index 50af053..9e740a4 100644 (file)
# 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 .domains import Domain
from .geometry import GeometricObject, Point
+from .islhelper import libisl, mainctx
from .linexprs import LinExpr, Rational
-from .domains import Domain

__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.
+    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__ = (
@@ -50,34 +56,38 @@ class Polyhedron(Domain):

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
+        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])
+        >>> 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:
+        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)
+        >>> 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.

-        >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+        >>> 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:
-
-        >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
-        >>> square2 = Polyhedron('2 <= x <= 4, 2 <= y <= 4')
-        >>> Polyhedron(square | square2)
+        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:
@@ -90,15 +100,23 @@ class Polyhedron(Domain):
sc_equalities = []
if equalities is not None:
for equality in equalities:
-                if not isinstance(equality, LinExpr):
-                    raise TypeError('equalities must be linear expressions')
-                sc_equalities.append(equality.scaleint())
+                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 not isinstance(inequality, LinExpr):
-                    raise TypeError('inequalities must be linear expressions')
-                sc_inequalities.append(inequality.scaleint())
+                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)
@@ -136,7 +154,7 @@ class Polyhedron(Domain):

def isuniverse(self):
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
@@ -168,12 +186,16 @@ class Polyhedron(Domain):

def subs(self, symbol, expression=None):
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)
@@ -188,8 +210,8 @@ class Polyhedron(Domain):
"""
if not isinstance(other, Polyhedron):
raise TypeError('argument must be a Polyhedron instance')
-        inequalities1 = self._asinequalities()
-        inequalities2 = other._asinequalities()
+        inequalities1 = self.asinequalities()
+        inequalities2 = other.asinequalities()
inequalities = []
for inequality1 in inequalities1:
if other <= Polyhedron(inequalities=[inequality1]):
@@ -206,6 +228,10 @@ class Polyhedron(Domain):

@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 = []
@@ -214,8 +240,8 @@ class Polyhedron(Domain):
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
@@ -240,26 +266,28 @@ class Polyhedron(Domain):
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()
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()
isleq = libisl.isl_constraint_set_constant_val(isleq, islval)
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()
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()
@@ -277,27 +305,44 @@ 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:
-            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):
-        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):
@@ -331,9 +376,6 @@ class EmptyType(Polyhedron):
def __repr__(self):
return 'Empty'

-    def _repr_latex_(self):
-        return '\$\$\\emptyset\$\$'
-
Empty = EmptyType()

@@ -354,69 +396,86 @@ class UniverseType(Polyhedron):
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, LinExpr):
-            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, LinExpr):
-            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):
"""
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):
"""
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):
"""
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):
"""
Create the domain such that expr1 != expr2 != expr3 ... The result is a
-    Domain, not a Polyhedron.
+    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 Ge(left, right):
+
+@_pseudoconstructor
+def Ge(*expressions):
"""
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)
+    return Polyhedron([], inequalities)
+

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