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 + 2y + 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 + 2*y + 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 Rational(self._coefficients.get(symbol, 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         for symbol, coefficient in self._coefficients.items():
 135             yield symbol, Rational(coefficient)
 136
 137     @property
 138     def constant(self):
 139         """
 140         The constant term of the expression.
 141         """
 142         return Rational(self._constant)
 143
 144     @property
 145     def symbols(self):
 146         """
 147         The tuple of symbols present in the expression, sorted according to
 148         Symbol.sortkey().
 149         """
 150         return self._symbols
 151
 152     @property
 153     def dimension(self):
 154         """
 155         The dimension of the expression, i.e. the number of symbols present in
 156         it.
 157         """
 158         return self._dimension
 159
 160     def __hash__(self):
 161         return hash((tuple(self._coefficients.items()), self._constant))
 162
 163     def isconstant(self):
 164         """
 165         Return True if the expression only consists of a constant term. In this
 166         case, it is a Rational instance.
 167         """
 168         return False
 169
 170     def issymbol(self):
 171         """
 172         Return True if an expression only consists of a symbol with coefficient
 173         1. In this case, it is a Symbol instance.
 174         """
 175         return False
 176
 177     def values(self):
 178         """
 179         Iterate over the coefficient values in the expression, and the constant
 180         term.
 181         """
 182         for coefficient in self._coefficients.values():
 183             yield Rational(coefficient)
 184         yield Rational(self._constant)
 185
 186     def __bool__(self):
 187         return True
 188
 189     def __pos__(self):
 190         return self
 191
 192     def __neg__(self):
 193         return self * -1
 194
 195     @_polymorphic
 196     def __add__(self, other):
 197         """
 198         Return the sum of two linear expressions.
 199         """
 200         coefficients = defaultdict(Fraction, self._coefficients)
 201         for symbol, coefficient in other._coefficients.items():
 202             coefficients[symbol] += coefficient
 203         constant = self._constant + other._constant
 204         return LinExpr(coefficients, constant)
 205
 206     __radd__ = __add__
 207
 208     @_polymorphic
 209     def __sub__(self, other):
 210         """
 211         Return the difference between two linear expressions.
 212         """
 213         coefficients = defaultdict(Fraction, self._coefficients)
 214         for symbol, coefficient in other._coefficients.items():
 215             coefficients[symbol] -= coefficient
 216         constant = self._constant - other._constant
 217         return LinExpr(coefficients, constant)
 218
 219     @_polymorphic
 220     def __rsub__(self, other):
 221         return other - self
 222
 223     def __mul__(self, other):
 224         """
 225         Return the product of the linear expression by a rational.
 226         """
 227         if isinstance(other, numbers.Rational):
 228             coefficients = ((symbol, coefficient * other)
 229                 for symbol, coefficient in self._coefficients.items())
 230             constant = self._constant * other
 231             return LinExpr(coefficients, constant)
 232         return NotImplemented
 233
 234     __rmul__ = __mul__
 235
 236     def __truediv__(self, other):
 237         """
 238         Return the quotient of the linear expression by a rational.
 239         """
 240         if isinstance(other, numbers.Rational):
 241             coefficients = ((symbol, coefficient / other)
 242                 for symbol, coefficient in self._coefficients.items())
 243             constant = self._constant / other
 244             return LinExpr(coefficients, constant)
 245         return NotImplemented
 246
 247     @_polymorphic
 248     def __eq__(self, other):
 249         """
 250         Test whether two linear expressions are equal.
 251         """
 252         return isinstance(other, LinExpr) and \
 253             self._coefficients == other._coefficients and \
 254             self._constant == other._constant
 255
 256     def __le__(self, other):
 257         from .polyhedra import Le
 258         return Le(self, other)
 259
 260     def __lt__(self, other):
 261         from .polyhedra import Lt
 262         return Lt(self, other)
 263
 264     def __ge__(self, other):
 265         from .polyhedra import Ge
 266         return Ge(self, other)
 267
 268     def __gt__(self, other):
 269         from .polyhedra import Gt
 270         return Gt(self, other)
 271
 272     def scaleint(self):
 273         """
 274         Return the expression multiplied by its lowest common denominator to
 275         make all values integer.
 276         """
 277         lcm = functools.reduce(lambda a, b: a*b // gcd(a, b),
 278             [value.denominator for value in self.values()])
 279         return self * lcm
 280
 281     def subs(self, symbol, expression=None):
 282         """
 283         Substitute the given symbol by an expression and return the resulting
 284         expression. Raise TypeError if the resulting expression is not linear.
 285
 286         >>> x, y = symbols('x y')
 287         >>> e = x + 2*y + 1
 288         >>> e.subs(y, x - 1)
 289         3*x - 1
 290
 291         To perform multiple substitutions at once, pass a sequence or a
 292         dictionary of (old, new) pairs to subs.
 293
 294         >>> e.subs({x: y, y: x})
 295         2*x + y + 1
 296         """
 297         if expression is None:
 298             if isinstance(symbol, Mapping):
 299                 symbol = symbol.items()
 300             substitutions = symbol
 301         else:
 302             substitutions = [(symbol, expression)]
 303         result = self
 304         for symbol, expression in substitutions:
 305             if not isinstance(symbol, Symbol):
 306                 raise TypeError('symbols must be Symbol instances')
 307             coefficients = [(othersymbol, coefficient)
 308                 for othersymbol, coefficient in result._coefficients.items()
 309                 if othersymbol != symbol]
 310             coefficient = result._coefficients.get(symbol, 0)
 311             constant = result._constant
 312             result = LinExpr(coefficients, constant) + 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         expr = cls._fromast(tree)
 352         if not isinstance(expr, cls):
 353             raise SyntaxError('invalid syntax')
 354         return expr
 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 _repr_latex_(self):
 381         string = ''
 382         for i, (symbol, coefficient) in enumerate(self.coefficients()):
 383             if coefficient == 1:
 384                 if i != 0:
 385                     string += ' + '
 386             elif coefficient == -1:
 387                 string += '-' if i == 0 else ' - '
 388             elif i == 0:
 389                 string += '{}'.format(coefficient._repr_latex_().strip('$'))
 390             elif coefficient > 0:
 391                 string += ' + {}'.format(coefficient._repr_latex_().strip('$'))
 392             elif coefficient < 0:
 393                 string += ' - {}'.format((-coefficient)._repr_latex_().strip('$'))
 394             string += '{}'.format(symbol._repr_latex_().strip('$'))
 395         constant = self.constant
 396         if len(string) == 0:
 397             string += '{}'.format(constant._repr_latex_().strip('$'))
 398         elif constant > 0:
 399             string += ' + {}'.format(constant._repr_latex_().strip('$'))
 400         elif constant < 0:
 401             string += ' - {}'.format((-constant)._repr_latex_().strip('$'))
 402         return '$${}$$'.format(string)
 403
 404     def _parenstr(self, always=False):
 405         string = str(self)
 406         if not always and (self.isconstant() or self.issymbol()):
 407             return string
 408         else:
 409             return '({})'.format(string)
 410
 411     @classmethod
 412     def fromsympy(cls, expr):
 413         """
 414         Create a linear expression from a sympy expression. Raise TypeError is
 415         the sympy expression is not linear.
 416         """
 417         import sympy
 418         coefficients = []
 419         constant = 0
 420         for symbol, coefficient in expr.as_coefficients_dict().items():
 421             coefficient = Fraction(coefficient.p, coefficient.q)
 422             if symbol == sympy.S.One:
 423                 constant = coefficient
 424             elif isinstance(symbol, sympy.Dummy):
 425                 # we cannot properly convert dummy symbols
 426                 raise TypeError('cannot convert dummy symbols')
 427             elif isinstance(symbol, sympy.Symbol):
 428                 symbol = Symbol(symbol.name)
 429                 coefficients.append((symbol, coefficient))
 430             else:
 431                 raise TypeError('non-linear expression: {!r}'.format(expr))
 432         expr = LinExpr(coefficients, constant)
 433         if not isinstance(expr, cls):
 434             raise TypeError('cannot convert to a {} instance'.format(cls.__name__))
 435         return expr
 436
 437     def tosympy(self):
 438         """
 439         Convert the linear expression to a sympy expression.
 440         """
 441         import sympy
 442         expr = 0
 443         for symbol, coefficient in self.coefficients():
 444             term = coefficient * sympy.Symbol(symbol.name)
 445             expr += term
 446         expr += self.constant
 447         return expr
 448
 449
 450 class Symbol(LinExpr):
 451     """
 452     Symbols are the basic components to build expressions and constraints.
 453     They correspond to mathematical variables. Symbols are instances of
 454     class LinExpr and inherit its functionalities.
 455
 456     Two instances of Symbol are equal if they have the same name.
 457     """
 458
 459     def __new__(cls, name):
 460         """
 461         Return a symbol with the name string given in argument.
 462         """
 463         if not isinstance(name, str):
 464             raise TypeError('name must be a string')
 465         node = ast.parse(name)
 466         try:
 467             name = node.body[0].value.id
 468         except (AttributeError, SyntaxError):
 469             raise SyntaxError('invalid syntax')
 470         self = object().__new__(cls)
 471         self._name = name
 472         self._coefficients = {self: Fraction(1)}
 473         self._constant = Fraction(0)
 474         self._symbols = (self,)
 475         self._dimension = 1
 476         return self
 477
 478     @property
 479     def name(self):
 480         """
 481         The name of the symbol.
 482         """
 483         return self._name
 484
 485     def __hash__(self):
 486         return hash(self.sortkey())
 487
 488     def sortkey(self):
 489         """
 490         Return a sorting key for the symbol. It is useful to sort a list of
 491         symbols in a consistent order, as comparison functions are overridden
 492         (see the documentation of class LinExpr).
 493
 494         >>> sort(symbols, key=Symbol.sortkey)
 495         """
 496         return self.name,
 497
 498     def issymbol(self):
 499         return True
 500
 501     def __eq__(self, other):
 502         return self.sortkey() == other.sortkey()
 503
 504     def asdummy(self):
 505         """
 506         Return a new Dummy symbol instance with the same name.
 507         """
 508         return Dummy(self.name)
 509
 510     def __repr__(self):
 511         return self.name
 512
 513     def _repr_latex_(self):
 514         return '$${}$$'.format(self.name)
 515
 516
 517 def symbols(names):
 518     """
 519     This function returns a tuple of symbols whose names are taken from a comma
 520     or whitespace delimited string, or a sequence of strings. It is useful to
 521     define several symbols at once.
 522
 523     >>> x, y = symbols('x y')
 524     >>> x, y = symbols('x, y')
 525     >>> x, y = symbols(['x', 'y'])
 526     """
 527     if isinstance(names, str):
 528         names = names.replace(',', ' ').split()
 529     return tuple(Symbol(name) for name in names)
 530
 531
 532 class Dummy(Symbol):
 533     """
 534     A variation of Symbol in which all symbols are unique and identified by
 535     an internal count index. If a name is not supplied then a string value
 536     of the count index will be used. This is useful when a unique, temporary
 537     variable is needed and the name of the variable used in the expression
 538     is not important.
 539
 540     Unlike Symbol, Dummy instances with the same name are not equal:
 541
 542     >>> x = Symbol('x')
 543     >>> x1, x2 = Dummy('x'), Dummy('x')
 544     >>> x == x1
 545     False
 546     >>> x1 == x2
 547     False
 548     >>> x1 == x1
 549     True
 550     """
 551
 552     _count = 0
 553
 554     def __new__(cls, name=None):
 555         """
 556         Return a fresh dummy symbol with the name string given in argument.
 557         """
 558         if name is None:
 559             name = 'Dummy_{}'.format(Dummy._count)
 560         elif not isinstance(name, str):
 561             raise TypeError('name must be a string')
 562         self = object().__new__(cls)
 563         self._index = Dummy._count
 564         self._name = name.strip()
 565         self._coefficients = {self: Fraction(1)}
 566         self._constant = Fraction(0)
 567         self._symbols = (self,)
 568         self._dimension = 1
 569         Dummy._count += 1
 570         return self
 571
 572     def __hash__(self):
 573         return hash(self.sortkey())
 574
 575     def sortkey(self):
 576         return self._name, self._index
 577
 578     def __repr__(self):
 579         return '_{}'.format(self.name)
 580
 581     def _repr_latex_(self):
 582         return '$${}_{{{}}}$$'.format(self.name, self._index)
 583
 584
 585 class Rational(LinExpr, Fraction):
 586     """
 587     A particular case of linear expressions are rational values, i.e. linear
 588     expressions consisting only of a constant term, with no symbol. They are
 589     implemented by the Rational class, that inherits from both LinExpr and
 590     fractions.Fraction classes.
 591     """
 592
 593     def __new__(cls, numerator=0, denominator=None):
 594         self = object().__new__(cls)
 595         self._coefficients = {}
 596         self._constant = Fraction(numerator, denominator)
 597         self._symbols = ()
 598         self._dimension = 0
 599         self._numerator = self._constant.numerator
 600         self._denominator = self._constant.denominator
 601         return self
 602
 603     def __hash__(self):
 604         return Fraction.__hash__(self)
 605
 606     @property
 607     def constant(self):
 608         return self
 609
 610     def isconstant(self):
 611         return True
 612
 613     def __bool__(self):
 614         return Fraction.__bool__(self)
 615
 616     def __repr__(self):
 617         if self.denominator == 1:
 618             return '{!r}'.format(self.numerator)
 619         else:
 620             return '{!r}/{!r}'.format(self.numerator, self.denominator)
 621
 622     def _repr_latex_(self):
 623         if self.denominator == 1:
 624             return '$${}$$'.format(self.numerator)
 625         elif self.numerator < 0:
 626             return '$$-\\frac{{{}}}{{{}}}$$'.format(-self.numerator,
 627                 self.denominator)
 628         else:
 629             return '$$\\frac{{{}}}{{{}}}$$'.format(self.numerator,
 630                 self.denominator)