Mention that we use Z-polyhedra

[linpy.git] / linpy / polyhedra.py

diff --git a/linpy/polyhedra.py b/linpy/polyhedra.py
index e9226f2..f802151 100644 (file)
--- a/linpy/polyhedra.py
+++ b/linpy/polyhedra.py
@@ -23,7 +23,7 @@ from . import islhelper
 
 from .islhelper import mainctx, libisl
 from .geometry import GeometricObject, Point
 
 from .islhelper import mainctx, libisl
 from .geometry import GeometricObject, Point

-from .linexprs import Expression, Rational
+from .linexprs import LinExpr, Rational

 from .domains import Domain
 
 
 from .domains import Domain
 
 


@@ -35,16 +35,54 @@ __all__ = [
 
 
 class Polyhedron(Domain):
 
 
 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',
 
     __slots__ = (
         '_equalities',
         '_inequalities',

-        '_constraints',

         '_symbols',
         '_dimension',
     )
 
     def __new__(cls, equalities=None, inequalities=None):
         '_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 isinstance(equalities, str):
             if inequalities is not None:
                 raise TypeError('too many arguments')


@@ -53,59 +91,54 @@ class Polyhedron(Domain):
             if inequalities is not None:
                 raise TypeError('too many arguments')
             return equalities.aspolyhedron()
             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')
                     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')
                     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 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 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 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,
 
 
     @property
     def polyhedra(self):
         return self,
 

-    def disjoint(self):
-        """
-        Return a set as disjoint.
-        """
+    def make_disjoint(self):

         return self
 
     def isuniverse(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))
         islbset = self._toislbasicset(self.equalities, self.inequalities,
             self.symbols)
         universe = bool(libisl.isl_basic_set_is_universe(islbset))


@@ -113,10 +146,16 @@ class Polyhedron(Domain):
         return universe
 
     def aspolyhedron(self):
         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):
 
     def __contains__(self, point):
         if not isinstance(point, Point):


@@ -132,27 +171,33 @@ class Polyhedron(Domain):
         return True
 
     def subs(self, symbol, expression=None):
         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)
 
         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):
         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):
         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]):
         inequalities = []
         for inequality1 in inequalities1:
             if other <= Polyhedron(inequalities=[inequality1]):


@@ -182,7 +227,7 @@ class Polyhedron(Domain):
                 coefficient = islhelper.isl_val_to_int(coefficient)
                 if coefficient != 0:
                     coefficients[symbol] = coefficient
                 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:
             if libisl.isl_constraint_is_equality(islconstraint):
                 equalities.append(expression)
             else:


@@ -191,8 +236,7 @@ class Polyhedron(Domain):
         self = object().__new__(Polyhedron)
         self._equalities = tuple(equalities)
         self._inequalities = tuple(inequalities)
         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
 
         self._dimension = len(self._symbols)
         return self
 


@@ -241,15 +285,38 @@ class Polyhedron(Domain):
     def __repr__(self):
         strings = []
         for equality in self.equalities:
     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:
         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))
 
         if len(strings) == 1:
             return strings[0]
         else:
             return 'And({})'.format(', '.join(strings))
 

-

     def _repr_latex_(self):
         strings = []
         for equality in self.equalities:
     def _repr_latex_(self):
         strings = []
         for equality in self.equalities:


@@ -260,18 +327,12 @@ class Polyhedron(Domain):
 
     @classmethod
     def fromsympy(cls, expr):
 
     @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):
         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:
         import sympy
         constraints = []
         for equality in self.equalities:


@@ -282,14 +343,14 @@ class Polyhedron(Domain):
 
 
 class EmptyType(Polyhedron):
 
 
 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 = ()
 
     def __new__(cls):
         self = object().__new__(cls)
         self._equalities = (Rational(1),)
         self._inequalities = ()

-        self._constraints = self._equalities

         self._symbols = ()
         self._dimension = 0
         return self
         self._symbols = ()
         self._dimension = 0
         return self


@@ -309,14 +370,15 @@ Empty = EmptyType()
 
 
 class UniverseType(Polyhedron):
 
 
 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 = ()
 
     def __new__(cls):
         self = object().__new__(cls)
         self._equalities = ()
         self._inequalities = ()

-        self._constraints = ()

         self._symbols = ()
         self._dimension = ()
         return self
         self._symbols = ()
         self._dimension = ()
         return self


@@ -330,62 +392,77 @@ class UniverseType(Polyhedron):
 Universe = UniverseType()
 
 
 Universe = UniverseType()
 
 

-def _polymorphic(func):
+def _pseudoconstructor(func):

     @functools.wraps(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
 
     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)