# You should have received a copy of the GNU General Public License
# along with LinPy. If not, see <http://www.gnu.org/licenses/>.
-"""
-Polyhedral domains
-
-This module provides classes and functions to deal with polyhedral
-domains, i.e. unions of polyhedra.
-"""
-
import ast
import functools
import re
from fractions import Fraction
from . import islhelper
-from .islhelper import mainctx, libisl
-from .linexprs import Expression, Symbol, Rational
from .geometry import GeometricObject, Point, Vector
+from .islhelper import libisl
+from .linexprs import LinExpr, Symbol
__all__ = [
+ 'And',
'Domain',
- 'And', 'Or', 'Not',
+ 'Not',
+ 'Or',
]
@functools.total_ordering
class Domain(GeometricObject):
"""
- This class represents polyhedral domains, i.e. unions of polyhedra.
+ A domain is a union of polyhedra. Unlike polyhedra, domains allow exact
+ computation of union, subtraction and complementary operations.
+
+ A domain with a unique polyhedron is automatically subclassed as a
+ Polyhedron instance.
"""
__slots__ = (
def __new__(cls, *polyhedra):
"""
- Create and return a new domain from a string or a list of polyhedra.
+ Return a domain from a sequence of polyhedra.
+
+ >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+ >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
+ >>> dom = Domain(square1, square2)
+ >>> dom
+ Or(And(x <= 2, 0 <= x, y <= 2, 0 <= y),
+ And(x <= 3, 1 <= x, y <= 3, 1 <= y))
+
+ It is also possible to build domains from polyhedra using arithmetic
+ operators Domain.__or__(), Domain.__invert__() or functions Or() and
+ Not(), using one of the following instructions:
+
+ >>> dom = square1 | square2
+ >>> dom = Or(square1, square2)
+
+ Alternatively, a domain can be built from a string:
+
+ >>> dom = Domain('0 <= x <= 2, 0 <= y <= 2; 1 <= x <= 3, 1 <= y <= 3')
+
+ Finally, a domain can be built from a GeometricObject instance, calling
+ the GeometricObject.asdomain() method.
"""
from .polyhedra import Polyhedron
if len(polyhedra) == 1:
return argument.aspolyhedron()
else:
raise TypeError('argument must be a string '
- 'or a GeometricObject instance')
+ 'or a GeometricObject instance')
else:
for polyhedron in polyhedra:
if not isinstance(polyhedron, Polyhedron):
@property
def polyhedra(self):
"""
- The tuple of polyhedra which constitute the domain.
+ The tuple of polyhedra present in the domain.
"""
return self._polyhedra
@property
def symbols(self):
"""
- The tuple of symbols present in the domain equations.
+ The tuple of symbols present in the domain equations, sorted according
+ to Symbol.sortkey().
"""
return self._symbols
@property
def dimension(self):
"""
- The dimension of the domain, i.e. the number of symbols.
+ The dimension of the domain, i.e. the number of symbols present in it.
"""
return self._dimension
- def make_disjoint(self):
- """
- Return an equivalent domain, whose polyhedra are disjoint.
- """
- islset = self._toislset(self.polyhedra, self.symbols)
- islset = libisl.isl_set_make_disjoint(mainctx, islset)
- return self._fromislset(islset, self.symbols)
-
def isempty(self):
"""
- Return True if the domain is empty.
+ Return True if the domain is empty, that is, equal to Empty.
"""
islset = self._toislset(self.polyhedra, self.symbols)
empty = bool(libisl.isl_set_is_empty(islset))
def isuniverse(self):
"""
- Return True if the domain is universal, i.e. with no constraint.
+ Return True if the domain is universal, that is, equal to Universe.
"""
islset = self._toislset(self.polyhedra, self.symbols)
universe = bool(libisl.isl_set_plain_is_universe(islset))
def __eq__(self, other):
"""
- Return True if the two domains are equal.
+ Return True if two domains are equal.
"""
- symbols = self._xsymbols([self, other])
- islset1 = self._toislset(self.polyhedra, symbols)
- islset2 = other._toislset(other.polyhedra, symbols)
- equal = bool(libisl.isl_set_is_equal(islset1, islset2))
- libisl.isl_set_free(islset1)
- libisl.isl_set_free(islset2)
- return equal
+ if isinstance(other, Domain):
+ symbols = self._xsymbols([self, other])
+ islset1 = self._toislset(self.polyhedra, symbols)
+ islset2 = other._toislset(other.polyhedra, symbols)
+ equal = bool(libisl.isl_set_is_equal(islset1, islset2))
+ libisl.isl_set_free(islset1)
+ libisl.isl_set_free(islset2)
+ return equal
+ return NotImplemented
def isdisjoint(self, other):
"""
Return True if two domains have a null intersection.
"""
+ if not isinstance(other, Domain):
+ raise TypeError('other must be a Domain instance')
symbols = self._xsymbols([self, other])
islset1 = self._toislset(self.polyhedra, symbols)
islset2 = self._toislset(other.polyhedra, symbols)
"""
Report whether another domain contains the domain.
"""
- symbols = self._xsymbols([self, other])
- islset1 = self._toislset(self.polyhedra, symbols)
- islset2 = self._toislset(other.polyhedra, symbols)
- equal = bool(libisl.isl_set_is_subset(islset1, islset2))
- libisl.isl_set_free(islset1)
- libisl.isl_set_free(islset2)
- return equal
+ return self < other
def __le__(self, other):
- return self.issubset(other)
+ if isinstance(other, Domain):
+ symbols = self._xsymbols([self, other])
+ islset1 = self._toislset(self.polyhedra, symbols)
+ islset2 = self._toislset(other.polyhedra, symbols)
+ equal = bool(libisl.isl_set_is_subset(islset1, islset2))
+ libisl.isl_set_free(islset1)
+ libisl.isl_set_free(islset2)
+ return equal
+ return NotImplemented
__le__.__doc__ = issubset.__doc__
def __lt__(self, other):
"""
Report whether another domain is contained within the domain.
"""
- symbols = self._xsymbols([self, other])
- islset1 = self._toislset(self.polyhedra, symbols)
- islset2 = self._toislset(other.polyhedra, symbols)
- equal = bool(libisl.isl_set_is_strict_subset(islset1, islset2))
- libisl.isl_set_free(islset1)
- libisl.isl_set_free(islset2)
- return equal
+ if isinstance(other, Domain):
+ symbols = self._xsymbols([self, other])
+ islset1 = self._toislset(self.polyhedra, symbols)
+ islset2 = self._toislset(other.polyhedra, symbols)
+ equal = bool(libisl.isl_set_is_strict_subset(islset1, islset2))
+ libisl.isl_set_free(islset1)
+ libisl.isl_set_free(islset2)
+ return equal
+ return NotImplemented
def complement(self):
"""
return self.complement()
__invert__.__doc__ = complement.__doc__
+ def make_disjoint(self):
+ """
+ Return an equivalent domain, whose polyhedra are disjoint.
+ """
+ islset = self._toislset(self.polyhedra, self.symbols)
+ islset = libisl.isl_set_make_disjoint(islset)
+ return self._fromislset(islset, self.symbols)
+
def coalesce(self):
"""
Simplify the representation of the domain by trying to combine pairs of
- polyhedra into a single polyhedron.
+ polyhedra into a single polyhedron, and return the resulting domain.
"""
islset = self._toislset(self.polyhedra, self.symbols)
islset = libisl.isl_set_coalesce(islset)
def detect_equalities(self):
"""
Simplify the representation of the domain by detecting implicit
- equalities.
+ equalities, and return the resulting domain.
"""
islset = self._toislset(self.polyhedra, self.symbols)
islset = libisl.isl_set_detect_equalities(islset)
def remove_redundancies(self):
"""
- Remove redundant constraints in the domain.
+ Remove redundant constraints in the domain, and return the resulting
+ domain.
"""
islset = self._toislset(self.polyhedra, self.symbols)
islset = libisl.isl_set_remove_redundancies(islset)
return self._fromislset(islset, self.symbols)
def aspolyhedron(self):
- """
- Return the polyhedral hull of the domain.
- """
from .polyhedra import Polyhedron
islset = self._toislset(self.polyhedra, self.symbols)
islbset = libisl.isl_set_polyhedral_hull(islset)
def project(self, symbols):
"""
- Project out the symbols given in arguments.
+ Project out the sequence of symbols given in arguments, and return the
+ resulting domain.
"""
+ symbols = list(symbols)
+ for symbol in symbols:
+ if not isinstance(symbol, Symbol):
+ raise TypeError('symbols must be Symbol instances')
islset = self._toislset(self.polyhedra, self.symbols)
n = 0
for index, symbol in reversed(list(enumerate(self.symbols))):
if symbol in symbols:
n += 1
elif n > 0:
- islset = libisl.isl_set_project_out(islset,
- libisl.isl_dim_set, index + 1, n)
+ islset = libisl.isl_set_project_out(
+ islset, libisl.isl_dim_set, index + 1, n)
n = 0
if n > 0:
- islset = libisl.isl_set_project_out(islset, libisl.isl_dim_set, 0, n)
+ islset = libisl.isl_set_project_out(
+ islset, libisl.isl_dim_set, 0, n)
symbols = [symbol for symbol in self.symbols if symbol not in symbols]
return Domain._fromislset(islset, symbols)
def sample(self):
"""
- Return a sample of the domain.
+ Return a sample of the domain, as an integer instance of Point. If the
+ domain is empty, a ValueError exception is raised.
"""
islset = self._toislset(self.polyhedra, self.symbols)
islpoint = libisl.isl_set_sample_point(islset)
raise ValueError('domain must be non-empty')
point = {}
for index, symbol in enumerate(self.symbols):
- coordinate = libisl.isl_point_get_coordinate_val(islpoint,
- libisl.isl_dim_set, index)
+ coordinate = libisl.isl_point_get_coordinate_val(
+ islpoint, libisl.isl_dim_set, index)
coordinate = islhelper.isl_val_to_int(coordinate)
point[symbol] = coordinate
libisl.isl_point_free(islpoint)
def intersection(self, *others):
"""
- Return the intersection of two domains as a new domain.
+ Return the intersection of two or more domains as a new domain. As an
+ alternative, function And() can be used.
"""
- if len(others) == 0:
- return self
- symbols = self._xsymbols((self,) + others)
- islset1 = self._toislset(self.polyhedra, symbols)
+ result = self
for other in others:
- islset2 = other._toislset(other.polyhedra, symbols)
- islset1 = libisl.isl_set_intersect(islset1, islset2)
- return self._fromislset(islset1, symbols)
+ result &= other
+ return result
def __and__(self, other):
- return self.intersection(other)
+ if isinstance(other, Domain):
+ symbols = self._xsymbols([self, other])
+ islset1 = self._toislset(self.polyhedra, symbols)
+ islset2 = other._toislset(other.polyhedra, symbols)
+ islset = libisl.isl_set_intersect(islset1, islset2)
+ return self._fromislset(islset, symbols)
+ return NotImplemented
__and__.__doc__ = intersection.__doc__
def union(self, *others):
"""
- Return the union of two domains as a new domain.
+ Return the union of two or more domains as a new domain. As an
+ alternative, function Or() can be used.
"""
- if len(others) == 0:
- return self
- symbols = self._xsymbols((self,) + others)
- islset1 = self._toislset(self.polyhedra, symbols)
+ result = self
for other in others:
- islset2 = other._toislset(other.polyhedra, symbols)
- islset1 = libisl.isl_set_union(islset1, islset2)
- return self._fromislset(islset1, symbols)
+ result |= other
+ return result
def __or__(self, other):
- return self.union(other)
+ if isinstance(other, Domain):
+ symbols = self._xsymbols([self, other])
+ islset1 = self._toislset(self.polyhedra, symbols)
+ islset2 = other._toislset(other.polyhedra, symbols)
+ islset = libisl.isl_set_union(islset1, islset2)
+ return self._fromislset(islset, symbols)
+ return NotImplemented
__or__.__doc__ = union.__doc__
def __add__(self, other):
- return self.union(other)
+ return self | other
__add__.__doc__ = union.__doc__
def difference(self, other):
"""
Return the difference of two domains as a new domain.
"""
- symbols = self._xsymbols([self, other])
- islset1 = self._toislset(self.polyhedra, symbols)
- islset2 = other._toislset(other.polyhedra, symbols)
- islset = libisl.isl_set_subtract(islset1, islset2)
- return self._fromislset(islset, symbols)
+ return self - other
def __sub__(self, other):
- return self.difference(other)
+ if isinstance(other, Domain):
+ symbols = self._xsymbols([self, other])
+ islset1 = self._toislset(self.polyhedra, symbols)
+ islset2 = other._toislset(other.polyhedra, symbols)
+ islset = libisl.isl_set_subtract(islset1, islset2)
+ return self._fromislset(islset, symbols)
+ return NotImplemented
__sub__.__doc__ = difference.__doc__
def lexmin(self):
islset = libisl.isl_set_lexmax(islset)
return self._fromislset(islset, self.symbols)
- _RE_COORDINATE = re.compile(r'\((?P<num>\-?\d+)\)(/(?P<den>\d+))?')
+ if islhelper.isl_version >= '0.13':
+ _RE_COORDINATE = re.compile(r'\((?P<num>\-?\d+)\)(/(?P<den>\d+))?')
+ else:
+ _RE_COORDINATE = None
def vertices(self):
"""
- Return the vertices of the domain.
+ Return the vertices of the domain, as a list of rational instances of
+ Point.
"""
- from .polyhedra import Polyhedron
if not self.isbounded():
raise ValueError('domain must be bounded')
islbset = self._toislbasicset(self.equalities, self.inequalities,
- self.symbols)
- vertices = libisl.isl_basic_set_compute_vertices(islbset);
+ self.symbols)
+ vertices = libisl.isl_basic_set_compute_vertices(islbset)
vertices = islhelper.isl_vertices_vertices(vertices)
points = []
for vertex in vertices:
- expr = libisl.isl_vertex_get_expr(vertex)
+ expression = libisl.isl_vertex_get_expr(vertex)
coordinates = []
- if islhelper.isl_version < '0.13':
- constraints = islhelper.isl_basic_set_constraints(expr)
+ if self._RE_COORDINATE is None:
+ constraints = islhelper.isl_basic_set_constraints(expression)
for constraint in constraints:
- constant = libisl.isl_constraint_get_constant_val(constraint)
+ constant = libisl.isl_constraint_get_constant_val(
+ constraint)
constant = islhelper.isl_val_to_int(constant)
for index, symbol in enumerate(self.symbols):
- coefficient = libisl.isl_constraint_get_coefficient_val(constraint,
- libisl.isl_dim_set, index)
+ coefficient = \
+ libisl.isl_constraint_get_coefficient_val(
+ constraint, libisl.isl_dim_set, index)
coefficient = islhelper.isl_val_to_int(coefficient)
if coefficient != 0:
coordinate = -Fraction(constant, coefficient)
coordinates.append((symbol, coordinate))
else:
- string = islhelper.isl_multi_aff_to_str(expr)
+ string = islhelper.isl_multi_aff_to_str(expression)
matches = self._RE_COORDINATE.finditer(string)
for symbol, match in zip(self.symbols, matches):
numerator = int(match.group('num'))
denominator = match.group('den')
- denominator = 1 if denominator is None else int(denominator)
+ denominator = \
+ 1 if denominator is None else int(denominator)
coordinate = Fraction(numerator, denominator)
coordinates.append((symbol, coordinate))
points.append(Point(coordinates))
def points(self):
"""
- Return the points with integer coordinates contained in the domain.
+ Return the integer points of a bounded domain, as a list of integer
+ instances of Point. If the domain is not bounded, a ValueError
+ exception is raised.
"""
if not self.isbounded():
raise ValueError('domain must be bounded')
- from .polyhedra import Universe, Eq
islset = self._toislset(self.polyhedra, self.symbols)
islpoints = islhelper.isl_set_points(islset)
points = []
for islpoint in islpoints:
coordinates = {}
for index, symbol in enumerate(self.symbols):
- coordinate = libisl.isl_point_get_coordinate_val(islpoint,
- libisl.isl_dim_set, index)
+ coordinate = libisl.isl_point_get_coordinate_val(
+ islpoint, libisl.isl_dim_set, index)
coordinate = islhelper.isl_val_to_int(coordinate)
coordinates[symbol] = coordinate
points.append(Point(coordinates))
return points
+ def __contains__(self, point):
+ """
+ Return True if the point if contained within the domain.
+ """
+ for polyhedron in self.polyhedra:
+ if point in polyhedron:
+ return True
+ return False
+
@classmethod
def _polygon_inner_point(cls, points):
symbols = points[0].symbols
def faces(self):
"""
- Return the vertices of the domain, grouped by face.
+ Return the list of faces of a bounded domain. Each face is represented
+ by a list of vertices, in the form of rational instances of Point. If
+ the domain is not bounded, a ValueError exception is raised.
"""
faces = []
for polyhedron in self.polyhedra:
def plot(self, plot=None, **kwargs):
"""
- Plot the domain using matplotlib.
+ Plot a 2D or 3D domain using matplotlib. Draw it to the current plot
+ object if present, otherwise create a new one. options are keyword
+ arguments passed to the matplotlib drawing functions, they can be used
+ to set the drawing color for example. Raise ValueError is the domain is
+ not 2D or 3D.
"""
if not self.isbounded():
raise ValueError('domain must be bounded')
elif self.dimension == 3:
return self._plot_3d(plot=plot, **kwargs)
else:
- raise ValueError('polyhedron must be 2 or 3-dimensional')
-
- def __contains__(self, point):
- """
- Return True if point if contained within the domain.
- """
- for polyhedron in self.polyhedra:
- if point in polyhedron:
- return True
- return False
+ raise ValueError('domain must be two or three-dimensional')
def subs(self, symbol, expression=None):
"""
- Subsitute symbol by expression in equations and return the resulting
- domain.
+ Substitute the given symbol by an expression in the domain constraints.
+ To perform multiple substitutions at once, pass a sequence or a
+ dictionary of (old, new) pairs to subs. The syntax of this function is
+ similar to LinExpr.subs().
"""
polyhedra = [polyhedron.subs(symbol, expression)
- for polyhedron in self.polyhedra]
+ for polyhedron in self.polyhedra]
return Domain(*polyhedra)
@classmethod
@classmethod
def _toislset(cls, polyhedra, symbols):
polyhedron = polyhedra[0]
- islbset = polyhedron._toislbasicset(polyhedron.equalities,
- polyhedron.inequalities, symbols)
+ islbset = polyhedron._toislbasicset(
+ polyhedron.equalities, polyhedron.inequalities, symbols)
islset1 = libisl.isl_set_from_basic_set(islbset)
for polyhedron in polyhedra[1:]:
- islbset = polyhedron._toislbasicset(polyhedron.equalities,
- polyhedron.inequalities, symbols)
+ islbset = polyhedron._toislbasicset(
+ polyhedron.equalities, polyhedron.inequalities, symbols)
islset2 = libisl.isl_set_from_basic_set(islbset)
islset1 = libisl.isl_set_union(islset1, islset2)
return islset1
elif isinstance(node, ast.Compare):
equalities = []
inequalities = []
- left = Expression._fromast(node.left)
+ left = LinExpr._fromast(node.left)
for i in range(len(node.ops)):
op = node.ops[i]
- right = Expression._fromast(node.comparators[i])
+ right = LinExpr._fromast(node.comparators[i])
if isinstance(op, ast.Lt):
inequalities.append(right - left - 1)
elif isinstance(op, ast.LtE):
_RE_AND = re.compile(r'\band\b|,|&&|/\\|∧|∩')
_RE_OR = re.compile(r'\bor\b|;|\|\||\\/|∨|∪')
_RE_NOT = re.compile(r'\bnot\b|!|¬')
- _RE_NUM_VAR = Expression._RE_NUM_VAR
+ _RE_NUM_VAR = LinExpr._RE_NUM_VAR
_RE_OPERATORS = re.compile(r'(&|\||~)')
@classmethod
def fromstring(cls, string):
"""
- Convert a string into a domain.
+ Create a domain from a string. Raise SyntaxError if the string is not
+ properly formatted.
"""
- # remove curly brackets
+ # Remove curly brackets.
string = cls._RE_BRACES.sub(r'', string)
- # replace '=' by '=='
+ # Replace '=' by '=='.
string = cls._RE_EQ.sub(r'\1==\2', string)
- # replace 'and', 'or', 'not'
+ # Replace 'and', 'or', 'not'.
string = cls._RE_AND.sub(r' & ', string)
string = cls._RE_OR.sub(r' | ', string)
string = cls._RE_NOT.sub(r' ~', string)
- # add implicit multiplication operators, e.g. '5x' -> '5*x'
+ # Add implicit multiplication operators, e.g. '5x' -> '5*x'.
string = cls._RE_NUM_VAR.sub(r'\1*\2', string)
- # add parentheses to force precedence
+ # Add parentheses to force precedence.
tokens = cls._RE_OPERATORS.split(string)
for i, token in enumerate(tokens):
if i % 2 == 0:
return cls._fromast(tree)
def __repr__(self):
- """
- Return repr(self).
- """
assert len(self.polyhedra) >= 2
strings = [repr(polyhedron) for polyhedron in self.polyhedra]
return 'Or({})'.format(', '.join(strings))
- def _repr_latex_(self):
- strings = []
- for polyhedron in self.polyhedra:
- strings.append('({})'.format(polyhedron._repr_latex_().strip('$')))
- return '${}$'.format(' \\vee '.join(strings))
-
@classmethod
- def fromsympy(cls, expr):
+ def fromsympy(cls, expression):
"""
- Convert a SymPy expression into a domain.
+ Create a domain from a SymPy expression.
"""
import sympy
from .polyhedra import Lt, Le, Eq, Ne, Ge, Gt
sympy.Eq: Eq, sympy.Ne: Ne,
sympy.Ge: Ge, sympy.Gt: Gt,
}
- if expr.func in funcmap:
- args = [Domain.fromsympy(arg) for arg in expr.args]
- return funcmap[expr.func](*args)
- elif isinstance(expr, sympy.Expr):
- return Expression.fromsympy(expr)
- raise ValueError('non-domain expression: {!r}'.format(expr))
+ if expression.func in funcmap:
+ args = [Domain.fromsympy(arg) for arg in expression.args]
+ return funcmap[expression.func](*args)
+ elif isinstance(expression, sympy.Expr):
+ return LinExpr.fromsympy(expression)
+ raise ValueError('non-domain expression: {!r}'.format(expression))
def tosympy(self):
"""
- Convert a domain into a SymPy expression.
+ Convert the domain to a SymPy expression.
"""
import sympy
- polyhedra = [polyhedron.tosympy() for polyhedron in polyhedra]
+ polyhedra = [polyhedron.tosympy() for polyhedron in self.polyhedra]
return sympy.Or(*polyhedra)
def And(*domains):
+ """
+ Create the intersection domain of the domains given in arguments.
+ """
if len(domains) == 0:
from .polyhedra import Universe
return Universe
else:
return domains[0].intersection(*domains[1:])
-And.__doc__ = Domain.intersection.__doc__
+
def Or(*domains):
+ """
+ Create the union domain of the domains given in arguments.
+ """
if len(domains) == 0:
from .polyhedra import Empty
return Empty
else:
return domains[0].union(*domains[1:])
-Or.__doc__ = Domain.union.__doc__
+
def Not(domain):
+ """
+ Create the complementary domain of the domain given in argument.
+ """
return ~domain
-Not.__doc__ = Domain.complement.__doc__
projects / linpy.git / blobdiff