# along with LinPy. If not, see <http://www.gnu.org/licenses/>.
import functools
-import math
import numbers
from . import islhelper
-from .islhelper import mainctx, libisl
-from .geometry import GeometricObject, Point
-from .linexprs import Expression, Rational
from .domains import Domain
+from .geometry import GeometricObject, Point
+from .islhelper import libisl, mainctx
+from .linexprs import LinExpr, Rational
__all__ = [
+ 'Empty',
+ 'Eq',
+ 'Ge',
+ 'Gt',
+ 'Le',
+ 'Lt',
+ 'Ne',
'Polyhedron',
- 'Lt', 'Le', 'Eq', 'Ne', 'Ge', 'Gt',
- 'Empty', 'Universe',
+ 'Universe',
]
class Polyhedron(Domain):
+ """
+ A convex polyhedron (or simply "polyhedron") is the space defined by a
+ system of linear equalities and inequalities. This space can be unbounded.
+ A Z-polyhedron (simply called "polyhedron" in LinPy) is the set of integer
+ points in a convex polyhedron.
+ """
__slots__ = (
'_equalities',
'_inequalities',
- '_constraints',
'_symbols',
'_dimension',
)
def __new__(cls, equalities=None, inequalities=None):
+ """
+ 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')
+ >>> square1 = Polyhedron([], [x, 2 - x, y, 2 - y])
+ >>> square1
+ And(0 <= x, x <= 2, 0 <= y, y <= 2)
+
+ 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')
+ >>> square1 = (0 <= x) & (x <= 2) & (0 <= y) & (y <= 2)
+ >>> square1 = Le(0, x, 2) & Le(0, y, 2)
+
+ It is also possible to build a polyhedron from a string.
+
+ >>> square1 = 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:
+
+ >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+ >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
+ >>> Polyhedron(square1 | square2)
+ And(0 <= x, 0 <= y, x <= y + 2, y <= x + 2, x <= 3, y <= 3)
+ """
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):
- if not isinstance(equality, Expression):
- raise TypeError('equalities must be linear expressions')
- equalities[i] = equality.scaleint()
- if inequalities is None:
- inequalities = []
- else:
- for i, inequality in enumerate(inequalities):
- if not isinstance(inequality, Expression):
- raise TypeError('inequalities must be linear expressions')
- inequalities[i] = inequality.scaleint()
- symbols = cls._xsymbols(equalities + inequalities)
- islbset = cls._toislbasicset(equalities, inequalities, symbols)
+ sc_equalities = []
+ if equalities is not None:
+ for equality in equalities:
+ if isinstance(equality, LinExpr):
+ sc_equalities.append(equality.scaleint())
+ elif isinstance(equality, numbers.Rational):
+ sc_equalities.append(Rational(equality).scaleint())
+ else:
+ raise TypeError('equalities must be linear expressions '
+ 'or rational numbers')
+ sc_inequalities = []
+ if inequalities is not None:
+ for inequality in inequalities:
+ if isinstance(inequality, LinExpr):
+ sc_inequalities.append(inequality.scaleint())
+ elif isinstance(inequality, numbers.Rational):
+ sc_inequalities.append(Rational(inequality).scaleint())
+ else:
+ raise TypeError('inequalities must be linear expressions '
+ 'or rational numbers')
+ 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 set.
+ 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 set.
+ 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 ta list of the constraints of a set.
+ 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 disjoint(self):
- """
- Return a set as disjoint.
- """
+ def make_disjoint(self):
return self
def isuniverse(self):
- """
- Return true if a set is the Universe set.
- """
islbset = self._toislbasicset(self.equalities, self.inequalities,
- self.symbols)
+ self.symbols)
universe = bool(libisl.isl_basic_set_is_universe(islbset))
libisl.isl_basic_set_free(islbset)
return universe
def aspolyhedron(self):
+ return self
+
+ def convex_union(self, *others):
"""
- Return polyhedral hull of a set.
+ Return the convex union of two or more polyhedra.
"""
- return self
+ for other in others:
+ if not isinstance(other, Polyhedron):
+ raise TypeError('arguments must be Polyhedron instances')
+ return Polyhedron(self.union(*others))
def __contains__(self, point):
if not isinstance(point, Point):
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]
+ for equality in self.equalities]
inequalities = [inequality.subs(symbol, expression)
- for inequality in self.inequalities]
+ for inequality in self.inequalities]
return Polyhedron(equalities, inequalities)
- def _asinequalities(self):
+ def asinequalities(self):
+ """
+ Express the polyhedron using inequalities, given as a list of
+ expressions greater or equal to 0.
+ """
inequalities = list(self.equalities)
inequalities.extend([-expression for expression in self.equalities])
inequalities.extend(self.inequalities)
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()
- inequalities2 = other._asinequalities()
+ raise TypeError('argument must be a Polyhedron instance')
+ inequalities1 = self.asinequalities()
+ inequalities2 = other.asinequalities()
inequalities = []
for inequality1 in inequalities1:
if other <= Polyhedron(inequalities=[inequality1]):
@classmethod
def _fromislbasicset(cls, islbset, symbols):
+ if bool(libisl.isl_basic_set_is_empty(islbset)):
+ return Empty
+ if bool(libisl.isl_basic_set_is_universe(islbset)):
+ return Universe
islconstraints = islhelper.isl_basic_set_constraints(islbset)
equalities = []
inequalities = []
constant = islhelper.isl_val_to_int(constant)
coefficients = {}
for index, symbol in enumerate(symbols):
- coefficient = libisl.isl_constraint_get_coefficient_val(islconstraint,
- libisl.isl_dim_set, index)
+ coefficient = libisl.isl_constraint_get_coefficient_val(
+ islconstraint, libisl.isl_dim_set, index)
coefficient = islhelper.isl_val_to_int(coefficient)
if coefficient != 0:
coefficients[symbol] = coefficient
- expression = Expression(coefficients, constant)
+ expression = LinExpr(coefficients, constant)
if libisl.isl_constraint_is_equality(islconstraint):
equalities.append(expression)
else:
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
islbset = libisl.isl_basic_set_universe(libisl.isl_space_copy(islsp))
islls = libisl.isl_local_space_from_space(islsp)
for equality in equalities:
- isleq = libisl.isl_equality_alloc(libisl.isl_local_space_copy(islls))
+ isleq = libisl.isl_equality_alloc(
+ libisl.isl_local_space_copy(islls))
for symbol, coefficient in equality.coefficients():
islval = str(coefficient).encode()
islval = libisl.isl_val_read_from_str(mainctx, islval)
index = indices[symbol]
- isleq = libisl.isl_constraint_set_coefficient_val(isleq,
- libisl.isl_dim_set, index, islval)
+ isleq = libisl.isl_constraint_set_coefficient_val(
+ isleq, libisl.isl_dim_set, index, islval)
if equality.constant != 0:
islval = str(equality.constant).encode()
islval = libisl.isl_val_read_from_str(mainctx, islval)
isleq = libisl.isl_constraint_set_constant_val(isleq, islval)
islbset = libisl.isl_basic_set_add_constraint(islbset, isleq)
for inequality in inequalities:
- islin = libisl.isl_inequality_alloc(libisl.isl_local_space_copy(islls))
+ islin = libisl.isl_inequality_alloc(
+ libisl.isl_local_space_copy(islls))
for symbol, coefficient in inequality.coefficients():
islval = str(coefficient).encode()
islval = libisl.isl_val_read_from_str(mainctx, islval)
index = indices[symbol]
- islin = libisl.isl_constraint_set_coefficient_val(islin,
- libisl.isl_dim_set, index, islval)
+ islin = libisl.isl_constraint_set_coefficient_val(
+ islin, libisl.isl_dim_set, index, islval)
if inequality.constant != 0:
islval = str(inequality.constant).encode()
islval = libisl.isl_val_read_from_str(mainctx, islval)
def __repr__(self):
strings = []
for equality in self.equalities:
- strings.append('Eq({}, 0)'.format(equality))
+ left, right, swap = 0, 0, False
+ for i, (symbol, coefficient) in enumerate(equality.coefficients()):
+ if coefficient > 0:
+ left += coefficient * symbol
+ else:
+ right -= coefficient * symbol
+ if i == 0:
+ swap = True
+ if equality.constant > 0:
+ left += equality.constant
+ else:
+ right -= equality.constant
+ if swap:
+ left, right = right, left
+ strings.append('{} == {}'.format(left, right))
for inequality in self.inequalities:
- strings.append('Ge({}, 0)'.format(inequality))
+ left, right = 0, 0
+ for symbol, coefficient in inequality.coefficients():
+ if coefficient < 0:
+ left -= coefficient * symbol
+ else:
+ right += coefficient * symbol
+ if inequality.constant < 0:
+ left -= inequality.constant
+ else:
+ right += inequality.constant
+ strings.append('{} <= {}'.format(left, right))
if len(strings) == 1:
return strings[0]
else:
return 'And({})'.format(', '.join(strings))
-
- def _repr_latex_(self):
- strings = []
- for equality in self.equalities:
- strings.append('{} = 0'.format(equality._repr_latex_().strip('$')))
- for inequality in self.inequalities:
- strings.append('{} \\ge 0'.format(inequality._repr_latex_().strip('$')))
- return '$${}$$'.format(' \\wedge '.join(strings))
-
@classmethod
- def fromsympy(cls, expr):
- """
- Convert a sympy object to an expression.
- """
- domain = Domain.fromsympy(expr)
+ def fromsympy(cls, expression):
+ domain = Domain.fromsympy(expression)
if not isinstance(domain, Polyhedron):
- raise ValueError('non-polyhedral expression: {!r}'.format(expr))
+ raise ValueError('non-polyhedral expression: {!r}'.format(
+ expression))
return domain
def tosympy(self):
- """
- Return an expression 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
def __repr__(self):
return 'Empty'
- def _repr_latex_(self):
- return '$$\\emptyset$$'
-
Empty = EmptyType()
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
def __repr__(self):
return 'Universe'
- def _repr_latex_(self):
- return '$$\\Omega$$'
-
Universe = UniverseType()
-def _polymorphic(func):
+def _pseudoconstructor(func):
@functools.wraps(func)
- def wrapper(left, right):
- if not isinstance(left, Expression):
- if isinstance(left, numbers.Rational):
- left = Rational(left)
- else:
- raise TypeError('left must be a a rational number '
- 'or a linear expression')
- if not isinstance(right, Expression):
- if isinstance(right, numbers.Rational):
- right = Rational(right)
- else:
- raise TypeError('right must be a a rational number '
- 'or a linear expression')
- return func(left, right)
+ def wrapper(expression1, expression2, *expressions):
+ expressions = (expression1, expression2) + expressions
+ for expression in expressions:
+ if not isinstance(expression, LinExpr):
+ if isinstance(expression, numbers.Rational):
+ expression = Rational(expression)
+ else:
+ raise TypeError('arguments must be rational numbers '
+ 'or linear expressions')
+ return func(*expressions)
return wrapper
-@_polymorphic
-def Lt(left, right):
+
+@_pseudoconstructor
+def Lt(*expressions):
"""
- Assert first set is less than the second set.
+ Create the polyhedron with constraints expr1 < expr2 < expr3 ...
"""
- return Polyhedron([], [right - left - 1])
+ inequalities = []
+ for left, right in zip(expressions, expressions[1:]):
+ inequalities.append(right - left - 1)
+ return Polyhedron([], inequalities)
+
-@_polymorphic
-def Le(left, right):
+@_pseudoconstructor
+def Le(*expressions):
"""
- Assert first set is less than or equal to the second set.
+ Create the polyhedron with constraints expr1 <= expr2 <= expr3 ...
"""
- return Polyhedron([], [right - left])
+ inequalities = []
+ for left, right in zip(expressions, expressions[1:]):
+ inequalities.append(right - left)
+ return Polyhedron([], inequalities)
+
-@_polymorphic
-def Eq(left, right):
+@_pseudoconstructor
+def Eq(*expressions):
"""
- Assert first set is equal to the second set.
+ Create the polyhedron with constraints expr1 == expr2 == expr3 ...
"""
- return Polyhedron([left - right], [])
+ equalities = []
+ for left, right in zip(expressions, expressions[1:]):
+ equalities.append(left - right)
+ return Polyhedron(equalities, [])
-@_polymorphic
-def Ne(left, right):
+
+@_pseudoconstructor
+def Ne(*expressions):
"""
- Assert first set is not equal to the second set.
+ Create the domain such that expr1 != expr2 != expr3 ... The result is a
+ Domain object, not a Polyhedron.
"""
- return ~Eq(left, right)
+ domain = Universe
+ for left, right in zip(expressions, expressions[1:]):
+ domain &= ~Eq(left, right)
+ return domain
+
-@_polymorphic
-def Gt(left, right):
+@_pseudoconstructor
+def Ge(*expressions):
"""
- Assert first set is greater than the second set.
+ Create the polyhedron with constraints expr1 >= expr2 >= expr3 ...
"""
- return Polyhedron([], [left - right - 1])
+ inequalities = []
+ for left, right in zip(expressions, expressions[1:]):
+ inequalities.append(left - right)
+ return Polyhedron([], inequalities)
+
-@_polymorphic
-def Ge(left, right):
+@_pseudoconstructor
+def Gt(*expressions):
"""
- Assert first set is greater than or equal to the second set.
+ Create the polyhedron with constraints expr1 > expr2 > expr3 ...
"""
- return Polyhedron([], [left - right])
+ inequalities = []
+ for left, right in zip(expressions, expressions[1:]):
+ inequalities.append(left - right - 1)
+ return Polyhedron([], inequalities)
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