linpy/linexprs.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 numbers
  21 import re
  22
  23 from collections import OrderedDict, defaultdict, Mapping
  24 from fractions import Fraction, gcd
  25
  26
  27 __all__ = [
  28     'LinExpr',
  29     'Symbol', 'Dummy', 'symbols',
  30     'Rational',
  31 ]
  32
  33
  34 def _polymorphic(func):
  35     @functools.wraps(func)
  36     def wrapper(left, right):
  37         if isinstance(right, LinExpr):
  38             return func(left, right)
  39         elif isinstance(right, numbers.Rational):
  40             right = Rational(right)
  41             return func(left, right)
  42         return NotImplemented
  43     return wrapper
  44
  45
  46 class LinExpr:
  47     """
  48     A linear expression consists of a list of coefficient-variable pairs
  49     that capture the linear terms, plus a constant term. Linear expressions
  50     are used to build constraints. They are temporary objects that typically
  51     have short lifespans.
  52
  53     Linear expressions are generally built using overloaded operators. For
  54     example, if x is a Symbol, then x + 1 is an instance of LinExpr.
  55
  56     LinExpr instances are hashable, and should be treated as immutable.
  57     """
  58
  59     def __new__(cls, coefficients=None, constant=0):
  60         """
  61         Return a linear expression from a dictionary or a sequence, that maps
  62         symbols to their coefficients, and a constant term. The coefficients and
  63         the constant term must be rational numbers.
  64
  65         For example, the linear expression x + 2*y + 1 can be constructed using
  66         one of the following instructions:
  67
  68         >>> x, y = symbols('x y')
  69         >>> LinExpr({x: 1, y: 2}, 1)
  70         >>> LinExpr([(x, 1), (y, 2)], 1)
  71
  72         However, it may be easier to use overloaded operators:
  73
  74         >>> x, y = symbols('x y')
  75         >>> x + 2*y + 1
  76
  77         Alternatively, linear expressions can be constructed from a string:
  78
  79         >>> LinExpr('x + 2y + 1')
  80
  81         A linear expression with a single symbol of coefficient 1 and no
  82         constant term is automatically subclassed as a Symbol instance. A linear
  83         expression with no symbol, only a constant term, is automatically
  84         subclassed as a Rational instance.
  85         """
  86         if isinstance(coefficients, str):
  87             if constant != 0:
  88                 raise TypeError('too many arguments')
  89             return LinExpr.fromstring(coefficients)
  90         if coefficients is None:
  91             return Rational(constant)
  92         if isinstance(coefficients, Mapping):
  93             coefficients = coefficients.items()
  94         coefficients = list(coefficients)
  95         for symbol, coefficient in coefficients:
  96             if not isinstance(symbol, Symbol):
  97                 raise TypeError('symbols must be Symbol instances')
  98             if not isinstance(coefficient, numbers.Rational):
  99                 raise TypeError('coefficients must be rational numbers')
 100         if not isinstance(constant, numbers.Rational):
 101             raise TypeError('constant must be a rational number')
 102         if len(coefficients) == 0:
 103             return Rational(constant)
 104         if len(coefficients) == 1 and constant == 0:
 105             symbol, coefficient = coefficients[0]
 106             if coefficient == 1:
 107                 return symbol
 108         coefficients = [(symbol, Fraction(coefficient))
 109             for symbol, coefficient in coefficients if coefficient != 0]
 110         coefficients.sort(key=lambda item: item[0].sortkey())
 111         self = object().__new__(cls)
 112         self._coefficients = OrderedDict(coefficients)
 113         self._constant = Fraction(constant)
 114         self._symbols = tuple(self._coefficients)
 115         self._dimension = len(self._symbols)
 116         return self
 117
 118     def coefficient(self, symbol):
 119         """
 120         Return the coefficient value of the given symbol, or 0 if the symbol
 121         does not appear in the expression.
 122         """
 123         if not isinstance(symbol, Symbol):
 124             raise TypeError('symbol must be a Symbol instance')
 125         return self._coefficients.get(symbol, Fraction(0))
 126
 127     __getitem__ = coefficient
 128
 129     def coefficients(self):
 130         """
 131         Iterate over the pairs (symbol, value) of linear terms in the
 132         expression. The constant term is ignored.
 133         """
 134         yield from self._coefficients.items()
 135
 136     @property
 137     def constant(self):
 138         """
 139         The constant term of the expression.
 140         """
 141         return self._constant
 142
 143     @property
 144     def symbols(self):
 145         """
 146         The tuple of symbols present in the expression, sorted according to
 147         Symbol.sortkey().
 148         """
 149         return self._symbols
 150
 151     @property
 152     def dimension(self):
 153         """
 154         The dimension of the expression, i.e. the number of symbols present in
 155         it.
 156         """
 157         return self._dimension
 158
 159     def __hash__(self):
 160         return hash((tuple(self._coefficients.items()), self._constant))
 161
 162     def isconstant(self):
 163         """
 164         Return True if the expression only consists of a constant term. In this
 165         case, it is a Rational instance.
 166         """
 167         return False
 168
 169     def issymbol(self):
 170         """
 171         Return True if an expression only consists of a symbol with coefficient
 172         1. In this case, it is a Symbol instance.
 173         """
 174         return False
 175
 176     def values(self):
 177         """
 178         Iterate over the coefficient values in the expression, and the constant
 179         term.
 180         """
 181         yield from self._coefficients.values()
 182         yield self._constant
 183
 184     def __bool__(self):
 185         return True
 186
 187     def __pos__(self):
 188         return self
 189
 190     def __neg__(self):
 191         return self * -1
 192
 193     @_polymorphic
 194     def __add__(self, other):
 195         """
 196         Return the sum of two linear expressions.
 197         """
 198         coefficients = defaultdict(Fraction, self._coefficients)
 199         for symbol, coefficient in other._coefficients.items():
 200             coefficients[symbol] += coefficient
 201         constant = self._constant + other._constant
 202         return LinExpr(coefficients, constant)
 203
 204     __radd__ = __add__
 205
 206     @_polymorphic
 207     def __sub__(self, other):
 208         """
 209         Return the difference between two linear expressions.
 210         """
 211         coefficients = defaultdict(Fraction, self._coefficients)
 212         for symbol, coefficient in other._coefficients.items():
 213             coefficients[symbol] -= coefficient
 214         constant = self._constant - other._constant
 215         return LinExpr(coefficients, constant)
 216
 217     @_polymorphic
 218     def __rsub__(self, other):
 219         return other - self
 220
 221     def __mul__(self, other):
 222         """
 223         Return the product of the linear expression by a rational.
 224         """
 225         if isinstance(other, numbers.Rational):
 226             coefficients = ((symbol, coefficient * other)
 227                 for symbol, coefficient in self._coefficients.items())
 228             constant = self._constant * other
 229             return LinExpr(coefficients, constant)
 230         return NotImplemented
 231
 232     __rmul__ = __mul__
 233
 234     def __truediv__(self, other):
 235         """
 236         Return the quotient of the linear expression by a rational.
 237         """
 238         if isinstance(other, numbers.Rational):
 239             coefficients = ((symbol, coefficient / other)
 240                 for symbol, coefficient in self._coefficients.items())
 241             constant = self._constant / other
 242             return LinExpr(coefficients, constant)
 243         return NotImplemented
 244
 245     @_polymorphic
 246     def __eq__(self, other):
 247         """
 248         Test whether two linear expressions are equal. Unlike methods
 249         LinExpr.__lt__(), LinExpr.__le__(), LinExpr.__ge__(), LinExpr.__gt__(),
 250         the result is a boolean value, not a polyhedron. To express that two
 251         linear expressions are equal or not equal, use functions Eq() and Ne()
 252         instead.
 253         """
 254         return self._coefficients == other._coefficients and \
 255             self._constant == other._constant
 256
 257     @_polymorphic
 258     def __lt__(self, other):
 259         from .polyhedra import Polyhedron
 260         return Polyhedron([], [other - self - 1])
 261
 262     @_polymorphic
 263     def __le__(self, other):
 264         from .polyhedra import Polyhedron
 265         return Polyhedron([], [other - self])
 266
 267     @_polymorphic
 268     def __ge__(self, other):
 269         from .polyhedra import Polyhedron
 270         return Polyhedron([], [self - other])
 271
 272     @_polymorphic
 273     def __gt__(self, other):
 274         from .polyhedra import Polyhedron
 275         return Polyhedron([], [self - other - 1])
 276
 277     def scaleint(self):
 278         """
 279         Return the expression multiplied by its lowest common denominator to
 280         make all values integer.
 281         """
 282         lcd = functools.reduce(lambda a, b: a*b // gcd(a, b),
 283             [value.denominator for value in self.values()])
 284         return self * lcd
 285
 286     def subs(self, symbol, expression=None):
 287         """
 288         Substitute the given symbol by an expression and return the resulting
 289         expression. Raise TypeError if the resulting expression is not linear.
 290
 291         >>> x, y = symbols('x y')
 292         >>> e = x + 2*y + 1
 293         >>> e.subs(y, x - 1)
 294         3*x - 1
 295
 296         To perform multiple substitutions at once, pass a sequence or a
 297         dictionary of (old, new) pairs to subs.
 298
 299         >>> e.subs({x: y, y: x})
 300         2*x + y + 1
 301         """
 302         if expression is None:
 303             substitutions = dict(symbol)
 304         else:
 305             substitutions = {symbol: expression}
 306         for symbol in substitutions:
 307             if not isinstance(symbol, Symbol):
 308                 raise TypeError('symbols must be Symbol instances')
 309         result = Rational(self._constant)
 310         for symbol, coefficient in self._coefficients.items():
 311             expression = substitutions.get(symbol, symbol)
 312             result += coefficient * expression
 313         return result
 314
 315     @classmethod
 316     def _fromast(cls, node):
 317         if isinstance(node, ast.Module) and len(node.body) == 1:
 318             return cls._fromast(node.body[0])
 319         elif isinstance(node, ast.Expr):
 320             return cls._fromast(node.value)
 321         elif isinstance(node, ast.Name):
 322             return Symbol(node.id)
 323         elif isinstance(node, ast.Num):
 324             return Rational(node.n)
 325         elif isinstance(node, ast.UnaryOp) and isinstance(node.op, ast.USub):
 326             return -cls._fromast(node.operand)
 327         elif isinstance(node, ast.BinOp):
 328             left = cls._fromast(node.left)
 329             right = cls._fromast(node.right)
 330             if isinstance(node.op, ast.Add):
 331                 return left + right
 332             elif isinstance(node.op, ast.Sub):
 333                 return left - right
 334             elif isinstance(node.op, ast.Mult):
 335                 return left * right
 336             elif isinstance(node.op, ast.Div):
 337                 return left / right
 338         raise SyntaxError('invalid syntax')
 339
 340     _RE_NUM_VAR = re.compile(r'(\d+|\))\s*([^\W\d]\w*|\()')
 341
 342     @classmethod
 343     def fromstring(cls, string):
 344         """
 345         Create an expression from a string. Raise SyntaxError if the string is
 346         not properly formatted.
 347         """
 348         # Add implicit multiplication operators, e.g. '5x' -> '5*x'.
 349         string = LinExpr._RE_NUM_VAR.sub(r'\1*\2', string)
 350         tree = ast.parse(string, 'eval')
 351         expression = cls._fromast(tree)
 352         if not isinstance(expression, cls):
 353             raise SyntaxError('invalid syntax')
 354         return expression
 355
 356     def __repr__(self):
 357         string = ''
 358         for i, (symbol, coefficient) in enumerate(self.coefficients()):
 359             if coefficient == 1:
 360                 if i != 0:
 361                     string += ' + '
 362             elif coefficient == -1:
 363                 string += '-' if i == 0 else ' - '
 364             elif i == 0:
 365                 string += '{}*'.format(coefficient)
 366             elif coefficient > 0:
 367                 string += ' + {}*'.format(coefficient)
 368             else:
 369                 string += ' - {}*'.format(-coefficient)
 370             string += '{}'.format(symbol)
 371         constant = self.constant
 372         if len(string) == 0:
 373             string += '{}'.format(constant)
 374         elif constant > 0:
 375             string += ' + {}'.format(constant)
 376         elif constant < 0:
 377             string += ' - {}'.format(-constant)
 378         return string
 379
 380     def _parenstr(self, always=False):
 381         string = str(self)
 382         if not always and (self.isconstant() or self.issymbol()):
 383             return string
 384         else:
 385             return '({})'.format(string)
 386
 387     @classmethod
 388     def fromsympy(cls, expression):
 389         """
 390         Create a linear expression from a SymPy expression. Raise TypeError is
 391         the sympy expression is not linear.
 392         """
 393         import sympy
 394         coefficients = []
 395         constant = 0
 396         for symbol, coefficient in expression.as_coefficients_dict().items():
 397             coefficient = Fraction(coefficient.p, coefficient.q)
 398             if symbol == sympy.S.One:
 399                 constant = coefficient
 400             elif isinstance(symbol, sympy.Dummy):
 401                 # We cannot properly convert dummy symbols with respect to
 402                 # symbol equalities.
 403                 raise TypeError('cannot convert dummy symbols')
 404             elif isinstance(symbol, sympy.Symbol):
 405                 symbol = Symbol(symbol.name)
 406                 coefficients.append((symbol, coefficient))
 407             else:
 408                 raise TypeError('non-linear expression: {!r}'.format(expression))
 409         expression = LinExpr(coefficients, constant)
 410         if not isinstance(expression, cls):
 411             raise TypeError('cannot convert to a {} instance'.format(cls.__name__))
 412         return expression
 413
 414     def tosympy(self):
 415         """
 416         Convert the linear expression to a SymPy expression.
 417         """
 418         import sympy
 419         expression = 0
 420         for symbol, coefficient in self.coefficients():
 421             term = coefficient * sympy.Symbol(symbol.name)
 422             expression += term
 423         expression += self.constant
 424         return expression
 425
 426
 427 class Symbol(LinExpr):
 428     """
 429     Symbols are the basic components to build expressions and constraints.
 430     They correspond to mathematical variables. Symbols are instances of
 431     class LinExpr and inherit its functionalities.
 432
 433     Two instances of Symbol are equal if they have the same name.
 434     """
 435
 436     __slots__ = (
 437         '_name',
 438         '_constant',
 439         '_symbols',
 440         '_dimension',
 441     )
 442
 443     def __new__(cls, name):
 444         """
 445         Return a symbol with the name string given in argument.
 446         """
 447         if not isinstance(name, str):
 448             raise TypeError('name must be a string')
 449         node = ast.parse(name)
 450         try:
 451             name = node.body[0].value.id
 452         except (AttributeError, SyntaxError):
 453             raise SyntaxError('invalid syntax')
 454         self = object().__new__(cls)
 455         self._name = name
 456         self._constant = Fraction(0)
 457         self._symbols = (self,)
 458         self._dimension = 1
 459         return self
 460
 461     @property
 462     def _coefficients(self):
 463         # This is not implemented as an attribute, because __hash__ is not
 464         # callable in __new__ in class Dummy.
 465         return {self: Fraction(1)}
 466
 467     @property
 468     def name(self):
 469         """
 470         The name of the symbol.
 471         """
 472         return self._name
 473
 474     def __hash__(self):
 475         return hash(self.sortkey())
 476
 477     def sortkey(self):
 478         """
 479         Return a sorting key for the symbol. It is useful to sort a list of
 480         symbols in a consistent order, as comparison functions are overridden
 481         (see the documentation of class LinExpr).
 482
 483         >>> sort(symbols, key=Symbol.sortkey)
 484         """
 485         return self.name,
 486
 487     def issymbol(self):
 488         return True
 489
 490     def __eq__(self, other):
 491         if isinstance(other, Symbol):
 492             return self.sortkey() == other.sortkey()
 493         return NotImplemented
 494
 495     def asdummy(self):
 496         """
 497         Return a new Dummy symbol instance with the same name.
 498         """
 499         return Dummy(self.name)
 500
 501     def __repr__(self):
 502         return self.name
 503
 504
 505 def symbols(names):
 506     """
 507     This function returns a tuple of symbols whose names are taken from a comma
 508     or whitespace delimited string, or a sequence of strings. It is useful to
 509     define several symbols at once.
 510
 511     >>> x, y = symbols('x y')
 512     >>> x, y = symbols('x, y')
 513     >>> x, y = symbols(['x', 'y'])
 514     """
 515     if isinstance(names, str):
 516         names = names.replace(',', ' ').split()
 517     return tuple(Symbol(name) for name in names)
 518
 519
 520 class Dummy(Symbol):
 521     """
 522     A variation of Symbol in which all symbols are unique and identified by
 523     an internal count index. If a name is not supplied then a string value
 524     of the count index will be used. This is useful when a unique, temporary
 525     variable is needed and the name of the variable used in the expression
 526     is not important.
 527
 528     Unlike Symbol, Dummy instances with the same name are not equal:
 529
 530     >>> x = Symbol('x')
 531     >>> x1, x2 = Dummy('x'), Dummy('x')
 532     >>> x == x1
 533     False
 534     >>> x1 == x2
 535     False
 536     >>> x1 == x1
 537     True
 538     """
 539
 540     _count = 0
 541
 542     def __new__(cls, name=None):
 543         """
 544         Return a fresh dummy symbol with the name string given in argument.
 545         """
 546         if name is None:
 547             name = 'Dummy_{}'.format(Dummy._count)
 548         self = super().__new__(cls, name)
 549         self._index = Dummy._count
 550         Dummy._count += 1
 551         return self
 552
 553     def __hash__(self):
 554         return hash(self.sortkey())
 555
 556     def sortkey(self):
 557         return self._name, self._index
 558
 559     def __repr__(self):
 560         return '_{}'.format(self.name)
 561
 562
 563 class Rational(LinExpr, Fraction):
 564     """
 565     A particular case of linear expressions are rational values, i.e. linear
 566     expressions consisting only of a constant term, with no symbol. They are
 567     implemented by the Rational class, that inherits from both LinExpr and
 568     fractions.Fraction classes.
 569     """
 570
 571     __slots__ = (
 572         '_coefficients',
 573         '_constant',
 574         '_symbols',
 575         '_dimension',
 576     ) + Fraction.__slots__
 577
 578     def __new__(cls, numerator=0, denominator=None):
 579         self = object().__new__(cls)
 580         self._coefficients = {}
 581         self._constant = Fraction(numerator, denominator)
 582         self._symbols = ()
 583         self._dimension = 0
 584         self._numerator = self._constant.numerator
 585         self._denominator = self._constant.denominator
 586         return self
 587
 588     def __hash__(self):
 589         return Fraction.__hash__(self)
 590
 591     @property
 592     def constant(self):
 593         return self
 594
 595     def isconstant(self):
 596         return True
 597
 598     def __bool__(self):
 599         return Fraction.__bool__(self)
 600
 601     def __repr__(self):
 602         if self.denominator == 1:
 603             return '{!r}'.format(self.numerator)
 604         else:
 605             return '{!r}/{!r}'.format(self.numerator, self.denominator)