index 346ffff..50af053 100644 (file)
@@ -36,21 +36,49 @@ __all__ = [

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')
@@ -59,59 +87,54 @@ class Polyhedron(Domain):
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))
@@ -119,15 +142,18 @@ class Polyhedron(Domain):
return universe

def aspolyhedron(self):
-        """
-        Return the polyhedral hull of a polyhedron.
-        """
return self

-    def __contains__(self, point):
+    def convex_union(self, *others):
"""
-        Report whether a polyhedron constains an integer point
+        Return the convex union of two or more polyhedra.
"""
+        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):
raise TypeError('point must be a Point instance')
if self.symbols != point.symbols:
@@ -141,10 +167,6 @@ class Polyhedron(Domain):
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)
@@ -158,8 +180,14 @@ class Polyhedron(Domain):
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')
+            raise TypeError('argument must be a Polyhedron instance')
inequalities1 = self._asinequalities()
inequalities2 = other._asinequalities()
inequalities = []
@@ -200,8 +228,7 @@ class Polyhedron(Domain):
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

@@ -242,9 +269,6 @@ class Polyhedron(Domain):

@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))
@@ -261,7 +285,6 @@ class Polyhedron(Domain):
else:
return 'And({})'.format(', '.join(strings))

-
def _repr_latex_(self):
strings = []
for equality in self.equalities:
@@ -272,18 +295,12 @@ class Polyhedron(Domain):

@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:
@@ -294,14 +311,14 @@ class Polyhedron(Domain):

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
@@ -321,14 +338,15 @@ 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
@@ -363,41 +381,42 @@ def _polymorphic(func):
@_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])