class Polyhedron(Domain):
"""
- Polyhedron class allows users to build and inspect polyherons. Polyhedron inherits from Domain.
+ A convex polyhedron (or simply "polyhedron") is the space defined by a
+ system of linear equalities and inequalities. This space can be
+ unbounded.
"""
+
__slots__ = (
'_equalities',
'_inequalities',
- '_constraints',
'_symbols',
'_dimension',
)
def __new__(cls, equalities=None, inequalities=None):
"""
- Create and return a new Polyhedron from a string or list of equalities and inequalities.
- """
+ Return a polyhedron from two sequences of linear expressions: equalities
+ is a list of expressions equal to 0, and inequalities is a list of
+ expressions greater or equal to 0. For example, the polyhedron
+ 0 <= x <= 2, 0 <= y <= 2 can be constructed with:
+
+ >>> x, y = symbols('x y')
+ >>> square = Polyhedron([], [x, 2 - x, y, 2 - y])
+
+ It may be easier to use comparison operators LinExpr.__lt__(),
+ LinExpr.__le__(), LinExpr.__ge__(), LinExpr.__gt__(), or functions Lt(),
+ Le(), Eq(), Ge() and Gt(), using one of the following instructions:
+
+ >>> x, y = symbols('x y')
+ >>> square = (0 <= x) & (x <= 2) & (0 <= y) & (y <= 2)
+ >>> square = Le(0, x, 2) & Le(0, y, 2)
+
+ It is also possible to build a polyhedron from a string.
+ >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+
+ Finally, a polyhedron can be constructed from a GeometricObject
+ instance, calling the GeometricObject.aspolyedron() method. This way, it
+ is possible to compute the polyhedral hull of a Domain instance, i.e.,
+ the convex hull of two polyhedra:
+
+ >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+ >>> square2 = Polyhedron('2 <= x <= 4, 2 <= y <= 4')
+ >>> Polyhedron(square | square2)
+ """
if isinstance(equalities, str):
if inequalities is not None:
raise TypeError('too many arguments')
if inequalities is not None:
raise TypeError('too many arguments')
return equalities.aspolyhedron()
- if equalities is None:
- equalities = []
- else:
- for i, equality in enumerate(equalities):
+ sc_equalities = []
+ if equalities is not None:
+ for equality in equalities:
if not isinstance(equality, LinExpr):
raise TypeError('equalities must be linear expressions')
- equalities[i] = equality.scaleint()
- if inequalities is None:
- inequalities = []
- else:
- for i, inequality in enumerate(inequalities):
+ sc_equalities.append(equality.scaleint())
+ sc_inequalities = []
+ if inequalities is not None:
+ for inequality in inequalities:
if not isinstance(inequality, LinExpr):
raise TypeError('inequalities must be linear expressions')
- inequalities[i] = inequality.scaleint()
- symbols = cls._xsymbols(equalities + inequalities)
- islbset = cls._toislbasicset(equalities, inequalities, symbols)
+ sc_inequalities.append(inequality.scaleint())
+ symbols = cls._xsymbols(sc_equalities + sc_inequalities)
+ islbset = cls._toislbasicset(sc_equalities, sc_inequalities, symbols)
return cls._fromislbasicset(islbset, symbols)
@property
def equalities(self):
"""
- Return a list of the equalities in a polyhedron.
+ The tuple of equalities. This is a list of LinExpr instances that are
+ equal to 0 in the polyhedron.
"""
return self._equalities
@property
def inequalities(self):
"""
- Return a list of the inequalities in a polyhedron.
+ The tuple of inequalities. This is a list of LinExpr instances that are
+ greater or equal to 0 in the polyhedron.
"""
return self._inequalities
@property
def constraints(self):
"""
- Return the list of the constraints of a polyhedron.
+ The tuple of constraints, i.e., equalities and inequalities. This is
+ semantically equivalent to: equalities + inequalities.
"""
- return self._constraints
+ return self._equalities + self._inequalities
@property
def polyhedra(self):
return self,
def make_disjoint(self):
- """
- Return a polyhedron as disjoint.
- """
return self
def isuniverse(self):
- """
- Return true if a polyhedron is the Universe set.
- """
islbset = self._toislbasicset(self.equalities, self.inequalities,
self.symbols)
universe = bool(libisl.isl_basic_set_is_universe(islbset))
return universe
def aspolyhedron(self):
- """
- Return the polyhedral hull of a polyhedron.
- """
return self
def __contains__(self, point):
- """
- Report whether a polyhedron constains an integer point
- """
if not isinstance(point, Point):
raise TypeError('point must be a Point instance')
if self.symbols != point.symbols:
return True
def subs(self, symbol, expression=None):
- """
- Subsitute the given value into an expression and return the resulting
- expression.
- """
equalities = [equality.subs(symbol, expression)
for equality in self.equalities]
inequalities = [inequality.subs(symbol, expression)
return inequalities
def widen(self, other):
+ """
+ Compute the standard widening of two polyhedra, à la Halbwachs.
+
+ In its current implementation, this method is slow and should not be
+ used on large polyhedra.
+ """
if not isinstance(other, Polyhedron):
raise ValueError('argument must be a Polyhedron instance')
inequalities1 = self._asinequalities()
self = object().__new__(Polyhedron)
self._equalities = tuple(equalities)
self._inequalities = tuple(inequalities)
- self._constraints = tuple(equalities + inequalities)
- self._symbols = cls._xsymbols(self._constraints)
+ self._symbols = cls._xsymbols(self.constraints)
self._dimension = len(self._symbols)
return self
@classmethod
def fromstring(cls, string):
- """
- Create and return a Polyhedron from a string.
- """
domain = Domain.fromstring(string)
if not isinstance(domain, Polyhedron):
raise ValueError('non-polyhedral expression: {!r}'.format(string))
else:
return 'And({})'.format(', '.join(strings))
-
def _repr_latex_(self):
strings = []
for equality in self.equalities:
@classmethod
def fromsympy(cls, expr):
- """
- Convert a sympy object to a polyhedron.
- """
domain = Domain.fromsympy(expr)
if not isinstance(domain, Polyhedron):
raise ValueError('non-polyhedral expression: {!r}'.format(expr))
return domain
def tosympy(self):
- """
- Return a polyhedron as a sympy object.
- """
import sympy
constraints = []
for equality in self.equalities:
class EmptyType(Polyhedron):
-
- __slots__ = Polyhedron.__slots__
+ """
+ The empty polyhedron, whose set of constraints is not satisfiable.
+ """
def __new__(cls):
self = object().__new__(cls)
self._equalities = (Rational(1),)
self._inequalities = ()
- self._constraints = self._equalities
self._symbols = ()
self._dimension = 0
return self
class UniverseType(Polyhedron):
-
- __slots__ = Polyhedron.__slots__
+ """
+ The universe polyhedron, whose set of constraints is always satisfiable,
+ i.e. is empty.
+ """
def __new__(cls):
self = object().__new__(cls)
self._equalities = ()
self._inequalities = ()
- self._constraints = ()
self._symbols = ()
self._dimension = ()
return self
@_polymorphic
def Lt(left, right):
"""
- Returns a Polyhedron instance with a single constraint as left less than right.
+ Create the polyhedron with constraints expr1 < expr2 < expr3 ...
"""
return Polyhedron([], [right - left - 1])
@_polymorphic
def Le(left, right):
"""
- Returns a Polyhedron instance with a single constraint as left less than or equal to right.
+ Create the polyhedron with constraints expr1 <= expr2 <= expr3 ...
"""
return Polyhedron([], [right - left])
@_polymorphic
def Eq(left, right):
"""
- Returns a Polyhedron instance with a single constraint as left equal to right.
+ Create the polyhedron with constraints expr1 == expr2 == expr3 ...
"""
return Polyhedron([left - right], [])
@_polymorphic
def Ne(left, right):
"""
- Returns a Polyhedron instance with a single constraint as left not equal to right.
+ Create the domain such that expr1 != expr2 != expr3 ... The result is a
+ Domain, not a Polyhedron.
"""
return ~Eq(left, right)
@_polymorphic
-def Gt(left, right):
+def Ge(left, right):
"""
- Returns a Polyhedron instance with a single constraint as left greater than right.
+ Create the polyhedron with constraints expr1 >= expr2 >= expr3 ...
"""
- return Polyhedron([], [left - right - 1])
+ return Polyhedron([], [left - right])
@_polymorphic
-def Ge(left, right):
+def Gt(left, right):
"""
- Returns a Polyhedron instance with a single constraint as left greater than or equal to right.
+ Create the polyhedron with constraints expr1 > expr2 > expr3 ...
"""
- return Polyhedron([], [left - right])
+ return Polyhedron([], [left - right - 1])
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