linpy/polyhedra.py

   1 # Copyright 2014 MINES ParisTech
   2 #
   3 # This file is part of LinPy.
   4 #
   5 # LinPy is free software: you can redistribute it and/or modify
   6 # it under the terms of the GNU General Public License as published by
   7 # the Free Software Foundation, either version 3 of the License, or
   8 # (at your option) any later version.
   9 #
  10 # LinPy is distributed in the hope that it will be useful,
  11 # but WITHOUT ANY WARRANTY; without even the implied warranty of
  12 # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  13 # GNU General Public License for more details.
  14 #
  15 # You should have received a copy of the GNU General Public License
  16 # along with LinPy.  If not, see <http://www.gnu.org/licenses/>.
  17
  18 import functools
  19 import math
  20 import numbers
  21
  22 from . import islhelper
  23
  24 from .islhelper import mainctx, libisl
  25 from .geometry import GeometricObject, Point
  26 from .linexprs import LinExpr, Rational
  27 from .domains import Domain
  28
  29
  30 __all__ = [
  31     'Polyhedron',
  32     'Lt', 'Le', 'Eq', 'Ne', 'Ge', 'Gt',
  33     'Empty', 'Universe',
  34 ]
  35
  36
  37 class Polyhedron(Domain):
  38     """
  39     A convex polyhedron (or simply "polyhedron") is the space defined by a
  40     system of linear equalities and inequalities. This space can be
  41     unbounded.
  42     """
  43
  44     __slots__ = (
  45         '_equalities',
  46         '_inequalities',
  47         '_symbols',
  48         '_dimension',
  49     )
  50
  51     def __new__(cls, equalities=None, inequalities=None):
  52         """
  53         Return a polyhedron from two sequences of linear expressions: equalities
  54         is a list of expressions equal to 0, and inequalities is a list of
  55         expressions greater or equal to 0. For example, the polyhedron
  56         0 <= x <= 2, 0 <= y <= 2 can be constructed with:
  57
  58         >>> x, y = symbols('x y')
  59         >>> square = Polyhedron([], [x, 2 - x, y, 2 - y])
  60
  61         It may be easier to use comparison operators LinExpr.__lt__(),
  62         LinExpr.__le__(), LinExpr.__ge__(), LinExpr.__gt__(), or functions Lt(),
  63         Le(), Eq(), Ge() and Gt(), using one of the following instructions:
  64
  65         >>> x, y = symbols('x y')
  66         >>> square = (0 <= x) & (x <= 2) & (0 <= y) & (y <= 2)
  67         >>> square = Le(0, x, 2) & Le(0, y, 2)
  68
  69         It is also possible to build a polyhedron from a string.
  70
  71         >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
  72
  73         Finally, a polyhedron can be constructed from a GeometricObject
  74         instance, calling the GeometricObject.aspolyedron() method. This way, it
  75         is possible to compute the polyhedral hull of a Domain instance, i.e.,
  76         the convex hull of two polyhedra:
  77
  78         >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
  79         >>> square2 = Polyhedron('2 <= x <= 4, 2 <= y <= 4')
  80         >>> Polyhedron(square | square2)
  81         """
  82         if isinstance(equalities, str):
  83             if inequalities is not None:
  84                 raise TypeError('too many arguments')
  85             return cls.fromstring(equalities)
  86         elif isinstance(equalities, GeometricObject):
  87             if inequalities is not None:
  88                 raise TypeError('too many arguments')
  89             return equalities.aspolyhedron()
  90         if equalities is None:
  91             equalities = []
  92         else:
  93             for i, equality in enumerate(equalities):
  94                 if not isinstance(equality, LinExpr):
  95                     raise TypeError('equalities must be linear expressions')
  96                 equalities[i] = equality.scaleint()
  97         if inequalities is None:
  98             inequalities = []
  99         else:
 100             for i, inequality in enumerate(inequalities):
 101                 if not isinstance(inequality, LinExpr):
 102                     raise TypeError('inequalities must be linear expressions')
 103                 inequalities[i] = inequality.scaleint()
 104         symbols = cls._xsymbols(equalities + inequalities)
 105         islbset = cls._toislbasicset(equalities, inequalities, symbols)
 106         return cls._fromislbasicset(islbset, symbols)
 107
 108     @property
 109     def equalities(self):
 110         """
 111         The tuple of equalities. This is a list of LinExpr instances that are
 112         equal to 0 in the polyhedron.
 113         """
 114         return self._equalities
 115
 116     @property
 117     def inequalities(self):
 118         """
 119         The tuple of inequalities. This is a list of LinExpr instances that are
 120         greater or equal to 0 in the polyhedron.
 121         """
 122         return self._inequalities
 123
 124     @property
 125     def constraints(self):
 126         """
 127         The tuple of constraints, i.e., equalities and inequalities. This is
 128         semantically equivalent to: equalities + inequalities.
 129         """
 130         return self._equalities + self._inequalities
 131
 132     @property
 133     def polyhedra(self):
 134         return self,
 135
 136     def make_disjoint(self):
 137         return self
 138
 139     def isuniverse(self):
 140         islbset = self._toislbasicset(self.equalities, self.inequalities,
 141             self.symbols)
 142         universe = bool(libisl.isl_basic_set_is_universe(islbset))
 143         libisl.isl_basic_set_free(islbset)
 144         return universe
 145
 146     def aspolyhedron(self):
 147         return self
 148
 149     def __contains__(self, point):
 150         if not isinstance(point, Point):
 151             raise TypeError('point must be a Point instance')
 152         if self.symbols != point.symbols:
 153             raise ValueError('arguments must belong to the same space')
 154         for equality in self.equalities:
 155             if equality.subs(point.coordinates()) != 0:
 156                 return False
 157         for inequality in self.inequalities:
 158             if inequality.subs(point.coordinates()) < 0:
 159                 return False
 160         return True
 161
 162     def subs(self, symbol, expression=None):
 163         equalities = [equality.subs(symbol, expression)
 164             for equality in self.equalities]
 165         inequalities = [inequality.subs(symbol, expression)
 166             for inequality in self.inequalities]
 167         return Polyhedron(equalities, inequalities)
 168
 169     def _asinequalities(self):
 170         inequalities = list(self.equalities)
 171         inequalities.extend([-expression for expression in self.equalities])
 172         inequalities.extend(self.inequalities)
 173         return inequalities
 174
 175     def widen(self, other):
 176         """
 177         Compute the standard widening of two polyhedra, à la Halbwachs.
 178         """
 179         if not isinstance(other, Polyhedron):
 180             raise ValueError('argument must be a Polyhedron instance')
 181         inequalities1 = self._asinequalities()
 182         inequalities2 = other._asinequalities()
 183         inequalities = []
 184         for inequality1 in inequalities1:
 185             if other <= Polyhedron(inequalities=[inequality1]):
 186                 inequalities.append(inequality1)
 187         for inequality2 in inequalities2:
 188             for i in range(len(inequalities1)):
 189                 inequalities3 = inequalities1[:i] + inequalities[i + 1:]
 190                 inequalities3.append(inequality2)
 191                 polyhedron3 = Polyhedron(inequalities=inequalities3)
 192                 if self == polyhedron3:
 193                     inequalities.append(inequality2)
 194                     break
 195         return Polyhedron(inequalities=inequalities)
 196
 197     @classmethod
 198     def _fromislbasicset(cls, islbset, symbols):
 199         islconstraints = islhelper.isl_basic_set_constraints(islbset)
 200         equalities = []
 201         inequalities = []
 202         for islconstraint in islconstraints:
 203             constant = libisl.isl_constraint_get_constant_val(islconstraint)
 204             constant = islhelper.isl_val_to_int(constant)
 205             coefficients = {}
 206             for index, symbol in enumerate(symbols):
 207                 coefficient = libisl.isl_constraint_get_coefficient_val(islconstraint,
 208                     libisl.isl_dim_set, index)
 209                 coefficient = islhelper.isl_val_to_int(coefficient)
 210                 if coefficient != 0:
 211                     coefficients[symbol] = coefficient
 212             expression = LinExpr(coefficients, constant)
 213             if libisl.isl_constraint_is_equality(islconstraint):
 214                 equalities.append(expression)
 215             else:
 216                 inequalities.append(expression)
 217         libisl.isl_basic_set_free(islbset)
 218         self = object().__new__(Polyhedron)
 219         self._equalities = tuple(equalities)
 220         self._inequalities = tuple(inequalities)
 221         self._symbols = cls._xsymbols(self.constraints)
 222         self._dimension = len(self._symbols)
 223         return self
 224
 225     @classmethod
 226     def _toislbasicset(cls, equalities, inequalities, symbols):
 227         dimension = len(symbols)
 228         indices = {symbol: index for index, symbol in enumerate(symbols)}
 229         islsp = libisl.isl_space_set_alloc(mainctx, 0, dimension)
 230         islbset = libisl.isl_basic_set_universe(libisl.isl_space_copy(islsp))
 231         islls = libisl.isl_local_space_from_space(islsp)
 232         for equality in equalities:
 233             isleq = libisl.isl_equality_alloc(libisl.isl_local_space_copy(islls))
 234             for symbol, coefficient in equality.coefficients():
 235                 islval = str(coefficient).encode()
 236                 islval = libisl.isl_val_read_from_str(mainctx, islval)
 237                 index = indices[symbol]
 238                 isleq = libisl.isl_constraint_set_coefficient_val(isleq,
 239                     libisl.isl_dim_set, index, islval)
 240             if equality.constant != 0:
 241                 islval = str(equality.constant).encode()
 242                 islval = libisl.isl_val_read_from_str(mainctx, islval)
 243                 isleq = libisl.isl_constraint_set_constant_val(isleq, islval)
 244             islbset = libisl.isl_basic_set_add_constraint(islbset, isleq)
 245         for inequality in inequalities:
 246             islin = libisl.isl_inequality_alloc(libisl.isl_local_space_copy(islls))
 247             for symbol, coefficient in inequality.coefficients():
 248                 islval = str(coefficient).encode()
 249                 islval = libisl.isl_val_read_from_str(mainctx, islval)
 250                 index = indices[symbol]
 251                 islin = libisl.isl_constraint_set_coefficient_val(islin,
 252                     libisl.isl_dim_set, index, islval)
 253             if inequality.constant != 0:
 254                 islval = str(inequality.constant).encode()
 255                 islval = libisl.isl_val_read_from_str(mainctx, islval)
 256                 islin = libisl.isl_constraint_set_constant_val(islin, islval)
 257             islbset = libisl.isl_basic_set_add_constraint(islbset, islin)
 258         return islbset
 259
 260     @classmethod
 261     def fromstring(cls, string):
 262         domain = Domain.fromstring(string)
 263         if not isinstance(domain, Polyhedron):
 264             raise ValueError('non-polyhedral expression: {!r}'.format(string))
 265         return domain
 266
 267     def __repr__(self):
 268         strings = []
 269         for equality in self.equalities:
 270             strings.append('Eq({}, 0)'.format(equality))
 271         for inequality in self.inequalities:
 272             strings.append('Ge({}, 0)'.format(inequality))
 273         if len(strings) == 1:
 274             return strings[0]
 275         else:
 276             return 'And({})'.format(', '.join(strings))
 277
 278     def _repr_latex_(self):
 279         strings = []
 280         for equality in self.equalities:
 281             strings.append('{} = 0'.format(equality._repr_latex_().strip('$')))
 282         for inequality in self.inequalities:
 283             strings.append('{} \\ge 0'.format(inequality._repr_latex_().strip('$')))
 284         return '$${}$$'.format(' \\wedge '.join(strings))
 285
 286     @classmethod
 287     def fromsympy(cls, expr):
 288         domain = Domain.fromsympy(expr)
 289         if not isinstance(domain, Polyhedron):
 290             raise ValueError('non-polyhedral expression: {!r}'.format(expr))
 291         return domain
 292
 293     def tosympy(self):
 294         import sympy
 295         constraints = []
 296         for equality in self.equalities:
 297             constraints.append(sympy.Eq(equality.tosympy(), 0))
 298         for inequality in self.inequalities:
 299             constraints.append(sympy.Ge(inequality.tosympy(), 0))
 300         return sympy.And(*constraints)
 301
 302
 303 class EmptyType(Polyhedron):
 304     """
 305     The empty polyhedron, whose set of constraints is not satisfiable.
 306     """
 307
 308     __slots__ = Polyhedron.__slots__
 309
 310     def __new__(cls):
 311         self = object().__new__(cls)
 312         self._equalities = (Rational(1),)
 313         self._inequalities = ()
 314         self._symbols = ()
 315         self._dimension = 0
 316         return self
 317
 318     def widen(self, other):
 319         if not isinstance(other, Polyhedron):
 320             raise ValueError('argument must be a Polyhedron instance')
 321         return other
 322
 323     def __repr__(self):
 324         return 'Empty'
 325
 326     def _repr_latex_(self):
 327         return '$$\\emptyset$$'
 328
 329 Empty = EmptyType()
 330
 331
 332 class UniverseType(Polyhedron):
 333     """
 334     The universe polyhedron, whose set of constraints is always satisfiable,
 335     i.e. is empty.
 336     """
 337
 338     __slots__ = Polyhedron.__slots__
 339
 340     def __new__(cls):
 341         self = object().__new__(cls)
 342         self._equalities = ()
 343         self._inequalities = ()
 344         self._symbols = ()
 345         self._dimension = ()
 346         return self
 347
 348     def __repr__(self):
 349         return 'Universe'
 350
 351     def _repr_latex_(self):
 352         return '$$\\Omega$$'
 353
 354 Universe = UniverseType()
 355
 356
 357 def _polymorphic(func):
 358     @functools.wraps(func)
 359     def wrapper(left, right):
 360         if not isinstance(left, LinExpr):
 361             if isinstance(left, numbers.Rational):
 362                 left = Rational(left)
 363             else:
 364                 raise TypeError('left must be a a rational number '
 365                     'or a linear expression')
 366         if not isinstance(right, LinExpr):
 367             if isinstance(right, numbers.Rational):
 368                 right = Rational(right)
 369             else:
 370                 raise TypeError('right must be a a rational number '
 371                     'or a linear expression')
 372         return func(left, right)
 373     return wrapper
 374
 375 @_polymorphic
 376 def Lt(left, right):
 377     """
 378     Create the polyhedron with constraints expr1 < expr2 < expr3 ...
 379     """
 380     return Polyhedron([], [right - left - 1])
 381
 382 @_polymorphic
 383 def Le(left, right):
 384     """
 385     Create the polyhedron with constraints expr1 <= expr2 <= expr3 ...
 386     """
 387     return Polyhedron([], [right - left])
 388
 389 @_polymorphic
 390 def Eq(left, right):
 391     """
 392     Create the polyhedron with constraints expr1 == expr2 == expr3 ...
 393     """
 394     return Polyhedron([left - right], [])
 395
 396 @_polymorphic
 397 def Ne(left, right):
 398     """
 399     Create the domain such that expr1 != expr2 != expr3 ... The result is a
 400     Domain, not a Polyhedron.
 401     """
 402     return ~Eq(left, right)
 403
 404 @_polymorphic
 405 def Gt(left, right):
 406     """
 407     Create the polyhedron with constraints expr1 > expr2 > expr3 ...
 408     """
 409     return Polyhedron([], [left - right - 1])
 410
 411 @_polymorphic
 412 def Ge(left, right):
 413     """
 414     Create the polyhedron with constraints expr1 >= expr2 >= expr3 ...
 415     """
 416     return Polyhedron([], [left - right])