""" This module contains the machinery handling assumptions. Do also consider the guide :ref:`assumptions-guide`. All symbolic objects have assumption attributes that can be accessed via ``.is_`` attribute. Assumptions determine certain properties of symbolic objects and can have 3 possible values: ``True``, ``False``, ``None``. ``True`` is returned if the object has the property and ``False`` is returned if it does not or cannot (i.e. does not make sense): >>> from sympy import I >>> I.is_algebraic True >>> I.is_real False >>> I.is_prime False When the property cannot be determined (or when a method is not implemented) ``None`` will be returned. For example, a generic symbol, ``x``, may or may not be positive so a value of ``None`` is returned for ``x.is_positive``. By default, all symbolic values are in the largest set in the given context without specifying the property. For example, a symbol that has a property being integer, is also real, complex, etc. Here follows a list of possible assumption names: .. glossary:: commutative object commutes with any other object with respect to multiplication operation. See [12]_. complex object can have only values from the set of complex numbers. See [13]_. imaginary object value is a number that can be written as a real number multiplied by the imaginary unit ``I``. See [3]_. Please note that ``0`` is not considered to be an imaginary number, see `issue #7649 `_. real object can have only values from the set of real numbers. extended_real object can have only values from the set of real numbers, ``oo`` and ``-oo``. integer object can have only values from the set of integers. odd even object can have only values from the set of odd (even) integers [2]_. prime object is a natural number greater than 1 that has no positive divisors other than 1 and itself. See [6]_. composite object is a positive integer that has at least one positive divisor other than 1 or the number itself. See [4]_. zero object has the value of 0. nonzero object is a real number that is not zero. rational object can have only values from the set of rationals. algebraic object can have only values from the set of algebraic numbers [11]_. transcendental object can have only values from the set of transcendental numbers [10]_. irrational object value cannot be represented exactly by :class:`~.Rational`, see [5]_. finite infinite object absolute value is bounded (arbitrarily large). See [7]_, [8]_, [9]_. negative nonnegative object can have only negative (nonnegative) values [1]_. positive nonpositive object can have only positive (nonpositive) values. extended_negative extended_nonnegative extended_positive extended_nonpositive extended_nonzero as without the extended part, but also including infinity with corresponding sign, e.g., extended_positive includes ``oo`` hermitian antihermitian object belongs to the field of Hermitian (antihermitian) operators. Examples ======== >>> from sympy import Symbol >>> x = Symbol('x', real=True); x x >>> x.is_real True >>> x.is_complex True See Also ======== .. seealso:: :py:class:`sympy.core.numbers.ImaginaryUnit` :py:class:`sympy.core.numbers.Zero` :py:class:`sympy.core.numbers.One` :py:class:`sympy.core.numbers.Infinity` :py:class:`sympy.core.numbers.NegativeInfinity` :py:class:`sympy.core.numbers.ComplexInfinity` Notes ===== The fully-resolved assumptions for any SymPy expression can be obtained as follows: >>> from sympy.core.assumptions import assumptions >>> x = Symbol('x',positive=True) >>> assumptions(x + I) {'commutative': True, 'complex': True, 'composite': False, 'even': False, 'extended_negative': False, 'extended_nonnegative': False, 'extended_nonpositive': False, 'extended_nonzero': False, 'extended_positive': False, 'extended_real': False, 'finite': True, 'imaginary': False, 'infinite': False, 'integer': False, 'irrational': False, 'negative': False, 'noninteger': False, 'nonnegative': False, 'nonpositive': False, 'nonzero': False, 'odd': False, 'positive': False, 'prime': False, 'rational': False, 'real': False, 'zero': False} Developers Notes ================ The current (and possibly incomplete) values are stored in the ``obj._assumptions dictionary``; queries to getter methods (with property decorators) or attributes of objects/classes will return values and update the dictionary. >>> eq = x**2 + I >>> eq._assumptions {} >>> eq.is_finite True >>> eq._assumptions {'finite': True, 'infinite': False} For a :class:`~.Symbol`, there are two locations for assumptions that may be of interest. The ``assumptions0`` attribute gives the full set of assumptions derived from a given set of initial assumptions. The latter assumptions are stored as ``Symbol._assumptions_orig`` >>> Symbol('x', prime=True, even=True)._assumptions_orig {'even': True, 'prime': True} The ``_assumptions_orig`` are not necessarily canonical nor are they filtered in any way: they records the assumptions used to instantiate a Symbol and (for storage purposes) represent a more compact representation of the assumptions needed to recreate the full set in ``Symbol.assumptions0``. References ========== .. [1] https://en.wikipedia.org/wiki/Negative_number .. [2] https://en.wikipedia.org/wiki/Parity_%28mathematics%29 .. [3] https://en.wikipedia.org/wiki/Imaginary_number .. [4] https://en.wikipedia.org/wiki/Composite_number .. [5] https://en.wikipedia.org/wiki/Irrational_number .. [6] https://en.wikipedia.org/wiki/Prime_number .. [7] https://en.wikipedia.org/wiki/Finite .. [8] https://docs.python.org/3/library/math.html#math.isfinite .. [9] https://numpy.org/doc/stable/reference/generated/numpy.isfinite.html .. [10] https://en.wikipedia.org/wiki/Transcendental_number .. [11] https://en.wikipedia.org/wiki/Algebraic_number .. [12] https://en.wikipedia.org/wiki/Commutative_property .. [13] https://en.wikipedia.org/wiki/Complex_number """ from sympy.utilities.exceptions import sympy_deprecation_warning from .facts import FactRules, FactKB from .sympify import sympify from sympy.core.random import _assumptions_shuffle as shuffle from sympy.core.assumptions_generated import generated_assumptions as _assumptions def _load_pre_generated_assumption_rules(): """ Load the assumption rules from pre-generated data To update the pre-generated data, see :method::`_generate_assumption_rules` """ _assume_rules=FactRules._from_python(_assumptions) return _assume_rules def _generate_assumption_rules(): """ Generate the default assumption rules This method should only be called to update the pre-generated assumption rules. To update the pre-generated assumptions run: bin/ask_update.py """ _assume_rules = FactRules([ 'integer -> rational', 'rational -> real', 'rational -> algebraic', 'algebraic -> complex', 'transcendental == complex & !algebraic', 'real -> hermitian', 'imaginary -> complex', 'imaginary -> antihermitian', 'extended_real -> commutative', 'complex -> commutative', 'complex -> finite', 'odd == integer & !even', 'even == integer & !odd', 'real -> complex', 'extended_real -> real | infinite', 'real == extended_real & finite', 'extended_real == extended_negative | zero | extended_positive', 'extended_negative == extended_nonpositive & extended_nonzero', 'extended_positive == extended_nonnegative & extended_nonzero', 'extended_nonpositive == extended_real & !extended_positive', 'extended_nonnegative == extended_real & !extended_negative', 'real == negative | zero | positive', 'negative == nonpositive & nonzero', 'positive == nonnegative & nonzero', 'nonpositive == real & !positive', 'nonnegative == real & !negative', 'positive == extended_positive & finite', 'negative == extended_negative & finite', 'nonpositive == extended_nonpositive & finite', 'nonnegative == extended_nonnegative & finite', 'nonzero == extended_nonzero & finite', 'zero -> even & finite', 'zero == extended_nonnegative & extended_nonpositive', 'zero == nonnegative & nonpositive', 'nonzero -> real', 'prime -> integer & positive', 'composite -> integer & positive & !prime', '!composite -> !positive | !even | prime', 'irrational == real & !rational', 'imaginary -> !extended_real', 'infinite == !finite', 'noninteger == extended_real & !integer', 'extended_nonzero == extended_real & !zero', ]) return _assume_rules _assume_rules = _load_pre_generated_assumption_rules() _assume_defined = _assume_rules.defined_facts.copy() _assume_defined.add('polar') _assume_defined = frozenset(_assume_defined) def assumptions(expr, _check=None): """return the T/F assumptions of ``expr``""" n = sympify(expr) if n.is_Symbol: rv = n.assumptions0 # are any important ones missing? if _check is not None: rv = {k: rv[k] for k in set(rv) & set(_check)} return rv rv = {} for k in _assume_defined if _check is None else _check: v = getattr(n, 'is_{}'.format(k)) if v is not None: rv[k] = v return rv def common_assumptions(exprs, check=None): """return those assumptions which have the same True or False value for all the given expressions. Examples ======== >>> from sympy.core import common_assumptions >>> from sympy import oo, pi, sqrt >>> common_assumptions([-4, 0, sqrt(2), 2, pi, oo]) {'commutative': True, 'composite': False, 'extended_real': True, 'imaginary': False, 'odd': False} By default, all assumptions are tested; pass an iterable of the assumptions to limit those that are reported: >>> common_assumptions([0, 1, 2], ['positive', 'integer']) {'integer': True} """ check = _assume_defined if check is None else set(check) if not check or not exprs: return {} # get all assumptions for each assume = [assumptions(i, _check=check) for i in sympify(exprs)] # focus on those of interest that are True for i, e in enumerate(assume): assume[i] = {k: e[k] for k in set(e) & check} # what assumptions are in common? common = set.intersection(*[set(i) for i in assume]) # which ones hold the same value a = assume[0] return {k: a[k] for k in common if all(a[k] == b[k] for b in assume)} def failing_assumptions(expr, **assumptions): """ Return a dictionary containing assumptions with values not matching those of the passed assumptions. Examples ======== >>> from sympy import failing_assumptions, Symbol >>> x = Symbol('x', positive=True) >>> y = Symbol('y') >>> failing_assumptions(6*x + y, positive=True) {'positive': None} >>> failing_assumptions(x**2 - 1, positive=True) {'positive': None} If *expr* satisfies all of the assumptions, an empty dictionary is returned. >>> failing_assumptions(x**2, positive=True) {} """ expr = sympify(expr) failed = {} for k in assumptions: test = getattr(expr, 'is_%s' % k, None) if test is not assumptions[k]: failed[k] = test return failed # {} or {assumption: value != desired} def check_assumptions(expr, against=None, **assume): """ Checks whether assumptions of ``expr`` match the T/F assumptions given (or possessed by ``against``). True is returned if all assumptions match; False is returned if there is a mismatch and the assumption in ``expr`` is not None; else None is returned. Explanation =========== *assume* is a dict of assumptions with True or False values Examples ======== >>> from sympy import Symbol, pi, I, exp, check_assumptions >>> check_assumptions(-5, integer=True) True >>> check_assumptions(pi, real=True, integer=False) True >>> check_assumptions(pi, negative=True) False >>> check_assumptions(exp(I*pi/7), real=False) True >>> x = Symbol('x', positive=True) >>> check_assumptions(2*x + 1, positive=True) True >>> check_assumptions(-2*x - 5, positive=True) False To check assumptions of *expr* against another variable or expression, pass the expression or variable as ``against``. >>> check_assumptions(2*x + 1, x) True To see if a number matches the assumptions of an expression, pass the number as the first argument, else its specific assumptions may not have a non-None value in the expression: >>> check_assumptions(x, 3) >>> check_assumptions(3, x) True ``None`` is returned if ``check_assumptions()`` could not conclude. >>> check_assumptions(2*x - 1, x) >>> z = Symbol('z') >>> check_assumptions(z, real=True) See Also ======== failing_assumptions """ expr = sympify(expr) if against is not None: if assume: raise ValueError( 'Expecting `against` or `assume`, not both.') assume = assumptions(against) known = True for k, v in assume.items(): if v is None: continue e = getattr(expr, 'is_' + k, None) if e is None: known = None elif v != e: return False return known class StdFactKB(FactKB): """A FactKB specialized for the built-in rules This is the only kind of FactKB that Basic objects should use. """ def __init__(self, facts=None): super().__init__(_assume_rules) # save a copy of the facts dict if not facts: self._generator = {} elif not isinstance(facts, FactKB): self._generator = facts.copy() else: self._generator = facts.generator if facts: self.deduce_all_facts(facts) def copy(self): return self.__class__(self) @property def generator(self): return self._generator.copy() def as_property(fact): """Convert a fact name to the name of the corresponding property""" return 'is_%s' % fact def make_property(fact): """Create the automagic property corresponding to a fact.""" def getit(self): try: return self._assumptions[fact] except KeyError: if self._assumptions is self.default_assumptions: self._assumptions = self.default_assumptions.copy() return _ask(fact, self) getit.func_name = as_property(fact) return property(getit) def _ask(fact, obj): """ Find the truth value for a property of an object. This function is called when a request is made to see what a fact value is. For this we use several techniques: First, the fact-evaluation function is tried, if it exists (for example _eval_is_integer). Then we try related facts. For example rational --> integer another example is joined rule: integer & !odd --> even so in the latter case if we are looking at what 'even' value is, 'integer' and 'odd' facts will be asked. In all cases, when we settle on some fact value, its implications are deduced, and the result is cached in ._assumptions. """ # FactKB which is dict-like and maps facts to their known values: assumptions = obj._assumptions # A dict that maps facts to their handlers: handler_map = obj._prop_handler # This is our queue of facts to check: facts_to_check = [fact] facts_queued = {fact} # Loop over the queue as it extends for fact_i in facts_to_check: # If fact_i has already been determined then we don't need to rerun the # handler. There is a potential race condition for multithreaded code # though because it's possible that fact_i was checked in another # thread. The main logic of the loop below would potentially skip # checking assumptions[fact] in this case so we check it once after the # loop to be sure. if fact_i in assumptions: continue # Now we call the associated handler for fact_i if it exists. fact_i_value = None handler_i = handler_map.get(fact_i) if handler_i is not None: fact_i_value = handler_i(obj) # If we get a new value for fact_i then we should update our knowledge # of fact_i as well as any related facts that can be inferred using the # inference rules connecting the fact_i and any other fact values that # are already known. if fact_i_value is not None: assumptions.deduce_all_facts(((fact_i, fact_i_value),)) # Usually if assumptions[fact] is now not None then that is because of # the call to deduce_all_facts above. The handler for fact_i returned # True or False and knowing fact_i (which is equal to fact in the first # iteration) implies knowing a value for fact. It is also possible # though that independent code e.g. called indirectly by the handler or # called in another thread in a multithreaded context might have # resulted in assumptions[fact] being set. Either way we return it. fact_value = assumptions.get(fact) if fact_value is not None: return fact_value # Extend the queue with other facts that might determine fact_i. Here # we randomise the order of the facts that are checked. This should not # lead to any non-determinism if all handlers are logically consistent # with the inference rules for the facts. Non-deterministic assumptions # queries can result from bugs in the handlers that are exposed by this # call to shuffle. These are pushed to the back of the queue meaning # that the inference graph is traversed in breadth-first order. new_facts_to_check = list(_assume_rules.prereq[fact_i] - facts_queued) shuffle(new_facts_to_check) facts_to_check.extend(new_facts_to_check) facts_queued.update(new_facts_to_check) # The above loop should be able to handle everything fine in a # single-threaded context but in multithreaded code it is possible that # this thread skipped computing a particular fact that was computed in # another thread (due to the continue). In that case it is possible that # fact was inferred and is now stored in the assumptions dict but it wasn't # checked for in the body of the loop. This is an obscure case but to make # sure we catch it we check once here at the end of the loop. if fact in assumptions: return assumptions[fact] # This query can not be answered. It's possible that e.g. another thread # has already stored None for fact but assumptions._tell does not mind if # we call _tell twice setting the same value. If this raises # InconsistentAssumptions then it probably means that another thread # attempted to compute this and got a value of True or False rather than # None. In that case there must be a bug in at least one of the handlers. # If the handlers are all deterministic and are consistent with the # inference rules then the same value should be computed for fact in all # threads. assumptions._tell(fact, None) return None def _prepare_class_assumptions(cls): """Precompute class level assumptions and generate handlers. This is called by Basic.__init_subclass__ each time a Basic subclass is defined. """ local_defs = {} for k in _assume_defined: attrname = as_property(k) v = cls.__dict__.get(attrname, '') if isinstance(v, (bool, int, type(None))): if v is not None: v = bool(v) local_defs[k] = v defs = {} for base in reversed(cls.__bases__): assumptions = getattr(base, '_explicit_class_assumptions', None) if assumptions is not None: defs.update(assumptions) defs.update(local_defs) cls._explicit_class_assumptions = defs cls.default_assumptions = StdFactKB(defs) cls._prop_handler = {} for k in _assume_defined: eval_is_meth = getattr(cls, '_eval_is_%s' % k, None) if eval_is_meth is not None: cls._prop_handler[k] = eval_is_meth # Put definite results directly into the class dict, for speed for k, v in cls.default_assumptions.items(): setattr(cls, as_property(k), v) # protection e.g. for Integer.is_even=F <- (Rational.is_integer=F) derived_from_bases = set() for base in cls.__bases__: default_assumptions = getattr(base, 'default_assumptions', None) # is an assumption-aware class if default_assumptions is not None: derived_from_bases.update(default_assumptions) for fact in derived_from_bases - set(cls.default_assumptions): pname = as_property(fact) if pname not in cls.__dict__: setattr(cls, pname, make_property(fact)) # Finally, add any missing automagic property (e.g. for Basic) for fact in _assume_defined: pname = as_property(fact) if not hasattr(cls, pname): setattr(cls, pname, make_property(fact)) # XXX: ManagedProperties used to be the metaclass for Basic but now Basic does # not use a metaclass. We leave this here for backwards compatibility for now # in case someone has been using the ManagedProperties class in downstream # code. The reason that it might have been used is that when subclassing a # class and wanting to use a metaclass the metaclass must be a subclass of the # metaclass for the class that is being subclassed. Anyone wanting to subclass # Basic and use a metaclass in their subclass would have needed to subclass # ManagedProperties. Here ManagedProperties is not the metaclass for Basic any # more but it should still be usable as a metaclass for Basic subclasses since # it is a subclass of type which is now the metaclass for Basic. class ManagedProperties(type): def __init__(cls, *args, **kwargs): msg = ("The ManagedProperties metaclass. " "Basic does not use metaclasses any more") sympy_deprecation_warning(msg, deprecated_since_version="1.12", active_deprecations_target='managedproperties') # Here we still call this function in case someone is using # ManagedProperties for something that is not a Basic subclass. For # Basic subclasses this function is now called by __init_subclass__ and # so this metaclass is not needed any more. _prepare_class_assumptions(cls)