You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
221 lines
7.4 KiB
221 lines
7.4 KiB
"""
|
|
Known facts in assumptions module.
|
|
|
|
This module defines the facts between unary predicates in ``get_known_facts()``,
|
|
and supports functions to generate the contents in
|
|
``sympy.assumptions.ask_generated`` file.
|
|
"""
|
|
|
|
from sympy.assumptions.ask import Q
|
|
from sympy.assumptions.assume import AppliedPredicate
|
|
from sympy.core.cache import cacheit
|
|
from sympy.core.symbol import Symbol
|
|
from sympy.logic.boolalg import (to_cnf, And, Not, Implies, Equivalent,
|
|
Exclusive,)
|
|
from sympy.logic.inference import satisfiable
|
|
|
|
|
|
@cacheit
|
|
def get_composite_predicates():
|
|
# To reduce the complexity of sat solver, these predicates are
|
|
# transformed into the combination of primitive predicates.
|
|
return {
|
|
Q.real : Q.negative | Q.zero | Q.positive,
|
|
Q.integer : Q.even | Q.odd,
|
|
Q.nonpositive : Q.negative | Q.zero,
|
|
Q.nonzero : Q.negative | Q.positive,
|
|
Q.nonnegative : Q.zero | Q.positive,
|
|
Q.extended_real : Q.negative_infinite | Q.negative | Q.zero | Q.positive | Q.positive_infinite,
|
|
Q.extended_positive: Q.positive | Q.positive_infinite,
|
|
Q.extended_negative: Q.negative | Q.negative_infinite,
|
|
Q.extended_nonzero: Q.negative_infinite | Q.negative | Q.positive | Q.positive_infinite,
|
|
Q.extended_nonpositive: Q.negative_infinite | Q.negative | Q.zero,
|
|
Q.extended_nonnegative: Q.zero | Q.positive | Q.positive_infinite,
|
|
Q.complex : Q.algebraic | Q.transcendental
|
|
}
|
|
|
|
|
|
@cacheit
|
|
def get_known_facts(x=None):
|
|
"""
|
|
Facts between unary predicates.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
x : Symbol, optional
|
|
Placeholder symbol for unary facts. Default is ``Symbol('x')``.
|
|
|
|
Returns
|
|
=======
|
|
|
|
fact : Known facts in conjugated normal form.
|
|
|
|
"""
|
|
if x is None:
|
|
x = Symbol('x')
|
|
|
|
fact = And(
|
|
# primitive predicates for extended real exclude each other.
|
|
Exclusive(Q.negative_infinite(x), Q.negative(x), Q.zero(x),
|
|
Q.positive(x), Q.positive_infinite(x)),
|
|
|
|
# build complex plane
|
|
Exclusive(Q.real(x), Q.imaginary(x)),
|
|
Implies(Q.real(x) | Q.imaginary(x), Q.complex(x)),
|
|
|
|
# other subsets of complex
|
|
Exclusive(Q.transcendental(x), Q.algebraic(x)),
|
|
Equivalent(Q.real(x), Q.rational(x) | Q.irrational(x)),
|
|
Exclusive(Q.irrational(x), Q.rational(x)),
|
|
Implies(Q.rational(x), Q.algebraic(x)),
|
|
|
|
# integers
|
|
Exclusive(Q.even(x), Q.odd(x)),
|
|
Implies(Q.integer(x), Q.rational(x)),
|
|
Implies(Q.zero(x), Q.even(x)),
|
|
Exclusive(Q.composite(x), Q.prime(x)),
|
|
Implies(Q.composite(x) | Q.prime(x), Q.integer(x) & Q.positive(x)),
|
|
Implies(Q.even(x) & Q.positive(x) & ~Q.prime(x), Q.composite(x)),
|
|
|
|
# hermitian and antihermitian
|
|
Implies(Q.real(x), Q.hermitian(x)),
|
|
Implies(Q.imaginary(x), Q.antihermitian(x)),
|
|
Implies(Q.zero(x), Q.hermitian(x) | Q.antihermitian(x)),
|
|
|
|
# define finity and infinity, and build extended real line
|
|
Exclusive(Q.infinite(x), Q.finite(x)),
|
|
Implies(Q.complex(x), Q.finite(x)),
|
|
Implies(Q.negative_infinite(x) | Q.positive_infinite(x), Q.infinite(x)),
|
|
|
|
# commutativity
|
|
Implies(Q.finite(x) | Q.infinite(x), Q.commutative(x)),
|
|
|
|
# matrices
|
|
Implies(Q.orthogonal(x), Q.positive_definite(x)),
|
|
Implies(Q.orthogonal(x), Q.unitary(x)),
|
|
Implies(Q.unitary(x) & Q.real_elements(x), Q.orthogonal(x)),
|
|
Implies(Q.unitary(x), Q.normal(x)),
|
|
Implies(Q.unitary(x), Q.invertible(x)),
|
|
Implies(Q.normal(x), Q.square(x)),
|
|
Implies(Q.diagonal(x), Q.normal(x)),
|
|
Implies(Q.positive_definite(x), Q.invertible(x)),
|
|
Implies(Q.diagonal(x), Q.upper_triangular(x)),
|
|
Implies(Q.diagonal(x), Q.lower_triangular(x)),
|
|
Implies(Q.lower_triangular(x), Q.triangular(x)),
|
|
Implies(Q.upper_triangular(x), Q.triangular(x)),
|
|
Implies(Q.triangular(x), Q.upper_triangular(x) | Q.lower_triangular(x)),
|
|
Implies(Q.upper_triangular(x) & Q.lower_triangular(x), Q.diagonal(x)),
|
|
Implies(Q.diagonal(x), Q.symmetric(x)),
|
|
Implies(Q.unit_triangular(x), Q.triangular(x)),
|
|
Implies(Q.invertible(x), Q.fullrank(x)),
|
|
Implies(Q.invertible(x), Q.square(x)),
|
|
Implies(Q.symmetric(x), Q.square(x)),
|
|
Implies(Q.fullrank(x) & Q.square(x), Q.invertible(x)),
|
|
Equivalent(Q.invertible(x), ~Q.singular(x)),
|
|
Implies(Q.integer_elements(x), Q.real_elements(x)),
|
|
Implies(Q.real_elements(x), Q.complex_elements(x)),
|
|
)
|
|
return fact
|
|
|
|
|
|
def generate_known_facts_dict(keys, fact):
|
|
"""
|
|
Computes and returns a dictionary which contains the relations between
|
|
unary predicates.
|
|
|
|
Each key is a predicate, and item is two groups of predicates.
|
|
First group contains the predicates which are implied by the key, and
|
|
second group contains the predicates which are rejected by the key.
|
|
|
|
All predicates in *keys* and *fact* must be unary and have same placeholder
|
|
symbol.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
keys : list of AppliedPredicate instances.
|
|
|
|
fact : Fact between predicates in conjugated normal form.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import Q, And, Implies
|
|
>>> from sympy.assumptions.facts import generate_known_facts_dict
|
|
>>> from sympy.abc import x
|
|
>>> keys = [Q.even(x), Q.odd(x), Q.zero(x)]
|
|
>>> fact = And(Implies(Q.even(x), ~Q.odd(x)),
|
|
... Implies(Q.zero(x), Q.even(x)))
|
|
>>> generate_known_facts_dict(keys, fact)
|
|
{Q.even: ({Q.even}, {Q.odd}),
|
|
Q.odd: ({Q.odd}, {Q.even, Q.zero}),
|
|
Q.zero: ({Q.even, Q.zero}, {Q.odd})}
|
|
"""
|
|
fact_cnf = to_cnf(fact)
|
|
mapping = single_fact_lookup(keys, fact_cnf)
|
|
|
|
ret = {}
|
|
for key, value in mapping.items():
|
|
implied = set()
|
|
rejected = set()
|
|
for expr in value:
|
|
if isinstance(expr, AppliedPredicate):
|
|
implied.add(expr.function)
|
|
elif isinstance(expr, Not):
|
|
pred = expr.args[0]
|
|
rejected.add(pred.function)
|
|
ret[key.function] = (implied, rejected)
|
|
return ret
|
|
|
|
|
|
@cacheit
|
|
def get_known_facts_keys():
|
|
"""
|
|
Return every unary predicates registered to ``Q``.
|
|
|
|
This function is used to generate the keys for
|
|
``generate_known_facts_dict``.
|
|
|
|
"""
|
|
exclude = set()
|
|
for pred in [Q.eq, Q.ne, Q.gt, Q.lt, Q.ge, Q.le]:
|
|
# exclude polyadic predicates
|
|
exclude.add(pred)
|
|
|
|
result = []
|
|
for attr in Q.__class__.__dict__:
|
|
if attr.startswith('__'):
|
|
continue
|
|
pred = getattr(Q, attr)
|
|
if pred in exclude:
|
|
continue
|
|
result.append(pred)
|
|
return result
|
|
|
|
|
|
def single_fact_lookup(known_facts_keys, known_facts_cnf):
|
|
# Return the dictionary for quick lookup of single fact
|
|
mapping = {}
|
|
for key in known_facts_keys:
|
|
mapping[key] = {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)
|
|
if ask_full_inference(~other_key, key, known_facts_cnf):
|
|
mapping[key].add(~other_key)
|
|
return mapping
|
|
|
|
|
|
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
|