1 import ast
2 import functools
3 import numbers
4 import re
6 from fractions import Fraction, gcd
8 from pypol import isl
9 from pypol.isl import libisl
12 __all__ = [
13 'Expression', 'Constant', 'Symbol', 'symbols',
14 'eq', 'le', 'lt', 'ge', 'gt',
15 'Polyhedron',
16 'Empty', 'Universe'
17 ]
20 def _polymorphic_method(func):
21 @functools.wraps(func)
22 def wrapper(a, b):
23 if isinstance(b, Expression):
24 return func(a, b)
25 if isinstance(b, numbers.Rational):
26 b = Constant(b)
27 return func(a, b)
28 return NotImplemented
29 return wrapper
31 def _polymorphic_operator(func):
32 # A polymorphic operator should call a polymorphic method, hence we just
33 # have to test the left operand.
34 @functools.wraps(func)
35 def wrapper(a, b):
36 if isinstance(a, numbers.Rational):
37 a = Constant(a)
38 return func(a, b)
39 elif isinstance(a, Expression):
40 return func(a, b)
41 raise TypeError('arguments must be linear expressions')
42 return wrapper
45 _main_ctx = isl.Context()
48 class Expression:
49 """
50 This class implements linear expressions.
51 """
53 __slots__ = (
54 '_coefficients',
55 '_constant',
56 '_symbols',
57 '_dimension'
58 )
60 def __new__(cls, coefficients=None, constant=0):
61 if isinstance(coefficients, str):
62 if constant:
63 raise TypeError('too many arguments')
64 return cls.fromstring(coefficients)
65 if isinstance(coefficients, dict):
66 coefficients = coefficients.items()
67 if coefficients is None:
68 return Constant(constant)
69 coefficients = [(symbol, coefficient)
70 for symbol, coefficient in coefficients if coefficient != 0]
71 if len(coefficients) == 0:
72 return Constant(constant)
73 elif len(coefficients) == 1 and constant == 0:
74 symbol, coefficient = coefficients[0]
75 if coefficient == 1:
76 return Symbol(symbol)
77 self = object().__new__(cls)
78 self._coefficients = {}
79 for symbol, coefficient in coefficients:
80 if isinstance(symbol, Symbol):
81 symbol = symbol.name
82 elif not isinstance(symbol, str):
83 raise TypeError('symbols must be strings or Symbol instances')
84 if isinstance(coefficient, Constant):
85 coefficient = coefficient.constant
86 if not isinstance(coefficient, numbers.Rational):
87 raise TypeError('coefficients must be rational numbers or Constant instances')
88 self._coefficients[symbol] = coefficient
89 if isinstance(constant, Constant):
90 constant = constant.constant
91 if not isinstance(constant, numbers.Rational):
92 raise TypeError('constant must be a rational number or a Constant instance')
93 self._constant = constant
94 self._symbols = tuple(sorted(self._coefficients))
95 self._dimension = len(self._symbols)
96 return self
98 @classmethod
99 def _fromast(cls, node):
100 if isinstance(node, ast.Module):
101 assert len(node.body) == 1
102 return cls._fromast(node.body[0])
103 elif isinstance(node, ast.Expr):
104 return cls._fromast(node.value)
105 elif isinstance(node, ast.Name):
106 return Symbol(node.id)
107 elif isinstance(node, ast.Num):
108 return Constant(node.n)
109 elif isinstance(node, ast.UnaryOp):
110 if isinstance(node.op, ast.USub):
111 return -cls._fromast(node.operand)
112 elif isinstance(node, ast.BinOp):
113 left = cls._fromast(node.left)
114 right = cls._fromast(node.right)
116 return left + right
117 elif isinstance(node.op, ast.Sub):
118 return left - right
119 elif isinstance(node.op, ast.Mult):
120 return left * right
121 elif isinstance(node.op, ast.Div):
122 return left / right
123 raise SyntaxError('invalid syntax')
125 @classmethod
126 def fromstring(cls, string):
127 string = re.sub(r'(\d+|\))\s*([^\W\d_]\w*|\()', r'\1*\2', string)
128 tree = ast.parse(string, 'eval')
129 return cls._fromast(tree)
131 @property
132 def symbols(self):
133 return self._symbols
135 @property
136 def dimension(self):
137 return self._dimension
139 def coefficient(self, symbol):
140 if isinstance(symbol, Symbol):
141 symbol = str(symbol)
142 elif not isinstance(symbol, str):
143 raise TypeError('symbol must be a string or a Symbol instance')
144 try:
145 return self._coefficients[symbol]
146 except KeyError:
147 return 0
149 __getitem__ = coefficient
151 def coefficients(self):
152 for symbol in self.symbols:
153 yield symbol, self.coefficient(symbol)
155 @property
156 def constant(self):
157 return self._constant
159 def isconstant(self):
160 return False
162 def values(self):
163 for symbol in self.symbols:
164 yield self.coefficient(symbol)
165 yield self.constant
167 def issymbol(self):
168 return False
170 def __bool__(self):
171 return True
173 def __pos__(self):
174 return self
176 def __neg__(self):
177 return self * -1
179 @_polymorphic_method
181 coefficients = dict(self.coefficients())
182 for symbol, coefficient in other.coefficients():
183 if symbol in coefficients:
184 coefficients[symbol] += coefficient
185 else:
186 coefficients[symbol] = coefficient
187 constant = self.constant + other.constant
188 return Expression(coefficients, constant)
192 @_polymorphic_method
193 def __sub__(self, other):
194 coefficients = dict(self.coefficients())
195 for symbol, coefficient in other.coefficients():
196 if symbol in coefficients:
197 coefficients[symbol] -= coefficient
198 else:
199 coefficients[symbol] = -coefficient
200 constant = self.constant - other.constant
201 return Expression(coefficients, constant)
203 def __rsub__(self, other):
204 return -(self - other)
206 @_polymorphic_method
207 def __mul__(self, other):
208 if other.isconstant():
209 coefficients = dict(self.coefficients())
210 for symbol in coefficients:
211 coefficients[symbol] *= other.constant
212 constant = self.constant * other.constant
213 return Expression(coefficients, constant)
214 if isinstance(other, Expression) and not self.isconstant():
215 raise ValueError('non-linear expression: '
216 '{} * {}'.format(self._parenstr(), other._parenstr()))
217 return NotImplemented
219 __rmul__ = __mul__
221 @_polymorphic_method
222 def __truediv__(self, other):
223 if other.isconstant():
224 coefficients = dict(self.coefficients())
225 for symbol in coefficients:
226 coefficients[symbol] = \
227 Fraction(coefficients[symbol], other.constant)
228 constant = Fraction(self.constant, other.constant)
229 return Expression(coefficients, constant)
230 if isinstance(other, Expression):
231 raise ValueError('non-linear expression: '
232 '{} / {}'.format(self._parenstr(), other._parenstr()))
233 return NotImplemented
235 def __rtruediv__(self, other):
236 if isinstance(other, self):
237 if self.isconstant():
238 constant = Fraction(other, self.constant)
239 return Expression(constant=constant)
240 else:
241 raise ValueError('non-linear expression: '
242 '{} / {}'.format(other._parenstr(), self._parenstr()))
243 return NotImplemented
245 def __str__(self):
246 string = ''
247 i = 0
248 for symbol in self.symbols:
249 coefficient = self.coefficient(symbol)
250 if coefficient == 1:
251 if i == 0:
252 string += symbol
253 else:
254 string += ' + {}'.format(symbol)
255 elif coefficient == -1:
256 if i == 0:
257 string += '-{}'.format(symbol)
258 else:
259 string += ' - {}'.format(symbol)
260 else:
261 if i == 0:
262 string += '{}*{}'.format(coefficient, symbol)
263 elif coefficient > 0:
264 string += ' + {}*{}'.format(coefficient, symbol)
265 else:
266 assert coefficient < 0
267 coefficient *= -1
268 string += ' - {}*{}'.format(coefficient, symbol)
269 i += 1
270 constant = self.constant
271 if constant != 0 and i == 0:
272 string += '{}'.format(constant)
273 elif constant > 0:
274 string += ' + {}'.format(constant)
275 elif constant < 0:
276 constant *= -1
277 string += ' - {}'.format(constant)
278 if string == '':
279 string = '0'
280 return string
282 def _parenstr(self, always=False):
283 string = str(self)
284 if not always and (self.isconstant() or self.issymbol()):
285 return string
286 else:
287 return '({})'.format(string)
289 def __repr__(self):
290 string = '{}({{'.format(self.__class__.__name__)
291 for i, (symbol, coefficient) in enumerate(self.coefficients()):
292 if i != 0:
293 string += ', '
294 string += '{!r}: {!r}'.format(symbol, coefficient)
295 string += '}}, {!r})'.format(self.constant)
296 return string
298 @_polymorphic_method
299 def __eq__(self, other):
300 # "normal" equality
301 # see http://docs.sympy.org/dev/tutorial/gotchas.html#equals-signs
302 return isinstance(other, Expression) and \
303 self._coefficients == other._coefficients and \
304 self.constant == other.constant
306 def __hash__(self):
307 return hash((tuple(sorted(self._coefficients.items())), self._constant))
309 def _toint(self):
310 lcm = functools.reduce(lambda a, b: a*b // gcd(a, b),
311 [value.denominator for value in self.values()])
312 return self * lcm
314 @_polymorphic_method
315 def _eq(self, other):
316 return Polyhedron(equalities=[(self - other)._toint()])
318 @_polymorphic_method
319 def __le__(self, other):
320 return Polyhedron(inequalities=[(other - self)._toint()])
322 @_polymorphic_method
323 def __lt__(self, other):
324 return Polyhedron(inequalities=[(other - self)._toint() - 1])
326 @_polymorphic_method
327 def __ge__(self, other):
328 return Polyhedron(inequalities=[(self - other)._toint()])
330 @_polymorphic_method
331 def __gt__(self, other):
332 return Polyhedron(inequalities=[(self - other)._toint() - 1])
335 class Constant(Expression):
337 def __new__(cls, numerator=0, denominator=None):
338 self = object().__new__(cls)
339 if denominator is None:
340 if isinstance(numerator, numbers.Rational):
341 self._constant = numerator
342 elif isinstance(numerator, Constant):
343 self._constant = numerator.constant
344 else:
345 raise TypeError('constant must be a rational number or a Constant instance')
346 else:
347 self._constant = Fraction(numerator, denominator)
348 self._coefficients = {}
349 self._symbols = ()
350 self._dimension = 0
351 return self
353 def isconstant(self):
354 return True
356 def __bool__(self):
357 return bool(self.constant)
359 def __repr__(self):
360 return '{}({!r})'.format(self.__class__.__name__, self._constant)
363 class Symbol(Expression):
365 def __new__(cls, name):
366 if isinstance(name, Symbol):
367 name = name.name
368 elif not isinstance(name, str):
369 raise TypeError('name must be a string or a Symbol instance')
370 self = object().__new__(cls)
371 self._coefficients = {name: 1}
372 self._constant = 0
373 self._symbols = tuple(name)
374 self._name = name
375 self._dimension = 1
376 return self
378 @property
379 def name(self):
380 return self._name
382 def issymbol(self):
383 return True
385 def __repr__(self):
386 return '{}({!r})'.format(self.__class__.__name__, self._name)
388 def symbols(names):
389 if isinstance(names, str):
390 names = names.replace(',', ' ').split()
391 return (Symbol(name) for name in names)
394 @_polymorphic_operator
395 def eq(a, b):
396 return a.__eq__(b)
398 @_polymorphic_operator
399 def le(a, b):
400 return a.__le__(b)
402 @_polymorphic_operator
403 def lt(a, b):
404 return a.__lt__(b)
406 @_polymorphic_operator
407 def ge(a, b):
408 return a.__ge__(b)
410 @_polymorphic_operator
411 def gt(a, b):
412 return a.__gt__(b)
415 class Polyhedron:
416 """
417 This class implements polyhedrons.
418 """
420 __slots__ = (
421 '_equalities',
422 '_inequalities',
423 '_constraints',
424 '_symbols'
425 )
427 def __new__(cls, equalities=None, inequalities=None):
428 if isinstance(equalities, str):
429 if inequalities is not None:
430 raise TypeError('too many arguments')
431 return cls.fromstring(equalities)
432 self = super().__new__(cls)
433 self._equalities = []
434 if equalities is not None:
435 for constraint in equalities:
436 for value in constraint.values():
437 if value.denominator != 1:
438 raise TypeError('non-integer constraint: '
439 '{} == 0'.format(constraint))
440 self._equalities.append(constraint)
441 self._equalities = tuple(self._equalities)
442 self._inequalities = []
443 if inequalities is not None:
444 for constraint in inequalities:
445 for value in constraint.values():
446 if value.denominator != 1:
447 raise TypeError('non-integer constraint: '
448 '{} <= 0'.format(constraint))
449 self._inequalities.append(constraint)
450 self._inequalities = tuple(self._inequalities)
451 self._constraints = self._equalities + self._inequalities
452 self._symbols = set()
453 for constraint in self._constraints:
454 self.symbols.update(constraint.symbols)
455 self._symbols = tuple(sorted(self._symbols))
456 return self
458 @classmethod
459 def fromstring(cls, string):
460 string = string.strip()
461 string = re.sub(r'^\{\s*|\s*\}\$', '', string)
462 string = re.sub(r'([^<=>])=([^<=>])', r'\1==\2', string)
463 string = re.sub(r'(\d+|\))\s*([^\W\d_]\w*|\()', r'\1*\2', string)
464 equalities = []
465 inequalities = []
466 for cstr in re.split(r',|;|and|&&|/\\|∧', string, flags=re.I):
467 tree = ast.parse(cstr.strip(), 'eval')
468 if not isinstance(tree, ast.Module) or len(tree.body) != 1:
469 raise SyntaxError('invalid syntax')
470 node = tree.body[0]
471 if not isinstance(node, ast.Expr):
472 raise SyntaxError('invalid syntax')
473 node = node.value
474 if not isinstance(node, ast.Compare):
475 raise SyntaxError('invalid syntax')
476 left = Expression._fromast(node.left)
477 for i in range(len(node.ops)):
478 op = node.ops[i]
479 right = Expression._fromast(node.comparators[i])
480 if isinstance(op, ast.Lt):
481 inequalities.append(right - left - 1)
482 elif isinstance(op, ast.LtE):
483 inequalities.append(right - left)
484 elif isinstance(op, ast.Eq):
485 equalities.append(left - right)
486 elif isinstance(op, ast.GtE):
487 inequalities.append(left - right)
488 elif isinstance(op, ast.Gt):
489 inequalities.append(left - right - 1)
490 else:
491 raise SyntaxError('invalid syntax')
492 left = right
493 return cls(equalities, inequalities)
495 @property
496 def equalities(self):
497 return self._equalities
499 @property
500 def inequalities(self):
501 return self._inequalities
503 @property
504 def constraints(self):
505 return self._constraints
507 @property
508 def symbols(self):
509 return self._symbols
511 @property
512 def dimension(self):
513 return len(self.symbols)
515 def __bool__(self):
516 return not self.is_empty()
518 def __contains__(self, value):
519 # is the value in the polyhedron?
520 raise NotImplementedError
522 def __eq__(self, other):
523 # works correctly when symbols is not passed
524 # should be equal if values are the same even if symbols are different
525 bset = self._toisl()
526 other = other._toisl()
527 return bool(libisl.isl_basic_set_plain_is_equal(bset, other))
529 def isempty(self):
530 bset = self._toisl()
531 return bool(libisl.isl_basic_set_is_empty(bset))
533 def isuniverse(self):
534 bset = self._toisl()
535 return bool(libisl.isl_basic_set_is_universe(bset))
537 def isdisjoint(self, other):
538 # return true if the polyhedron has no elements in common with other
539 #symbols = self._symbolunion(other)
540 bset = self._toisl()
541 other = other._toisl()
542 return bool(libisl.isl_set_is_disjoint(bset, other))
544 def issubset(self, other):
545 # check if self(bset) is a subset of other
546 symbols = self._symbolunion(other)
547 bset = self._toisl(symbols)
548 other = other._toisl(symbols)
549 return bool(libisl.isl_set_is_strict_subset(other, bset))
551 def __le__(self, other):
552 return self.issubset(other)
554 def __lt__(self, other):
555 symbols = self._symbolunion(other)
556 bset = self._toisl(symbols)
557 other = other._toisl(symbols)
558 return bool(libisl.isl_set_is_strict_subset(other, bset))
560 def issuperset(self, other):
561 # test whether every element in other is in the polyhedron
562 raise NotImplementedError
564 def __ge__(self, other):
565 return self.issuperset(other)
567 def __gt__(self, other):
568 symbols = self._symbolunion(other)
569 bset = self._toisl(symbols)
570 other = other._toisl(symbols)
571 bool(libisl.isl_set_is_strict_subset(other, bset))
572 raise NotImplementedError
574 def union(self, *others):
575 # return a new polyhedron with elements from the polyhedron and all
576 # others (convex union)
577 raise NotImplementedError
579 def __or__(self, other):
580 return self.union(other)
582 def intersection(self, *others):
583 # return a new polyhedron with elements common to the polyhedron and all
584 # others
585 # a poor man's implementation could be:
586 # equalities = list(self.equalities)
587 # inequalities = list(self.inequalities)
588 # for other in others:
589 # equalities.extend(other.equalities)
590 # inequalities.extend(other.inequalities)
591 # return self.__class__(equalities, inequalities)
592 raise NotImplementedError
594 def __and__(self, other):
595 return self.intersection(other)
597 def difference(self, other):
598 # return a new polyhedron with elements in the polyhedron that are not in the other
599 symbols = self._symbolunion(other)
600 bset = self._toisl(symbols)
601 other = other._toisl(symbols)
602 difference = libisl.isl_set_subtract(bset, other)
603 return difference
605 def __sub__(self, other):
606 return self.difference(other)
608 def __str__(self):
609 constraints = []
610 for constraint in self.equalities:
611 constraints.append('{} == 0'.format(constraint))
612 for constraint in self.inequalities:
613 constraints.append('{} >= 0'.format(constraint))
614 return '{{{}}}'.format(', '.join(constraints))
616 def __repr__(self):
617 if self.isempty():
618 return 'Empty'
619 elif self.isuniverse():
620 return 'Universe'
621 else:
622 equalities = list(self.equalities)
623 inequalities = list(self.inequalities)
624 return '{}(equalities={!r}, inequalities={!r})' \
625 ''.format(self.__class__.__name__, equalities, inequalities)
627 def _symbolunion(self, *others):
628 symbols = set(self.symbols)
629 for other in others:
630 symbols.update(other.symbols)
631 return sorted(symbols)
633 def _toisl(self, symbols=None):
634 if symbols is None:
635 symbols = self.symbols
636 dimension = len(symbols)
637 space = libisl.isl_space_set_alloc(_main_ctx, 0, dimension)
638 bset = libisl.isl_basic_set_universe(libisl.isl_space_copy(space))
639 ls = libisl.isl_local_space_from_space(space)
640 for equality in self.equalities:
641 ceq = libisl.isl_equality_alloc(libisl.isl_local_space_copy(ls))
642 for symbol, coefficient in equality.coefficients():
643 val = str(coefficient).encode()
645 dim = symbols.index(symbol)
646 ceq = libisl.isl_constraint_set_coefficient_val(ceq, libisl.isl_dim_set, dim, val)
647 if equality.constant != 0:
648 val = str(equality.constant).encode()
650 ceq = libisl.isl_constraint_set_constant_val(ceq, val)
652 for inequality in self.inequalities:
653 cin = libisl.isl_inequality_alloc(libisl.isl_local_space_copy(ls))
654 for symbol, coefficient in inequality.coefficients():
655 val = str(coefficient).encode()
657 dim = symbols.index(symbol)
658 cin = libisl.isl_constraint_set_coefficient_val(cin, libisl.isl_dim_set, dim, val)
659 if inequality.constant != 0:
660 val = str(inequality.constant).encode()
662 cin = libisl.isl_constraint_set_constant_val(cin, val)
664 bset = isl.BasicSet(bset)
665 return bset
667 @classmethod
668 def _fromisl(cls, bset, symbols):
669 raise NotImplementedError
670 equalities = ...
671 inequalities = ...
672 return cls(equalities, inequalities)
673 '''takes basic set in isl form and puts back into python version of polyhedron
674 isl example code gives isl form as:
675 "{[i] : exists (a : i = 2a and i >= 10 and i <= 42)}")
676 our printer is giving form as:
677 { [i0, i1] : 2i1 >= -2 - i0 } '''
679 Empty = eq(0,1)
680 Universe = Polyhedron()
682 if __name__ == '__main__':
683 p1 = Polyhedron('2a + 2b + 1 == 0') # empty
684 print(p1._toisl())
685 p2 = Polyhedron('3x + 2y + 3 == 0') # not empty
686 print(p2._toisl())