Installation to apply
Buchberger's first and second criterion to avoid useless pairs.
INPUT:
- ``F`` -- an ideal in a multivariate polynomial ring
OUTPUT: a Groebner basis for F
.. NOTE::
The verbosity of this function may be controlled with a
``set_verbose()`` call. Any value ``>=1`` will result in this
function printing intermediate Groebner bases.
EXAMPLES::
sage: from sage.rings.polynomial.toy_buchberger import buchberger_improved
sage: R.<x,y,z> = PolynomialRing(QQ)
sage: set_verbose(0)
sage: sorted(buchberger_improved(R.ideal([x^4 - y - z, x*y*z - 1])))
[x*y*z - 1, x^3 - y^2*z - y*z^2, y^3*z^2 + y^2*z^3 - x^2]
"""
F = inter_reduction(F.gens())
G = set()
B = set()
if get_verbose() >= 1:
reductions_to_zero = 0
while F:
f = min(F)
F.remove(f)
G, B = update(G, B, f)
while B:
g1, g2 = select(B)
B.remove((g1, g2))
h = spol(g1, g2).reduce(G)
if h != 0:
G, B = update(G, B, h)
if get_verbose() >= 1:
print("(%s, %s) => %s" % (g1, g2, h))
print("G: %s\n" % G)
if h == 0:
reductions_to_zero += 1
if get_verbose() >= 1:
print("%d reductions to zero." % reductions_to_zero)
return Sequence(inter_reduction(G))
def buchberger(F):
"""
Compute a Groebner basis using the original version of Buchberger's
algorithm as presented in [BW1993]_, page 214.
INPUT:
- ``F`` -- an ideal in a multivariate polynomial ring
OUTPUT: a Groebner basis for F
.. NOTE::
The verbosity of this function may be controlled with a
``set_verbose()`` call. Any value >=1 will result in this
function printing intermediate bases.
EXAMPLES::
sage: from sage.rings.polynomial.toy_buchberger import buchberger
sage: R.<x,y,z> = PolynomialRing(QQ)
sage: I = R.ideal([x^2 - z - 1, z^2 - y - 1, x*y^2 - x - 1])
sage: set_verbose(0)
sage: gb = buchberger(I)
sage: gb.is_groebner()
True
sage: gb.ideal() == I
True
"""
G = set(F.gens())
B = set((g1, g2) for g1 in G for g2 in G if g1 != g2)
if get_verbose() >= 1:
reductions_to_zero = 0
while B:
g1, g2 = select(B)
B.remove((g1, g2))
h = spol(g1, g2).reduce(G)
if h != 0:
B = B.union((g, h) for g in G)
G.add(h)
if get_verbose() >= 1:
print("(%s, %s) => %s" % (g1, g2, h))
print("G: %s\n" % G)
if h == 0:
reductions_to_zero += 1
if get_verbose() >= 1:
print("%d reductions to zero." % reductions_to_zero)
return Sequence(G)
def __call__(self, assumptions=None):
"""
Run 'command' and collect output.
INPUT:
- ``assumptions`` - ignored, accepted for compatibility with
other solvers (default: ``None``)
TESTS:
This class is not meant to be called directly::
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()
Traceback (most recent call last):
...
ValueError: No SAT solver command selected.
"""
from sage.misc.verbose import get_verbose
if assumptions is not None:
raise NotImplementedError("Assumptions are not supported for DIMACS based solvers.")
self.write()
output_filename = None
self._output = []
command = self._command.strip()
if not command:
raise ValueError("No SAT solver command selected.")
if "{output}" in command:
output_filename = tmp_filename()
command = command.format(input=self._headname, output=output_filename)
args = shlex.split(command)
try:
process = subprocess.Popen(args, stdout=subprocess.PIPE)
except OSError:
raise OSError("Could run '%s', perhaps you need to add your SAT solver to $PATH?"%(" ".join(args)))
try:
while process.poll() is None:
for line in iter(process.stdout.readline, b''):
if get_verbose() or self._verbosity:
print(line)
sys.stdout.flush()
self._output.append(line.decode('utf-8'))
sleep(0.1)
if output_filename:
self._output.extend(open(output_filename).readlines())
except BaseException:
process.kill()
raise