def check_activation(modal_implication, valuation):
"""
:param modal_implication: see active_modalities above
:param valuation: see active_modalities above
:return tuple representing modal implications (modal atom, classical atom) if modality is active.
otherwise, return False.
"""
assert len(
modal_implication.disjuncts
) <= 2, "The modal disjunction is not formed correctly: %s" % modal_implication
prop_atom, modal_atom = None, None
true_literals = valuation[0]
false_literals = valuation[1]
# given assertion there should be one prop atom and one modal atom
for disjunct in modal_implication.disjuncts:
if not is_complex(disjunct):
prop_atom = disjunct
elif my_isinstance(disjunct[0], Modality):
modal_atom = disjunct
# check valuation of prop_atom, to determine num_modal_atoms activity
# check if prop_atom is negated
if is_atomic(prop_atom) and str(prop_atom) not in true_literals:
# not negated
return modal_atom, prop_atom
if not is_atomic(prop_atom) and str(prop_atom[1]) not in false_literals:
return modal_atom, prop_atom
return False
def deactivate_modalities(active_modal_set, valuation):
"""
:param active_modal_set: set of active modal implications (modal atom, classical atom)
:return list of modal implications that remain active, after trying to deactivate by satisfying antecedent.
"""
active_modal = set()
s = Optimize()
# add current valuation to solver
s.add((And(set(Bool(lit) for lit in valuation[0]))))
s.add((And(set(Not(Bool(lit)) for lit in valuation[1]))))
for imp_modalities in active_modal_set:
s.add_soft(And(get_bool(imp_modalities[1])), 1)
s.check()
valuation = process_model(s.model())
for imp_modalities in active_modal_set:
prop_ante = imp_modalities[1]
if is_atomic(prop_ante) and (prop_ante not in valuation[0]):
active_modal.add(imp_modalities)
if not is_atomic(prop_ante) and prop_ante[1] not in valuation[1]:
active_modal.add(imp_modalities)
return active_modal
def get_bool(atom):
"""
:param atom: parsed classical atom, can be negated
:return z3 appropriate boolean value
"""
assert not is_complex(atom), "Error adding atom to sat solver: %s" % atom
# need to account for 'False' and 'True'
if is_atomic(atom):
if atom == TOP:
return True
elif atom == BOTTOM:
return False
else:
return Bool(str(atom))
elif str(atom[0]) == '~':
if atom[1] == TOP:
return False
elif atom[1] == BOTTOM:
return True
else:
return Not(Bool(str(atom[1])))
else: # error
sys.stderr.write('Error adding atom to sat solver: ' + str(atom) +
'\n')
raise SystemExit(1)
def get_tuple(atom):
""" if atom is negated or modal returns tuple representation
"""
if u.is_atomic(atom):
return atom
else:
return atom[0], get_tuple(atom[1])
def get_str(fml):
""" Returns string for unary operators in correct syntactic form
"""
if u.is_atomic(fml):
return str(fml)
else:
unary_connective = fml[0]
if repr(unary_connective) == '~':
return '~' + get_str(fml[1])
elif repr(unary_connective) == 'box':
return '[' + unary_connective.id + ']' + get_str(fml[1])
elif repr(unary_connective) == 'dia':
return '<' + unary_connective.id + '>' + get_str(fml[1])
def get_mc(fml, modal_context):
""" Returns a tuple, specifically the modal context (MC) and remaining expression.
Note, MC is a possibly empty sequence of universal box-like modal operators.
"""
if not u.is_atomic(fml):
connective = repr(fml[0])
if connective == 'box':
modal_context.append(fml[0])
return get_mc(fml[1], modal_context)
else:
return [modal_context, fml]
else:
return [modal_context, fml]
def wedge(self, other):
r"""
Compute the exterior product with another differential form.
INPUT:
- ``other``: another differential form
OUTPUT:
- the exterior product self/\\other.
EXAMPLES:
Exterior product of two 1-forms on a 3-dimensional manifold::
sage: m = Manifold(3, 'M')
sage: c_xyz.<x,y,z> = m.chart('x y z', 'coord_xyz')
sage: a = OneForm(m, 'A')
sage: a[:] = (x, y, z)
sage: b = OneForm(m, 'B')
sage: b[2] = z^2
sage: a.view() ; b.view()
A = x dx + y dy + z dz
B = z^2 dz
sage: h = a.wedge(b) ; h
2-form 'A/\B' on the 3-dimensional manifold 'M'
sage: h.view()
A/\B = x*z^2 dx/\dz + y*z^2 dy/\dz
sage: latex(h)
A\wedge B
sage: latex(h.view())
A\wedge B = x z^{2} \mathrm{d} x\wedge\mathrm{d} z + y z^{2} \mathrm{d} y\wedge\mathrm{d} z
The exterior product of two 1-forms is antisymmetric::
sage: (b.wedge(a)).view()
B/\A = -x*z^2 dx/\dz - y*z^2 dy/\dz
sage: a.wedge(b) == - b.wedge(a)
True
sage: (a.wedge(b) + b.wedge(a))[:] # for the skeptical mind
[0 0 0]
[0 0 0]
[0 0 0]
sage: a.wedge(a) == 0
True
The exterior product of a 2-form by a 1-form::
sage: c = OneForm(m, 'C')
sage: c[1] = y^3 ; c.view()
C = y^3 dy
sage: g = h.wedge(c) ; g
3-form 'A/\B/\C' on the 3-dimensional manifold 'M'
sage: g.view()
A/\B/\C = -x*y^3*z^2 dx/\dy/\dz
sage: g[:]
[[[0, 0, 0], [0, 0, -x*y^3*z^2], [0, x*y^3*z^2, 0]], [[0, 0, x*y^3*z^2], [0, 0, 0], [-x*y^3*z^2, 0, 0]], [[0, -x*y^3*z^2, 0], [x*y^3*z^2, 0, 0], [0, 0, 0]]]
The exterior product of a 2-form by a 1-form is symmetric::
sage: h.wedge(c) == c.wedge(h)
True
"""
from utilities import is_atomic
if not isinstance(other, DiffForm):
raise TypeError("The second argument for the exterior product " +
"must be a differential form.")
if other.rank == 0:
return other*self
if self.rank == 0:
return self*other
frame_name = self.common_frame(other)
if frame_name is None:
raise ValueError("No common frame for the exterior product.")
rank_r = self.rank + other.rank
cmp_s = self.components[frame_name]
cmp_o = other.components[frame_name]
cmp_r = CompFullyAntiSym(self.domain.frames[frame_name], rank_r)
for ind_s, val_s in cmp_s._comp.items():
for ind_o, val_o in cmp_o._comp.items():
ind_r = ind_s + ind_o
if len(ind_r) == len(set(ind_r)): # all indices are different
cmp_r[ind_r] += val_s * val_o
result = DiffForm(self.domain, rank_r)
result.components[frame_name] = cmp_r
if self.name is not None and other.name is not None:
sname = self.name
oname = other.name
if not is_atomic(sname):
sname = '(' + sname + ')'
if not is_atomic(oname):
oname = '(' + oname + ')'
result.name = sname + '/\\' + oname
if self.latex_name is not None and other.latex_name is not None:
slname = self.latex_name
olname = other.latex_name
if not is_atomic(slname):
slname = '(' + slname + ')'
if not is_atomic(olname):
olname = '(' + olname + ')'
result.latex_name = slname + r'\wedge ' + olname
return result
def view(self, frame_name=None, chart_name=None):
r"""
Displays the differential form in terms of its expansion onto a given
coframe.
The output is either text-formatted (console mode) or LaTeX-formatted
(notebook mode).
INPUT:
- ``frame_name`` -- (default: None) string containing the name of the
vector frame with respect to which the differential form is expanded;
if none is provided, the domain's default frame is assumed
- ``chart_name`` -- (default: None) string containing the name of the
chart with respect to which the components of the differential form in
the selected frame are expressed; if none is provided, the domain's
default chart is assumed
EXAMPLES:
Display of a 2-form on `\RR^3`::
sage: m = Manifold(3, 'R3', '\RR^3', start_index=1)
sage: c_cart.<x,y,z> = m.chart('x y z', 'cart')
sage: a = DiffForm(m, 2, 'A')
sage: a[1,2], a[2,3] = x*z, x^2+y^2
sage: a.view() # expansion on the manifold's default coframe (dx, dy, dz)
A = x*z dx/\dy + (x^2 + y^2) dy/\dz
sage: latex(a.view()) # output for the notebook
A = x z \mathrm{d} x\wedge\mathrm{d} y + \left( x^{2} + y^{2} \right) \mathrm{d} y\wedge\mathrm{d} z
Display in a coframe different from the default one::
sage: c_spher.<r,th,ph> = m.chart(r'r:[0,+oo) th:[0,pi]:\theta ph:[0,2*pi):\phi', 'spher') # new coordinates
sage: spher_to_cart = CoordChange(c_spher, c_cart, r*sin(th)*cos(ph), r*sin(th)*sin(ph), r*cos(th)) # the standard spherical coordinates
sage: cart_to_spher = spher_to_cart.set_inverse(sqrt(x^2+y^2+z^2), atan2(sqrt(x^2+y^2),z), atan2(y, x), check=False)
sage: a.view('spher_b') # expansion on the coframe (dr, dth, dph) with the coefficients expressed in terms of the default coordinates (x,y,z)
A = -sqrt(x^2 + y^2 + z^2)*sqrt(x^2 + y^2)*y dr/\dth + (x^3 + x*y^2 + x*z^2)*sqrt(x^2 + y^2) dth/\dph
sage: a.view('spher_b', 'spher') # expansion on the coframe (dr, dth, dph) with the coefficients expressed in terms of the spherical coordinates
A = -r^3*sin(ph)*sin(th)^2 dr/\dth + r^4*cos(ph)*sin(th)^2 dth/\dph
"""
from sage.misc.latex import latex
from utilities import is_atomic, FormattedExpansion
if frame_name is None:
frame_name = self.domain.def_frame.name
if chart_name is None:
chart_name = self.domain.def_chart.name
frame = self.domain.frames[frame_name]
coframe = frame.coframe
comp = self.comp(frame_name)
terms_txt = []
terms_latex = []
n_con = self.tensor_type[0]
for ind in comp.non_redundant_index_generator():
coef = comp[[ind]].expr(chart_name)
if coef != 0:
bases_txt = []
bases_latex = []
for k in range(self.rank):
bases_txt.append(coframe(ind[k]).name)
bases_latex.append(latex(coframe(ind[k])))
basis_term_txt = "/\\".join(bases_txt)
basis_term_latex = r"\wedge".join(bases_latex)
if coef == 1:
terms_txt.append(basis_term_txt)
terms_latex.append(basis_term_latex)
elif coef == -1:
terms_txt.append("-" + basis_term_txt)
terms_latex.append("-" + basis_term_latex)
else:
coef_txt = repr(coef)
coef_latex = latex(coef)
if is_atomic(coef_txt):
terms_txt.append(coef_txt + " " + basis_term_txt)
else:
terms_txt.append("(" + coef_txt + ") " +
basis_term_txt)
if is_atomic(coef_latex):
terms_latex.append(coef_latex + basis_term_latex)
else:
terms_latex.append(r"\left(" + coef_latex + r"\right)" +
basis_term_latex)
if terms_txt == []:
expansion_txt = "0"
else:
expansion_txt = terms_txt[0]
for term in terms_txt[1:]:
if term[0] == "-":
expansion_txt += " - " + term[1:]
else:
expansion_txt += " + " + term
if terms_latex == []:
expansion_latex = "0"
else:
expansion_latex = terms_latex[0]
for term in terms_latex[1:]:
if term[0] == "-":
expansion_latex += term
else:
expansion_latex += "+" + term
result = FormattedExpansion(self)
if self.name is None:
result.txt = expansion_txt
else:
result.txt = self.name + " = " + expansion_txt
if self.latex_name is None:
result.latex = expansion_latex
else:
result.latex = latex(self) + " = " + expansion_latex
return result
def wedge(self, other):
r"""
Compute the exterior product with another differential form.
INPUT:
- ``other``: another differential form
OUTPUT:
- the exterior product self/\\other.
EXAMPLES:
Exterior product of two 1-forms on a 3-dimensional manifold::
sage: m = Manifold(3, 'M')
sage: c_xyz = Chart(m, 'x y z', 'coord_xyz')
sage: a = OneForm(m, 'A')
sage: a[:] = (x, y, z)
sage: b = OneForm(m, 'B')
sage: b[2] = z^2
sage: a.show() ; b.show()
A = x dx + y dy + z dz
B = z^2 dz
sage: h = a.wedge(b) ; h
2-form 'A/\B' on the 3-dimensional manifold 'M'
sage: h.show()
A/\B = x*z^2 dx/\dz + y*z^2 dy/\dz
sage: latex(h)
A\wedge B
sage: latex(h.show())
A\wedge B = x z^{2} \mathrm{d} x\wedge\mathrm{d} z + y z^{2} \mathrm{d} y\wedge\mathrm{d} z
The exterior product of two 1-forms is antisymmetric::
sage: (b.wedge(a)).show()
B/\A = -x*z^2 dx/\dz - y*z^2 dy/\dz
sage: a.wedge(b) == - b.wedge(a)
True
sage: (a.wedge(b) + b.wedge(a))[:] # for the skeptical mind
[0 0 0]
[0 0 0]
[0 0 0]
sage: a.wedge(a) == 0
True
The exterior product of a 2-form by a 1-form::
sage: c = OneForm(m, 'C')
sage: c[1] = y^3 ; c.show()
C = y^3 dy
sage: g = h.wedge(c) ; g
3-form 'A/\B/\C' on the 3-dimensional manifold 'M'
sage: g.show()
A/\B/\C = -x*y^3*z^2 dx/\dy/\dz
sage: g[:]
[[[0, 0, 0], [0, 0, -x*y^3*z^2], [0, x*y^3*z^2, 0]], [[0, 0, x*y^3*z^2], [0, 0, 0], [-x*y^3*z^2, 0, 0]], [[0, -x*y^3*z^2, 0], [x*y^3*z^2, 0, 0], [0, 0, 0]]]
The exterior product of a 2-form by a 1-form is symmetric::
sage: h.wedge(c) == c.wedge(h)
True
"""
from utilities import is_atomic
if not isinstance(other, DiffForm):
raise TypeError("The second argument for the exterior product " +
"must be a differential form.")
if other.rank == 0:
return other * self
if self.rank == 0:
return self * other
frame_name = self.common_frame(other)
if frame_name is None:
raise ValueError("No common frame for the exterior product.")
rank_r = self.rank + other.rank
cmp_s = self.components[frame_name]
cmp_o = other.components[frame_name]
cmp_r = CompFullyAntiSym(self.manifold, rank_r, frame_name)
for ind_s, val_s in cmp_s._comp.items():
for ind_o, val_o in cmp_o._comp.items():
ind_r = ind_s + ind_o
if len(ind_r) == len(set(ind_r)): # all indices are different
cmp_r[ind_r] += val_s * val_o
result = DiffForm(self.manifold, rank_r)
result.components[frame_name] = cmp_r
if self.name is not None and other.name is not None:
sname = self.name
oname = other.name
if not is_atomic(sname):
sname = '(' + sname + ')'
if not is_atomic(oname):
oname = '(' + oname + ')'
result.name = sname + '/\\' + oname
if self.latex_name is not None and other.latex_name is not None:
slname = self.latex_name
olname = other.latex_name
if not is_atomic(slname):
slname = '(' + slname + ')'
if not is_atomic(olname):
olname = '(' + olname + ')'
result.latex_name = slname + r'\wedge ' + olname
return result
def show(self, frame_name=None, chart_name=None):
r"""
Displays the differential form in terms of its expansion onto a given
coframe.
The output is either text-formatted (console mode) or LaTeX-formatted
(notebook mode).
INPUT:
- ``frame_name`` -- (default: None) string containing the name of the
vector frame with respect to which the differential form is expanded;
if none is provided, the manifold's default frame is assumed
- ``chart_name`` -- (default: None) string containing the name of the
chart with respect to which the components of the differential form in
the selected frame are expressed; if none is provided, the manifold's
default chart is assumed
EXAMPLES:
Display of a 2-form on `\RR^3`::
sage: m = Manifold(3, 'R3', '\RR^3', start_index=1)
sage: c_cart = Chart(m, 'x y z', 'cart')
sage: a = DiffForm(m, 2, 'A')
sage: a[1,2], a[2,3] = x*z, x^2+y^2
sage: a.show() # expansion on the manifold's default coframe (dx, dy, dz)
A = x*z dx/\dy + (x^2 + y^2) dy/\dz
sage: latex(a.show()) # output for the notebook
A = x z \mathrm{d} x\wedge\mathrm{d} y + \left( x^{2} + y^{2} \right) \mathrm{d} y\wedge\mathrm{d} z
Display in a coframe different from the default one::
sage: c_spher = Chart(m, r'r:positive th:positive:\theta ph:\phi', 'spher') # new coordinates
sage: spher_to_cart = CoordChange(c_spher, c_cart, r*sin(th)*cos(ph), r*sin(th)*sin(ph), r*cos(th)) # the standard spherical coordinates
sage: cart_to_spher = spher_to_cart.set_inverse(sqrt(x^2+y^2+z^2), atan2(sqrt(x^2+y^2),z), atan2(y, x), check=False)
sage: a.show('spher_b') # expansion on the coframe (dr, dth, dph) with the coefficients expressed in terms of the default coordinates (x,y,z)
A = -sqrt(x^2 + y^2)*sqrt(x^2 + y^2 + z^2)*y dr/\dth + sqrt(x^2 + y^2)*(x^3 + x*y^2 + x*z^2) dth/\dph
sage: a.show('spher_b', 'spher') # expansion on the coframe (dr, dth, dph) with the coefficients expressed in terms of the spherical coordinates
A = -r^3*sin(ph)*sin(th)^2 dr/\dth + r^4*sin(th)^2*cos(ph) dth/\dph
"""
from sage.misc.latex import latex
from utilities import is_atomic, FormattedExpansion
if frame_name is None:
frame_name = self.manifold.def_frame.name
if chart_name is None:
chart_name = self.manifold.def_chart.name
frame = self.manifold.frames[frame_name]
coframe = frame.coframe
comp = self.comp(frame_name)
terms_txt = []
terms_latex = []
n_con = self.tensor_type[0]
for ind in comp.non_redundant_index_generator():
coef = comp[[ind]].expr(chart_name)
if coef != 0:
bases_txt = []
bases_latex = []
for k in range(self.rank):
bases_txt.append(coframe(ind[k]).name)
bases_latex.append(latex(coframe(ind[k])))
basis_term_txt = "/\\".join(bases_txt)
basis_term_latex = r"\wedge".join(bases_latex)
if coef == 1:
terms_txt.append(basis_term_txt)
terms_latex.append(basis_term_latex)
elif coef == -1:
terms_txt.append("-" + basis_term_txt)
terms_latex.append("-" + basis_term_latex)
else:
coef_txt = repr(coef)
coef_latex = latex(coef)
if is_atomic(coef_txt):
terms_txt.append(coef_txt + " " + basis_term_txt)
else:
terms_txt.append("(" + coef_txt + ") " +
basis_term_txt)
if is_atomic(coef_latex):
terms_latex.append(coef_latex + basis_term_latex)
else:
terms_latex.append(r"\left(" + coef_latex +
r"\right)" + basis_term_latex)
if terms_txt == []:
expansion_txt = "0"
else:
expansion_txt = terms_txt[0]
for term in terms_txt[1:]:
if term[0] == "-":
expansion_txt += " - " + term[1:]
else:
expansion_txt += " + " + term
if terms_latex == []:
expansion_latex = "0"
else:
expansion_latex = terms_latex[0]
for term in terms_latex[1:]:
if term[0] == "-":
expansion_latex += term
else:
expansion_latex += "+" + term
result = FormattedExpansion(self)
if self.name is None:
result.txt = expansion_txt
else:
result.txt = self.name + " = " + expansion_txt
if self.latex_name is None:
result.latex = expansion_latex
else:
result.latex = latex(self) + " = " + expansion_latex
return result