index e9226f2..bfc7efe 100644 (file)
@@ -23,7 +23,7 @@ from . import islhelper

from .islhelper import mainctx, libisl
from .geometry import GeometricObject, Point
-from .linexprs import Expression, Rational
+from .linexprs import LinExpr, Rational
from .domains import Domain

@@ -35,16 +35,50 @@ __all__ = [

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.
+    """

__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')
+        >>> 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')
@@ -53,59 +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):
-                if not isinstance(equality, Expression):
+        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):
-                if not isinstance(inequality, Expression):
+                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 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)
universe = bool(libisl.isl_basic_set_is_universe(islbset))
@@ -113,10 +142,16 @@ class Polyhedron(Domain):
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):
@@ -132,27 +167,33 @@ 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)
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]):
@@ -182,7 +223,7 @@ class Polyhedron(Domain):
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:
@@ -191,8 +232,7 @@ 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

@@ -241,15 +281,38 @@ class Polyhedron(Domain):
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
else:
return 'And({})'.format(', '.join(strings))

-
def _repr_latex_(self):
strings = []
for equality in self.equalities:
@@ -260,18 +323,12 @@ class Polyhedron(Domain):

@classmethod
def fromsympy(cls, expr):
-        """
-        Convert a sympy object to an expression.
-        """
domain = Domain.fromsympy(expr)
if not isinstance(domain, Polyhedron):
raise ValueError('non-polyhedral expression: {!r}'.format(expr))
return domain

def tosympy(self):
-        """
-        Return an expression as a sympy object.
-        """
import sympy
constraints = []
for equality in self.equalities:
@@ -282,14 +339,14 @@ 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
@@ -309,14 +366,15 @@ 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
@@ -330,62 +388,77 @@ class UniverseType(Polyhedron):
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(expr1, expr2, *exprs):
+        exprs = (expr1, expr2) + exprs
+        for expr in exprs:
+            if not isinstance(expr, LinExpr):
+                if isinstance(expr, numbers.Rational):
+                    expr = Rational(expr)
+                else:
+                    raise TypeError('arguments must be rational numbers '
+                        'or linear expressions')
+        return func(*exprs)
return wrapper

-@_polymorphic
-def Lt(left, right):
+@_pseudoconstructor
+def Lt(*exprs):
"""
-    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(exprs, exprs[1:]):
+        inequalities.append(right - left - 1)
+    return Polyhedron([], inequalities)

-@_polymorphic
-def Le(left, right):
+@_pseudoconstructor
+def Le(*exprs):
"""
-    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(exprs, exprs[1:]):
+        inequalities.append(right - left)
+    return Polyhedron([], inequalities)

-@_polymorphic
-def Eq(left, right):
+@_pseudoconstructor
+def Eq(*exprs):
"""
-    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(exprs, exprs[1:]):
+        equalities.append(left - right)
+    return Polyhedron(equalities, [])

-@_polymorphic
-def Ne(left, right):
+@_pseudoconstructor
+def Ne(*exprs):
"""
-    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(exprs, exprs[1:]):
+        domain &= ~Eq(left, right)
+    return domain

-@_polymorphic
-def Gt(left, right):
+@_pseudoconstructor
+def Ge(*exprs):
"""
-    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(exprs, exprs[1:]):
+        inequalities.append(left - right)
+    return Polyhedron([], inequalities)

-@_polymorphic
-def Ge(left, right):
+@_pseudoconstructor
+def Gt(*exprs):
"""
-    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(exprs, exprs[1:]):
+        inequalities.append(left - right - 1)
+    return Polyhedron([], inequalities)