linpy/domains.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 ast
  19 import functools
  20 import re
  21 import math
  22
  23 from fractions import Fraction
  24
  25 from . import islhelper
  26 from .islhelper import mainctx, libisl
  27 from .linexprs import Expression, Symbol, Rational
  28 from .geometry import GeometricObject, Point, Vector
  29
  30
  31 __all__ = [
  32     'Domain',
  33     'And', 'Or', 'Not',
  34 ]
  35
  36
  37 @functools.total_ordering
  38 class Domain(GeometricObject):
  39
  40     __slots__ = (
  41         '_polyhedra',
  42         '_symbols',
  43         '_dimension',
  44     )
  45
  46     def __new__(cls, *polyhedra):
  47         from .polyhedra import Polyhedron
  48         if len(polyhedra) == 1:
  49             argument = polyhedra[0]
  50             if isinstance(argument, str):
  51                 return cls.fromstring(argument)
  52             elif isinstance(argument, GeometricObject):
  53                 return argument.aspolyhedron()
  54             else:
  55                 raise TypeError('argument must be a string '
  56                     'or a GeometricObject instance')
  57         else:
  58             for polyhedron in polyhedra:
  59                 if not isinstance(polyhedron, Polyhedron):
  60                     raise TypeError('arguments must be Polyhedron instances')
  61             symbols = cls._xsymbols(polyhedra)
  62             islset = cls._toislset(polyhedra, symbols)
  63             return cls._fromislset(islset, symbols)
  64
  65     @classmethod
  66     def _xsymbols(cls, iterator):
  67         """
  68         Return the ordered tuple of symbols present in iterator.
  69         """
  70         symbols = set()
  71         for item in iterator:
  72             symbols.update(item.symbols)
  73         return tuple(sorted(symbols, key=Symbol.sortkey))
  74
  75     @property
  76     def polyhedra(self):
  77         return self._polyhedra
  78
  79     @property
  80     def symbols(self):
  81         return self._symbols
  82
  83     @property
  84     def dimension(self):
  85         return self._dimension
  86
  87     def disjoint(self):
  88         """
  89         Returns this set as disjoint.
  90         """
  91         islset = self._toislset(self.polyhedra, self.symbols)
  92         islset = libisl.isl_set_make_disjoint(mainctx, islset)
  93         return self._fromislset(islset, self.symbols)
  94
  95     def isempty(self):
  96         """
  97         Returns true if this set is an Empty set.
  98         """
  99         islset = self._toislset(self.polyhedra, self.symbols)
 100         empty = bool(libisl.isl_set_is_empty(islset))
 101         libisl.isl_set_free(islset)
 102         return empty
 103
 104     def __bool__(self):
 105         return not self.isempty()
 106
 107     def isuniverse(self):
 108         """
 109         Returns true if this set is the Universe set.
 110         """
 111         islset = self._toislset(self.polyhedra, self.symbols)
 112         universe = bool(libisl.isl_set_plain_is_universe(islset))
 113         libisl.isl_set_free(islset)
 114         return universe
 115
 116     def isbounded(self):
 117         """
 118         Returns true if this set is bounded.
 119         """
 120         islset = self._toislset(self.polyhedra, self.symbols)
 121         bounded = bool(libisl.isl_set_is_bounded(islset))
 122         libisl.isl_set_free(islset)
 123         return bounded
 124
 125     def __eq__(self, other):
 126         """
 127         Returns true if two sets are equal.
 128         """
 129         symbols = self._xsymbols([self, other])
 130         islset1 = self._toislset(self.polyhedra, symbols)
 131         islset2 = other._toislset(other.polyhedra, symbols)
 132         equal = bool(libisl.isl_set_is_equal(islset1, islset2))
 133         libisl.isl_set_free(islset1)
 134         libisl.isl_set_free(islset2)
 135         return equal
 136
 137     def isdisjoint(self, other):
 138         """
 139         Return True if two sets have a null intersection.
 140         """
 141         symbols = self._xsymbols([self, other])
 142         islset1 = self._toislset(self.polyhedra, symbols)
 143         islset2 = self._toislset(other.polyhedra, symbols)
 144         equal = bool(libisl.isl_set_is_disjoint(islset1, islset2))
 145         libisl.isl_set_free(islset1)
 146         libisl.isl_set_free(islset2)
 147         return equal
 148
 149     def issubset(self, other):
 150         """
 151         Report whether another set contains this set.
 152         """
 153         symbols = self._xsymbols([self, other])
 154         islset1 = self._toislset(self.polyhedra, symbols)
 155         islset2 = self._toislset(other.polyhedra, symbols)
 156         equal = bool(libisl.isl_set_is_subset(islset1, islset2))
 157         libisl.isl_set_free(islset1)
 158         libisl.isl_set_free(islset2)
 159         return equal
 160
 161     def __le__(self, other):
 162         """
 163         Returns true if this set is less than or equal to another set.
 164         """
 165         return self.issubset(other)
 166
 167     def __lt__(self, other):
 168         """
 169         Returns true if this set is less than another set.
 170         """
 171         symbols = self._xsymbols([self, other])
 172         islset1 = self._toislset(self.polyhedra, symbols)
 173         islset2 = self._toislset(other.polyhedra, symbols)
 174         equal = bool(libisl.isl_set_is_strict_subset(islset1, islset2))
 175         libisl.isl_set_free(islset1)
 176         libisl.isl_set_free(islset2)
 177         return equal
 178
 179     def complement(self):
 180         """
 181         Returns the complement of this set.
 182         """
 183         islset = self._toislset(self.polyhedra, self.symbols)
 184         islset = libisl.isl_set_complement(islset)
 185         return self._fromislset(islset, self.symbols)
 186
 187     def __invert__(self):
 188         """
 189         Returns the complement of this set.
 190         """
 191         return self.complement()
 192
 193     def simplify(self):
 194         """
 195         Returns a set without redundant constraints.
 196         """
 197         islset = self._toislset(self.polyhedra, self.symbols)
 198         islset = libisl.isl_set_remove_redundancies(islset)
 199         return self._fromislset(islset, self.symbols)
 200
 201     def aspolyhedron(self):
 202         """
 203         Returns polyhedral hull of set.
 204         """
 205         from .polyhedra import Polyhedron
 206         islset = self._toislset(self.polyhedra, self.symbols)
 207         islbset = libisl.isl_set_polyhedral_hull(islset)
 208         return Polyhedron._fromislbasicset(islbset, self.symbols)
 209
 210     def asdomain(self):
 211         return self
 212
 213     def project(self, dims):
 214         """
 215         Return new set with given dimensions removed.
 216         """
 217         islset = self._toislset(self.polyhedra, self.symbols)
 218         n = 0
 219         for index, symbol in reversed(list(enumerate(self.symbols))):
 220             if symbol in dims:
 221                 n += 1
 222             elif n > 0:
 223                 islset = libisl.isl_set_project_out(islset, libisl.isl_dim_set, index + 1, n)
 224                 n = 0
 225         if n > 0:
 226             islset = libisl.isl_set_project_out(islset, libisl.isl_dim_set, 0, n)
 227         dims = [symbol for symbol in self.symbols if symbol not in dims]
 228         return Domain._fromislset(islset, dims)
 229
 230     def sample(self):
 231         """
 232         Returns a single subset of the input.
 233         """
 234         islset = self._toislset(self.polyhedra, self.symbols)
 235         islpoint = libisl.isl_set_sample_point(islset)
 236         if bool(libisl.isl_point_is_void(islpoint)):
 237             libisl.isl_point_free(islpoint)
 238             raise ValueError('domain must be non-empty')
 239         point = {}
 240         for index, symbol in enumerate(self.symbols):
 241             coordinate = libisl.isl_point_get_coordinate_val(islpoint,
 242                 libisl.isl_dim_set, index)
 243             coordinate = islhelper.isl_val_to_int(coordinate)
 244             point[symbol] = coordinate
 245         libisl.isl_point_free(islpoint)
 246         return point
 247
 248     def intersection(self, *others):
 249         """
 250          Return the intersection of two sets as a new set.
 251         """
 252         if len(others) == 0:
 253             return self
 254         symbols = self._xsymbols((self,) + others)
 255         islset1 = self._toislset(self.polyhedra, symbols)
 256         for other in others:
 257             islset2 = other._toislset(other.polyhedra, symbols)
 258             islset1 = libisl.isl_set_intersect(islset1, islset2)
 259         return self._fromislset(islset1, symbols)
 260
 261     def __and__(self, other):
 262         """
 263          Return the intersection of two sets as a new set.
 264         """
 265         return self.intersection(other)
 266
 267     def union(self, *others):
 268         """
 269         Return the union of sets as a new set.
 270         """
 271         if len(others) == 0:
 272             return self
 273         symbols = self._xsymbols((self,) + others)
 274         islset1 = self._toislset(self.polyhedra, symbols)
 275         for other in others:
 276             islset2 = other._toislset(other.polyhedra, symbols)
 277             islset1 = libisl.isl_set_union(islset1, islset2)
 278         return self._fromislset(islset1, symbols)
 279
 280     def __or__(self, other):
 281         """
 282         Return a new set with elements from both sets.
 283         """
 284         return self.union(other)
 285
 286     def __add__(self, other):
 287         """
 288         Return new set containing all elements in both sets.
 289         """
 290         return self.union(other)
 291
 292     def difference(self, other):
 293         """
 294         Return the difference of two sets as a new set.
 295         """
 296         symbols = self._xsymbols([self, other])
 297         islset1 = self._toislset(self.polyhedra, symbols)
 298         islset2 = other._toislset(other.polyhedra, symbols)
 299         islset = libisl.isl_set_subtract(islset1, islset2)
 300         return self._fromislset(islset, symbols)
 301
 302     def __sub__(self, other):
 303         """
 304         Return the difference of two sets as a new set.
 305         """
 306         return self.difference(other)
 307
 308     def lexmin(self):
 309         """
 310         Return a new set containing the lexicographic minimum of the elements in the set.
 311         """
 312         islset = self._toislset(self.polyhedra, self.symbols)
 313         islset = libisl.isl_set_lexmin(islset)
 314         return self._fromislset(islset, self.symbols)
 315
 316     def lexmax(self):
 317         """
 318         Return a new set containing the lexicographic maximum of the elements in the set.
 319         """
 320         islset = self._toislset(self.polyhedra, self.symbols)
 321         islset = libisl.isl_set_lexmax(islset)
 322         return self._fromislset(islset, self.symbols)
 323
 324
 325     def involves_vars(self, vars):
 326         """
 327         Returns true if a set depends on given dimensions.
 328         """
 329         islset = self._toislset(self.polyhedra, self.symbols)
 330         dims = sorted(vars)
 331         symbols = sorted(list(self.symbols))
 332         n = 0
 333         if len(dims)>0:
 334             for dim in dims:
 335                 if dim in symbols:
 336                     first = symbols.index(dims[0])
 337                     n +=1
 338                 else:
 339                     first = 0
 340         else:
 341             return False
 342         value = bool(libisl.isl_set_involves_dims(islset, libisl.isl_dim_set, first, n))
 343         libisl.isl_set_free(islset)
 344         return value
 345
 346     _RE_COORDINATE = re.compile(r'\((?P<num>\-?\d+)\)(/(?P<den>\d+))?')
 347
 348     def vertices(self):
 349         """
 350         Return a list of vertices for this Polygon.
 351         """
 352         from .polyhedra import Polyhedron
 353         if not self.isbounded():
 354             raise ValueError('domain must be bounded')
 355         islbset = self._toislbasicset(self.equalities, self.inequalities, self.symbols)
 356         vertices = libisl.isl_basic_set_compute_vertices(islbset);
 357         vertices = islhelper.isl_vertices_vertices(vertices)
 358         points = []
 359         for vertex in vertices:
 360             expr = libisl.isl_vertex_get_expr(vertex)
 361             coordinates = []
 362             if islhelper.isl_version < '0.13':
 363                 constraints = islhelper.isl_basic_set_constraints(expr)
 364                 for constraint in constraints:
 365                     constant = libisl.isl_constraint_get_constant_val(constraint)
 366                     constant = islhelper.isl_val_to_int(constant)
 367                     for index, symbol in enumerate(self.symbols):
 368                         coefficient = libisl.isl_constraint_get_coefficient_val(constraint,
 369                             libisl.isl_dim_set, index)
 370                         coefficient = islhelper.isl_val_to_int(coefficient)
 371                         if coefficient != 0:
 372                             coordinate = -Fraction(constant, coefficient)
 373                             coordinates.append((symbol, coordinate))
 374             else:
 375                 string = islhelper.isl_multi_aff_to_str(expr)
 376                 matches = self._RE_COORDINATE.finditer(string)
 377                 for symbol, match in zip(self.symbols, matches):
 378                     numerator = int(match.group('num'))
 379                     denominator = match.group('den')
 380                     denominator = 1 if denominator is None else int(denominator)
 381                     coordinate = Fraction(numerator, denominator)
 382                     coordinates.append((symbol, coordinate))
 383             points.append(Point(coordinates))
 384         return points
 385
 386     def points(self):
 387         """
 388         Returns the points contained in the set.
 389         """
 390         if not self.isbounded():
 391             raise ValueError('domain must be bounded')
 392         from .polyhedra import Universe, Eq
 393         islset = self._toislset(self.polyhedra, self.symbols)
 394         islpoints = islhelper.isl_set_points(islset)
 395         points = []
 396         for islpoint in islpoints:
 397             coordinates = {}
 398             for index, symbol in enumerate(self.symbols):
 399                 coordinate = libisl.isl_point_get_coordinate_val(islpoint,
 400                     libisl.isl_dim_set, index)
 401                 coordinate = islhelper.isl_val_to_int(coordinate)
 402                 coordinates[symbol] = coordinate
 403             points.append(Point(coordinates))
 404         return points
 405
 406     @classmethod
 407     def _polygon_inner_point(cls, points):
 408         symbols = points[0].symbols
 409         coordinates = {symbol: 0 for symbol in symbols}
 410         for point in points:
 411             for symbol, coordinate in point.coordinates():
 412                 coordinates[symbol] += coordinate
 413         for symbol in symbols:
 414             coordinates[symbol] /= len(points)
 415         return Point(coordinates)
 416
 417     @classmethod
 418     def _sort_polygon_2d(cls, points):
 419         if len(points) <= 3:
 420             return points
 421         o = cls._polygon_inner_point(points)
 422         angles = {}
 423         for m in points:
 424             om = Vector(o, m)
 425             dx, dy = (coordinate for symbol, coordinate in om.coordinates())
 426             angle = math.atan2(dy, dx)
 427             angles[m] = angle
 428         return sorted(points, key=angles.get)
 429
 430     @classmethod
 431     def _sort_polygon_3d(cls, points):
 432         if len(points) <= 3:
 433             return points
 434         o = cls._polygon_inner_point(points)
 435         a = points[0]
 436         oa = Vector(o, a)
 437         norm_oa = oa.norm()
 438         for b in points[1:]:
 439             ob = Vector(o, b)
 440             u = oa.cross(ob)
 441             if not u.isnull():
 442                 u = u.asunit()
 443                 break
 444         else:
 445             raise ValueError('degenerate polygon')
 446         angles = {a: 0.}
 447         for m in points[1:]:
 448             om = Vector(o, m)
 449             normprod = norm_oa * om.norm()
 450             cosinus = max(oa.dot(om) / normprod, -1.)
 451             sinus = u.dot(oa.cross(om)) / normprod
 452             angle = math.acos(cosinus)
 453             angle = math.copysign(angle, sinus)
 454             angles[m] = angle
 455         return sorted(points, key=angles.get)
 456
 457     def faces(self):
 458         """
 459         Returns the vertices of the faces of a polyhedra.
 460         """
 461         faces = []
 462         for polyhedron in self.polyhedra:
 463             vertices = polyhedron.vertices()
 464             for constraint in polyhedron.constraints:
 465                 face = []
 466                 for vertex in vertices:
 467                     if constraint.subs(vertex.coordinates()) == 0:
 468                         face.append(vertex)
 469                 if len(face) >= 3:
 470                     faces.append(face)
 471         return faces
 472
 473     def _plot_2d(self, plot=None, **kwargs):
 474         import matplotlib.pyplot as plt
 475         from matplotlib.patches import Polygon
 476         if plot is None:
 477             fig = plt.figure()
 478             plot = fig.add_subplot(1, 1, 1)
 479         xmin, xmax = plot.get_xlim()
 480         ymin, ymax = plot.get_ylim()
 481         for polyhedron in self.polyhedra:
 482             vertices = polyhedron._sort_polygon_2d(polyhedron.vertices())
 483             xys = [tuple(vertex.values()) for vertex in vertices]
 484             xs, ys = zip(*xys)
 485             xmin, xmax = min(xmin, float(min(xs))), max(xmax, float(max(xs)))
 486             ymin, ymax = min(ymin, float(min(ys))), max(ymax, float(max(ys)))
 487             plot.add_patch(Polygon(xys, closed=True, **kwargs))
 488         plot.set_xlim(xmin, xmax)
 489         plot.set_ylim(ymin, ymax)
 490         return plot
 491
 492     def _plot_3d(self, plot=None, **kwargs):
 493         import matplotlib.pyplot as plt
 494         from mpl_toolkits.mplot3d import Axes3D
 495         from mpl_toolkits.mplot3d.art3d import Poly3DCollection
 496         if plot is None:
 497             fig = plt.figure()
 498             axes = Axes3D(fig)
 499         else:
 500             axes = plot
 501         xmin, xmax = axes.get_xlim()
 502         ymin, ymax = axes.get_ylim()
 503         zmin, zmax = axes.get_zlim()
 504         poly_xyzs = []
 505         for vertices in self.faces():
 506             vertices = self._sort_polygon_3d(vertices)
 507             vertices.append(vertices[0])
 508             face_xyzs = [tuple(vertex.values()) for vertex in vertices]
 509             xs, ys, zs = zip(*face_xyzs)
 510             xmin, xmax = min(xmin, float(min(xs))), max(xmax, float(max(xs)))
 511             ymin, ymax = min(ymin, float(min(ys))), max(ymax, float(max(ys)))
 512             zmin, zmax = min(zmin, float(min(zs))), max(zmax, float(max(zs)))
 513             poly_xyzs.append(face_xyzs)
 514         collection = Poly3DCollection(poly_xyzs, **kwargs)
 515         axes.add_collection3d(collection)
 516         axes.set_xlim(xmin, xmax)
 517         axes.set_ylim(ymin, ymax)
 518         axes.set_zlim(zmin, zmax)
 519         return axes
 520
 521
 522     def plot(self, plot=None, **kwargs):
 523         """
 524         Display plot of this set.
 525         """
 526         if not self.isbounded():
 527             raise ValueError('domain must be bounded')
 528         elif self.dimension == 2:
 529             return self._plot_2d(plot=plot, **kwargs)
 530         elif self.dimension == 3:
 531             return self._plot_3d(plot=plot, **kwargs)
 532         else:
 533             raise ValueError('polyhedron must be 2 or 3-dimensional')
 534
 535     def __contains__(self, point):
 536         for polyhedron in self.polyhedra:
 537             if point in polyhedron:
 538                 return True
 539         return False
 540
 541     def subs(self, symbol, expression=None):
 542         """
 543         Subsitute the given value into an expression and return the resulting
 544         expression.
 545         """
 546         polyhedra = [polyhedron.subs(symbol, expression)
 547             for polyhedron in self.polyhedra]
 548         return Domain(*polyhedra)
 549
 550     @classmethod
 551     def _fromislset(cls, islset, symbols):
 552         from .polyhedra import Polyhedron
 553         islset = libisl.isl_set_remove_divs(islset)
 554         islbsets = islhelper.isl_set_basic_sets(islset)
 555         libisl.isl_set_free(islset)
 556         polyhedra = []
 557         for islbset in islbsets:
 558             polyhedron = Polyhedron._fromislbasicset(islbset, symbols)
 559             polyhedra.append(polyhedron)
 560         if len(polyhedra) == 0:
 561             from .polyhedra import Empty
 562             return Empty
 563         elif len(polyhedra) == 1:
 564             return polyhedra[0]
 565         else:
 566             self = object().__new__(Domain)
 567             self._polyhedra = tuple(polyhedra)
 568             self._symbols = cls._xsymbols(polyhedra)
 569             self._dimension = len(self._symbols)
 570             return self
 571
 572     @classmethod
 573     def _toislset(cls, polyhedra, symbols):
 574         polyhedron = polyhedra[0]
 575         islbset = polyhedron._toislbasicset(polyhedron.equalities,
 576             polyhedron.inequalities, symbols)
 577         islset1 = libisl.isl_set_from_basic_set(islbset)
 578         for polyhedron in polyhedra[1:]:
 579             islbset = polyhedron._toislbasicset(polyhedron.equalities,
 580                 polyhedron.inequalities, symbols)
 581             islset2 = libisl.isl_set_from_basic_set(islbset)
 582             islset1 = libisl.isl_set_union(islset1, islset2)
 583         return islset1
 584
 585     @classmethod
 586     def _fromast(cls, node):
 587         from .polyhedra import Polyhedron
 588         if isinstance(node, ast.Module) and len(node.body) == 1:
 589             return cls._fromast(node.body[0])
 590         elif isinstance(node, ast.Expr):
 591             return cls._fromast(node.value)
 592         elif isinstance(node, ast.UnaryOp):
 593             domain = cls._fromast(node.operand)
 594             if isinstance(node.operand, ast.invert):
 595                 return Not(domain)
 596         elif isinstance(node, ast.BinOp):
 597             domain1 = cls._fromast(node.left)
 598             domain2 = cls._fromast(node.right)
 599             if isinstance(node.op, ast.BitAnd):
 600                 return And(domain1, domain2)
 601             elif isinstance(node.op, ast.BitOr):
 602                 return Or(domain1, domain2)
 603         elif isinstance(node, ast.Compare):
 604             equalities = []
 605             inequalities = []
 606             left = Expression._fromast(node.left)
 607             for i in range(len(node.ops)):
 608                 op = node.ops[i]
 609                 right = Expression._fromast(node.comparators[i])
 610                 if isinstance(op, ast.Lt):
 611                     inequalities.append(right - left - 1)
 612                 elif isinstance(op, ast.LtE):
 613                     inequalities.append(right - left)
 614                 elif isinstance(op, ast.Eq):
 615                     equalities.append(left - right)
 616                 elif isinstance(op, ast.GtE):
 617                     inequalities.append(left - right)
 618                 elif isinstance(op, ast.Gt):
 619                     inequalities.append(left - right - 1)
 620                 else:
 621                     break
 622                 left = right
 623             else:
 624                 return Polyhedron(equalities, inequalities)
 625         raise SyntaxError('invalid syntax')
 626
 627     _RE_BRACES = re.compile(r'^\{\s*|\s*\}$')
 628     _RE_EQ = re.compile(r'([^<=>])=([^<=>])')
 629     _RE_AND = re.compile(r'\band\b|,|&&|/\\|∧|∩')
 630     _RE_OR = re.compile(r'\bor\b|;|\|\||\\/|∨|∪')
 631     _RE_NOT = re.compile(r'\bnot\b|!|¬')
 632     _RE_NUM_VAR = Expression._RE_NUM_VAR
 633     _RE_OPERATORS = re.compile(r'(&|\||~)')
 634
 635     @classmethod
 636     def fromstring(cls, string):
 637         # remove curly brackets
 638         string = cls._RE_BRACES.sub(r'', string)
 639         # replace '=' by '=='
 640         string = cls._RE_EQ.sub(r'\1==\2', string)
 641         # replace 'and', 'or', 'not'
 642         string = cls._RE_AND.sub(r' & ', string)
 643         string = cls._RE_OR.sub(r' | ', string)
 644         string = cls._RE_NOT.sub(r' ~', string)
 645         # add implicit multiplication operators, e.g. '5x' -> '5*x'
 646         string = cls._RE_NUM_VAR.sub(r'\1*\2', string)
 647         # add parentheses to force precedence
 648         tokens = cls._RE_OPERATORS.split(string)
 649         for i, token in enumerate(tokens):
 650             if i % 2 == 0:
 651                 token = '({})'.format(token)
 652                 tokens[i] = token
 653         string = ''.join(tokens)
 654         tree = ast.parse(string, 'eval')
 655         return cls._fromast(tree)
 656
 657     def __repr__(self):
 658         assert len(self.polyhedra) >= 2
 659         strings = [repr(polyhedron) for polyhedron in self.polyhedra]
 660         return 'Or({})'.format(', '.join(strings))
 661
 662     def _repr_latex_(self):
 663         strings = []
 664         for polyhedron in self.polyhedra:
 665             strings.append('({})'.format(polyhedron._repr_latex_().strip('$')))
 666         return '${}$'.format(' \\vee '.join(strings))
 667
 668     @classmethod
 669     def fromsympy(cls, expr):
 670         import sympy
 671         from .polyhedra import Lt, Le, Eq, Ne, Ge, Gt
 672         funcmap = {
 673             sympy.And: And, sympy.Or: Or, sympy.Not: Not,
 674             sympy.Lt: Lt, sympy.Le: Le,
 675             sympy.Eq: Eq, sympy.Ne: Ne,
 676             sympy.Ge: Ge, sympy.Gt: Gt,
 677         }
 678         if expr.func in funcmap:
 679             args = [Domain.fromsympy(arg) for arg in expr.args]
 680             return funcmap[expr.func](*args)
 681         elif isinstance(expr, sympy.Expr):
 682             return Expression.fromsympy(expr)
 683         raise ValueError('non-domain expression: {!r}'.format(expr))
 684
 685     def tosympy(self):
 686         import sympy
 687         polyhedra = [polyhedron.tosympy() for polyhedron in polyhedra]
 688         return sympy.Or(*polyhedra)
 689
 690
 691 def And(*domains):
 692     """
 693     Return the intersection of two sets as a new set.
 694     """
 695     if len(domains) == 0:
 696         from .polyhedra import Universe
 697         return Universe
 698     else:
 699         return domains[0].intersection(*domains[1:])
 700
 701 def Or(*domains):
 702     """
 703     Return the union of sets as a new set.
 704     """
 705     if len(domains) == 0:
 706         from .polyhedra import Empty
 707         return Empty
 708     else:
 709         return domains[0].union(*domains[1:])
 710
 711 def Not(domain):
 712     """
 713     Returns the complement of this set.
 714     """
 715     return ~domain