Source code for sympy.assumptions.ask

"""Module for querying SymPy objects about assumptions."""
from sympy.core import sympify
from sympy.logic.boolalg import to_cnf, And, Not, Or, Implies, Equivalent, BooleanFunction
from sympy.logic.inference import satisfiable
from sympy.assumptions.assume import (global_assumptions, Predicate,
        AppliedPredicate)


[docs]class Q: """Supported ask keys.""" antihermitian = Predicate('antihermitian') bounded = Predicate('bounded') commutative = Predicate('commutative') complex = Predicate('complex') composite = Predicate('composite') even = Predicate('even') extended_real = Predicate('extended_real') hermitian = Predicate('hermitian') imaginary = Predicate('imaginary') infinitesimal = Predicate('infinitesimal') infinity = Predicate('infinity') integer = Predicate('integer') irrational = Predicate('irrational') rational = Predicate('rational') negative = Predicate('negative') nonzero = Predicate('nonzero') positive = Predicate('positive') prime = Predicate('prime') real = Predicate('real') odd = Predicate('odd') is_true = Predicate('is_true') symmetric = Predicate('symmetric') invertible = Predicate('invertible') singular = Predicate('singular') orthogonal = Predicate('orthogonal') unitary = Predicate('unitary') normal = Predicate('normal') positive_definite = Predicate('positive_definite') upper_triangular = Predicate('upper_triangular') lower_triangular = Predicate('lower_triangular') diagonal = Predicate('diagonal') triangular = Predicate('triangular') unit_triangular = Predicate('unit_triangular') fullrank = Predicate('fullrank') square = Predicate('square') real_elements = Predicate('real_elements') complex_elements = Predicate('complex_elements') integer_elements = Predicate('integer_elements')
def _extract_facts(expr, symbol): """ Helper for ask(). Extracts the facts relevant to the symbol from an assumption. Returns None if there is nothing to extract. """ if not expr.has(symbol): return None if isinstance(expr, AppliedPredicate): if expr.arg == symbol: return expr.func else: return args = [_extract_facts(arg, symbol) for arg in expr.args] if isinstance(expr, And): return expr.func(*filter(lambda x: x is not None, args)) if all(arg != None for arg in args): return expr.func(*args)
[docs]def ask(proposition, assumptions=True, context=global_assumptions): """ Method for inferring properties about objects. **Syntax** * ask(proposition) * ask(proposition, assumptions) where ``proposition`` is any boolean expression Examples ======== >>> from sympy import ask, Q, pi >>> from sympy.abc import x, y >>> ask(Q.rational(pi)) False >>> ask(Q.even(x*y), Q.even(x) & Q.integer(y)) True >>> ask(Q.prime(x*y), Q.integer(x) & Q.integer(y)) False **Remarks** Relations in assumptions are not implemented (yet), so the following will not give a meaningful result. >>> ask(Q.positive(x), Q.is_true(x > 0)) # doctest: +SKIP It is however a work in progress. """ if not isinstance(proposition, (BooleanFunction, AppliedPredicate, bool)): raise TypeError("proposition must be a valid logical expression") if not isinstance(assumptions, (BooleanFunction, AppliedPredicate, bool)): raise TypeError("assumptions must be a valid logical expression") assumptions = And(assumptions, And(*context)) if isinstance(proposition, AppliedPredicate): key, expr = proposition.func, sympify(proposition.arg) else: key, expr = Q.is_true, sympify(proposition) # direct resolution method, no logic res = key(expr)._eval_ask(assumptions) if res is not None: return res if assumptions is True: return local_facts = _extract_facts(assumptions, expr) if local_facts is None or local_facts is True: return # See if there's a straight-forward conclusion we can make for the inference if local_facts.is_Atom: if key in known_facts_dict[local_facts]: return True if Not(key) in known_facts_dict[local_facts]: return False elif local_facts.func is And and all(k in known_facts_dict for k in local_facts.args): for assum in local_facts.args: if assum.is_Atom: if key in known_facts_dict[assum]: return True if Not(key) in known_facts_dict[assum]: return False elif assum.func is Not and assum.args[0].is_Atom: if key in known_facts_dict[assum]: return False if Not(key) in known_facts_dict[assum]: return True elif (isinstance(key, Predicate) and local_facts.func is Not and local_facts.args[0].is_Atom): if local_facts.args[0] in known_facts_dict[key]: return False # Failing all else, we do a full logical inference return ask_full_inference(key, local_facts, known_facts_cnf)
[docs]def ask_full_inference(proposition, assumptions, known_facts_cnf): """ Method for inferring properties about objects. """ if not satisfiable(And(known_facts_cnf, assumptions, proposition)): return False if not satisfiable(And(known_facts_cnf, assumptions, Not(proposition))): return True return None
[docs]def register_handler(key, handler): """ Register a handler in the ask system. key must be a string and handler a class inheriting from AskHandler:: >>> from sympy.assumptions import register_handler, ask, Q >>> from sympy.assumptions.handlers import AskHandler >>> class MersenneHandler(AskHandler): ... # Mersenne numbers are in the form 2**n + 1, n integer ... @staticmethod ... def Integer(expr, assumptions): ... import math ... return ask(Q.integer(math.log(expr + 1, 2))) >>> register_handler('mersenne', MersenneHandler) >>> ask(Q.mersenne(7)) True """ if type(key) is Predicate: key = key.name try: getattr(Q, key).add_handler(handler) except AttributeError: setattr(Q, key, Predicate(key, handlers=[handler]))
[docs]def remove_handler(key, handler): """Removes a handler from the ask system. Same syntax as register_handler""" if type(key) is Predicate: key = key.name getattr(Q, key).remove_handler(handler)
def single_fact_lookup(known_facts_keys, known_facts_cnf): # Compute the quick lookup for single facts mapping = {} for key in known_facts_keys: mapping[key] = set([key]) for other_key in known_facts_keys: if other_key != key: if ask_full_inference(other_key, key, known_facts_cnf): mapping[key].add(other_key) return mapping
[docs]def compute_known_facts(known_facts, known_facts_keys): """Compute the various forms of knowledge compilation used by the assumptions system. This function is typically applied to the variables ``known_facts`` and ``known_facts_keys`` defined at the bottom of this file. """ from textwrap import dedent, wrap fact_string = dedent('''\ """ The contents of this file are the return value of ``sympy.assumptions.ask.compute_known_facts``. Do NOT manually edit this file. """ from sympy.logic.boolalg import And, Not, Or from sympy.assumptions.ask import Q # -{ Known facts in CNF }- known_facts_cnf = And( %s ) # -{ Known facts in compressed sets }- known_facts_dict = { %s } ''') # Compute the known facts in CNF form for logical inference LINE = ",\n " HANG = ' '*8 cnf = to_cnf(known_facts) c = LINE.join([str(a) for a in cnf.args]) mapping = single_fact_lookup(known_facts_keys, cnf) m = LINE.join(['\n'.join( wrap("%s: %s" % item, subsequent_indent=HANG, break_long_words=False)) for item in mapping.items()]) + ',' return fact_string % (c, m) # handlers_dict tells us what ask handler we should use # for a particular key
_val_template = 'sympy.assumptions.handlers.%s' _handlers = [ ("antihermitian", "sets.AskAntiHermitianHandler"), ("bounded", "calculus.AskBoundedHandler"), ("commutative", "AskCommutativeHandler"), ("complex", "sets.AskComplexHandler"), ("composite", "ntheory.AskCompositeHandler"), ("even", "ntheory.AskEvenHandler"), ("extended_real", "sets.AskExtendedRealHandler"), ("hermitian", "sets.AskHermitianHandler"), ("imaginary", "sets.AskImaginaryHandler"), ("infinitesimal", "calculus.AskInfinitesimalHandler"), ("integer", "sets.AskIntegerHandler"), ("irrational", "sets.AskIrrationalHandler"), ("rational", "sets.AskRationalHandler"), ("negative", "order.AskNegativeHandler"), ("nonzero", "order.AskNonZeroHandler"), ("positive", "order.AskPositiveHandler"), ("prime", "ntheory.AskPrimeHandler"), ("real", "sets.AskRealHandler"), ("odd", "ntheory.AskOddHandler"), ("algebraic", "sets.AskAlgebraicHandler"), ("is_true", "TautologicalHandler"), ("symmetric", "matrices.AskSymmetricHandler"), ("invertible", "matrices.AskInvertibleHandler"), ("orthogonal", "matrices.AskOrthogonalHandler"), ("unitary", "matrices.AskUnitaryHandler"), ("positive_definite", "matrices.AskPositiveDefiniteHandler"), ("upper_triangular", "matrices.AskUpperTriangularHandler"), ("lower_triangular", "matrices.AskLowerTriangularHandler"), ("diagonal", "matrices.AskDiagonalHandler"), ("fullrank", "matrices.AskFullRankHandler"), ("square", "matrices.AskSquareHandler"), ("integer_elements", "matrices.AskIntegerElementsHandler"), ("real_elements", "matrices.AskRealElementsHandler"), ("complex_elements", "matrices.AskComplexElementsHandler"), ] for name, value in _handlers: register_handler(name, _val_template % value) known_facts_keys = [getattr(Q, attr) for attr in Q.__dict__ if not attr.startswith('__')] known_facts = And( Implies(Q.real, Q.complex), Implies(Q.real, Q.hermitian), Equivalent(Q.even, Q.integer & ~Q.odd), Equivalent(Q.extended_real, Q.real | Q.infinity), Equivalent(Q.odd, Q.integer & ~Q.even), Equivalent(Q.prime, Q.integer & Q.positive & ~Q.composite), Implies(Q.integer, Q.rational), Implies(Q.rational, Q.algebraic), Implies(Q.algebraic, Q.complex), Implies(Q.imaginary, Q.complex & ~Q.real), Implies(Q.imaginary, Q.antihermitian), Implies(Q.antihermitian, ~Q.hermitian), Equivalent(Q.negative, Q.nonzero & ~Q.positive), Equivalent(Q.positive, Q.nonzero & ~Q.negative), Equivalent(Q.rational, Q.real & ~Q.irrational), Equivalent(Q.real, Q.rational | Q.irrational), Implies(Q.nonzero, Q.real), Equivalent(Q.nonzero, Q.positive | Q.negative), Implies(Q.orthogonal, Q.positive_definite), Implies(Q.orthogonal, Q.unitary), Implies(Q.unitary & Q.real, Q.orthogonal), Implies(Q.unitary, Q.normal), Implies(Q.unitary, Q.invertible), Implies(Q.normal, Q.square), Implies(Q.diagonal, Q.normal), Implies(Q.positive_definite, Q.invertible), Implies(Q.diagonal, Q.upper_triangular), Implies(Q.diagonal, Q.lower_triangular), Implies(Q.lower_triangular, Q.triangular), Implies(Q.upper_triangular, Q.triangular), Implies(Q.triangular, Q.upper_triangular | Q.lower_triangular), Implies(Q.upper_triangular & Q.lower_triangular, Q.diagonal), Implies(Q.diagonal, Q.symmetric), Implies(Q.unit_triangular, Q.triangular), Implies(Q.invertible, Q.fullrank), Implies(Q.invertible, Q.square), Implies(Q.symmetric, Q.square), Implies(Q.fullrank & Q.square, Q.invertible), Equivalent(Q.invertible, ~Q.singular), Implies(Q.integer_elements, Q.real_elements), Implies(Q.real_elements, Q.complex_elements), ) from sympy.assumptions.ask_generated import known_facts_dict, known_facts_cnf