362 lines
11 KiB
Python
362 lines
11 KiB
Python
|
"""
|
||
|
|
Module to evaluate the proposition with assumptions using SAT algorithm.
|
||
|
|
"""
|
||
|
|
|
||
|
|
from sympy.core.singleton import S
|
||
|
|
from sympy.core.symbol import Symbol
|
||
|
|
from sympy.assumptions.ask_generated import get_all_known_facts
|
||
|
|
from sympy.assumptions.assume import global_assumptions, AppliedPredicate
|
||
|
|
from sympy.assumptions.sathandlers import class_fact_registry
|
||
|
|
from sympy.core import oo
|
||
|
|
from sympy.logic.inference import satisfiable
|
||
|
|
from sympy.assumptions.cnf import CNF, EncodedCNF
|
||
|
|
|
||
|
|
|
||
|
|
def satask(proposition, assumptions=True, context=global_assumptions,
|
||
|
|
use_known_facts=True, iterations=oo):
|
||
|
|
"""
|
||
|
|
Function to evaluate the proposition with assumptions using SAT algorithm.
|
||
|
|
|
||
|
|
This function extracts every fact relevant to the expressions composing
|
||
|
|
proposition and assumptions. For example, if a predicate containing
|
||
|
|
``Abs(x)`` is proposed, then ``Q.zero(Abs(x)) | Q.positive(Abs(x))``
|
||
|
|
will be found and passed to SAT solver because ``Q.nonnegative`` is
|
||
|
|
registered as a fact for ``Abs``.
|
||
|
|
|
||
|
|
Proposition is evaluated to ``True`` or ``False`` if the truth value can be
|
||
|
|
determined. If not, ``None`` is returned.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
proposition : Any boolean expression.
|
||
|
|
Proposition which will be evaluated to boolean value.
|
||
|
|
|
||
|
|
assumptions : Any boolean expression, optional.
|
||
|
|
Local assumptions to evaluate the *proposition*.
|
||
|
|
|
||
|
|
context : AssumptionsContext, optional.
|
||
|
|
Default assumptions to evaluate the *proposition*. By default,
|
||
|
|
this is ``sympy.assumptions.global_assumptions`` variable.
|
||
|
|
|
||
|
|
use_known_facts : bool, optional.
|
||
|
|
If ``True``, facts from ``sympy.assumptions.ask_generated``
|
||
|
|
module are passed to SAT solver as well.
|
||
|
|
|
||
|
|
iterations : int, optional.
|
||
|
|
Number of times that relevant facts are recursively extracted.
|
||
|
|
Default is infinite times until no new fact is found.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
``True``, ``False``, or ``None``
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Abs, Q
|
||
|
|
>>> from sympy.assumptions.satask import satask
|
||
|
|
>>> from sympy.abc import x
|
||
|
|
>>> satask(Q.zero(Abs(x)), Q.zero(x))
|
||
|
|
True
|
||
|
|
|
||
|
|
"""
|
||
|
|
props = CNF.from_prop(proposition)
|
||
|
|
_props = CNF.from_prop(~proposition)
|
||
|
|
|
||
|
|
assumptions = CNF.from_prop(assumptions)
|
||
|
|
|
||
|
|
context_cnf = CNF()
|
||
|
|
if context:
|
||
|
|
context_cnf = context_cnf.extend(context)
|
||
|
|
|
||
|
|
sat = get_all_relevant_facts(props, assumptions, context_cnf,
|
||
|
|
use_known_facts=use_known_facts, iterations=iterations)
|
||
|
|
sat.add_from_cnf(assumptions)
|
||
|
|
if context:
|
||
|
|
sat.add_from_cnf(context_cnf)
|
||
|
|
|
||
|
|
return check_satisfiability(props, _props, sat)
|
||
|
|
|
||
|
|
|
||
|
|
def check_satisfiability(prop, _prop, factbase):
|
||
|
|
sat_true = factbase.copy()
|
||
|
|
sat_false = factbase.copy()
|
||
|
|
sat_true.add_from_cnf(prop)
|
||
|
|
sat_false.add_from_cnf(_prop)
|
||
|
|
can_be_true = satisfiable(sat_true)
|
||
|
|
can_be_false = satisfiable(sat_false)
|
||
|
|
|
||
|
|
if can_be_true and can_be_false:
|
||
|
|
return None
|
||
|
|
|
||
|
|
if can_be_true and not can_be_false:
|
||
|
|
return True
|
||
|
|
|
||
|
|
if not can_be_true and can_be_false:
|
||
|
|
return False
|
||
|
|
|
||
|
|
if not can_be_true and not can_be_false:
|
||
|
|
# TODO: Run additional checks to see which combination of the
|
||
|
|
# assumptions, global_assumptions, and relevant_facts are
|
||
|
|
# inconsistent.
|
||
|
|
raise ValueError("Inconsistent assumptions")
|
||
|
|
|
||
|
|
|
||
|
|
def extract_predargs(proposition, assumptions=None, context=None):
|
||
|
|
"""
|
||
|
|
Extract every expression in the argument of predicates from *proposition*,
|
||
|
|
*assumptions* and *context*.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
proposition : sympy.assumptions.cnf.CNF
|
||
|
|
|
||
|
|
assumptions : sympy.assumptions.cnf.CNF, optional.
|
||
|
|
|
||
|
|
context : sympy.assumptions.cnf.CNF, optional.
|
||
|
|
CNF generated from assumptions context.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Q, Abs
|
||
|
|
>>> from sympy.assumptions.cnf import CNF
|
||
|
|
>>> from sympy.assumptions.satask import extract_predargs
|
||
|
|
>>> from sympy.abc import x, y
|
||
|
|
>>> props = CNF.from_prop(Q.zero(Abs(x*y)))
|
||
|
|
>>> assump = CNF.from_prop(Q.zero(x) & Q.zero(y))
|
||
|
|
>>> extract_predargs(props, assump)
|
||
|
|
{x, y, Abs(x*y)}
|
||
|
|
|
||
|
|
"""
|
||
|
|
req_keys = find_symbols(proposition)
|
||
|
|
keys = proposition.all_predicates()
|
||
|
|
# XXX: We need this since True/False are not Basic
|
||
|
|
lkeys = set()
|
||
|
|
if assumptions:
|
||
|
|
lkeys |= assumptions.all_predicates()
|
||
|
|
if context:
|
||
|
|
lkeys |= context.all_predicates()
|
||
|
|
|
||
|
|
lkeys = lkeys - {S.true, S.false}
|
||
|
|
tmp_keys = None
|
||
|
|
while tmp_keys != set():
|
||
|
|
tmp = set()
|
||
|
|
for l in lkeys:
|
||
|
|
syms = find_symbols(l)
|
||
|
|
if (syms & req_keys) != set():
|
||
|
|
tmp |= syms
|
||
|
|
tmp_keys = tmp - req_keys
|
||
|
|
req_keys |= tmp_keys
|
||
|
|
keys |= {l for l in lkeys if find_symbols(l) & req_keys != set()}
|
||
|
|
|
||
|
|
exprs = set()
|
||
|
|
for key in keys:
|
||
|
|
if isinstance(key, AppliedPredicate):
|
||
|
|
exprs |= set(key.arguments)
|
||
|
|
else:
|
||
|
|
exprs.add(key)
|
||
|
|
return exprs
|
||
|
|
|
||
|
|
def find_symbols(pred):
|
||
|
|
"""
|
||
|
|
Find every :obj:`~.Symbol` in *pred*.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
pred : sympy.assumptions.cnf.CNF, or any Expr.
|
||
|
|
|
||
|
|
"""
|
||
|
|
if isinstance(pred, CNF):
|
||
|
|
symbols = set()
|
||
|
|
for a in pred.all_predicates():
|
||
|
|
symbols |= find_symbols(a)
|
||
|
|
return symbols
|
||
|
|
return pred.atoms(Symbol)
|
||
|
|
|
||
|
|
|
||
|
|
def get_relevant_clsfacts(exprs, relevant_facts=None):
|
||
|
|
"""
|
||
|
|
Extract relevant facts from the items in *exprs*. Facts are defined in
|
||
|
|
``assumptions.sathandlers`` module.
|
||
|
|
|
||
|
|
This function is recursively called by ``get_all_relevant_facts()``.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
exprs : set
|
||
|
|
Expressions whose relevant facts are searched.
|
||
|
|
|
||
|
|
relevant_facts : sympy.assumptions.cnf.CNF, optional.
|
||
|
|
Pre-discovered relevant facts.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
exprs : set
|
||
|
|
Candidates for next relevant fact searching.
|
||
|
|
|
||
|
|
relevant_facts : sympy.assumptions.cnf.CNF
|
||
|
|
Updated relevant facts.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
Here, we will see how facts relevant to ``Abs(x*y)`` are recursively
|
||
|
|
extracted. On the first run, set containing the expression is passed
|
||
|
|
without pre-discovered relevant facts. The result is a set containing
|
||
|
|
candidates for next run, and ``CNF()`` instance containing facts
|
||
|
|
which are relevant to ``Abs`` and its argument.
|
||
|
|
|
||
|
|
>>> from sympy import Abs
|
||
|
|
>>> from sympy.assumptions.satask import get_relevant_clsfacts
|
||
|
|
>>> from sympy.abc import x, y
|
||
|
|
>>> exprs = {Abs(x*y)}
|
||
|
|
>>> exprs, facts = get_relevant_clsfacts(exprs)
|
||
|
|
>>> exprs
|
||
|
|
{x*y}
|
||
|
|
>>> facts.clauses #doctest: +SKIP
|
||
|
|
{frozenset({Literal(Q.odd(Abs(x*y)), False), Literal(Q.odd(x*y), True)}),
|
||
|
|
frozenset({Literal(Q.zero(Abs(x*y)), False), Literal(Q.zero(x*y), True)}),
|
||
|
|
frozenset({Literal(Q.even(Abs(x*y)), False), Literal(Q.even(x*y), True)}),
|
||
|
|
frozenset({Literal(Q.zero(Abs(x*y)), True), Literal(Q.zero(x*y), False)}),
|
||
|
|
frozenset({Literal(Q.even(Abs(x*y)), False),
|
||
|
|
Literal(Q.odd(Abs(x*y)), False),
|
||
|
|
Literal(Q.odd(x*y), True)}),
|
||
|
|
frozenset({Literal(Q.even(Abs(x*y)), False),
|
||
|
|
Literal(Q.even(x*y), True),
|
||
|
|
Literal(Q.odd(Abs(x*y)), False)}),
|
||
|
|
frozenset({Literal(Q.positive(Abs(x*y)), False),
|
||
|
|
Literal(Q.zero(Abs(x*y)), False)})}
|
||
|
|
|
||
|
|
We pass the first run's results to the second run, and get the expressions
|
||
|
|
for next run and updated facts.
|
||
|
|
|
||
|
|
>>> exprs, facts = get_relevant_clsfacts(exprs, relevant_facts=facts)
|
||
|
|
>>> exprs
|
||
|
|
{x, y}
|
||
|
|
|
||
|
|
On final run, no more candidate is returned thus we know that all
|
||
|
|
relevant facts are successfully retrieved.
|
||
|
|
|
||
|
|
>>> exprs, facts = get_relevant_clsfacts(exprs, relevant_facts=facts)
|
||
|
|
>>> exprs
|
||
|
|
set()
|
||
|
|
|
||
|
|
"""
|
||
|
|
if not relevant_facts:
|
||
|
|
relevant_facts = CNF()
|
||
|
|
|
||
|
|
newexprs = set()
|
||
|
|
for expr in exprs:
|
||
|
|
for fact in class_fact_registry(expr):
|
||
|
|
newfact = CNF.to_CNF(fact)
|
||
|
|
relevant_facts = relevant_facts._and(newfact)
|
||
|
|
for key in newfact.all_predicates():
|
||
|
|
if isinstance(key, AppliedPredicate):
|
||
|
|
newexprs |= set(key.arguments)
|
||
|
|
|
||
|
|
return newexprs - exprs, relevant_facts
|
||
|
|
|
||
|
|
|
||
|
|
def get_all_relevant_facts(proposition, assumptions, context,
|
||
|
|
use_known_facts=True, iterations=oo):
|
||
|
|
"""
|
||
|
|
Extract all relevant facts from *proposition* and *assumptions*.
|
||
|
|
|
||
|
|
This function extracts the facts by recursively calling
|
||
|
|
``get_relevant_clsfacts()``. Extracted facts are converted to
|
||
|
|
``EncodedCNF`` and returned.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
proposition : sympy.assumptions.cnf.CNF
|
||
|
|
CNF generated from proposition expression.
|
||
|
|
|
||
|
|
assumptions : sympy.assumptions.cnf.CNF
|
||
|
|
CNF generated from assumption expression.
|
||
|
|
|
||
|
|
context : sympy.assumptions.cnf.CNF
|
||
|
|
CNF generated from assumptions context.
|
||
|
|
|
||
|
|
use_known_facts : bool, optional.
|
||
|
|
If ``True``, facts from ``sympy.assumptions.ask_generated``
|
||
|
|
module are encoded as well.
|
||
|
|
|
||
|
|
iterations : int, optional.
|
||
|
|
Number of times that relevant facts are recursively extracted.
|
||
|
|
Default is infinite times until no new fact is found.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
sympy.assumptions.cnf.EncodedCNF
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import Q
|
||
|
|
>>> from sympy.assumptions.cnf import CNF
|
||
|
|
>>> from sympy.assumptions.satask import get_all_relevant_facts
|
||
|
|
>>> from sympy.abc import x, y
|
||
|
|
>>> props = CNF.from_prop(Q.nonzero(x*y))
|
||
|
|
>>> assump = CNF.from_prop(Q.nonzero(x))
|
||
|
|
>>> context = CNF.from_prop(Q.nonzero(y))
|
||
|
|
>>> get_all_relevant_facts(props, assump, context) #doctest: +SKIP
|
||
|
|
<sympy.assumptions.cnf.EncodedCNF at 0x7f09faa6ccd0>
|
||
|
|
|
||
|
|
"""
|
||
|
|
# The relevant facts might introduce new keys, e.g., Q.zero(x*y) will
|
||
|
|
# introduce the keys Q.zero(x) and Q.zero(y), so we need to run it until
|
||
|
|
# we stop getting new things. Hopefully this strategy won't lead to an
|
||
|
|
# infinite loop in the future.
|
||
|
|
i = 0
|
||
|
|
relevant_facts = CNF()
|
||
|
|
all_exprs = set()
|
||
|
|
while True:
|
||
|
|
if i == 0:
|
||
|
|
exprs = extract_predargs(proposition, assumptions, context)
|
||
|
|
all_exprs |= exprs
|
||
|
|
exprs, relevant_facts = get_relevant_clsfacts(exprs, relevant_facts)
|
||
|
|
i += 1
|
||
|
|
if i >= iterations:
|
||
|
|
break
|
||
|
|
if not exprs:
|
||
|
|
break
|
||
|
|
|
||
|
|
if use_known_facts:
|
||
|
|
known_facts_CNF = CNF()
|
||
|
|
known_facts_CNF.add_clauses(get_all_known_facts())
|
||
|
|
kf_encoded = EncodedCNF()
|
||
|
|
kf_encoded.from_cnf(known_facts_CNF)
|
||
|
|
|
||
|
|
def translate_literal(lit, delta):
|
||
|
|
if lit > 0:
|
||
|
|
return lit + delta
|
||
|
|
else:
|
||
|
|
return lit - delta
|
||
|
|
|
||
|
|
def translate_data(data, delta):
|
||
|
|
return [{translate_literal(i, delta) for i in clause} for clause in data]
|
||
|
|
data = []
|
||
|
|
symbols = []
|
||
|
|
n_lit = len(kf_encoded.symbols)
|
||
|
|
for i, expr in enumerate(all_exprs):
|
||
|
|
symbols += [pred(expr) for pred in kf_encoded.symbols]
|
||
|
|
data += translate_data(kf_encoded.data, i * n_lit)
|
||
|
|
|
||
|
|
encoding = dict(list(zip(symbols, range(1, len(symbols)+1))))
|
||
|
|
ctx = EncodedCNF(data, encoding)
|
||
|
|
else:
|
||
|
|
ctx = EncodedCNF()
|
||
|
|
|
||
|
|
ctx.add_from_cnf(relevant_facts)
|
||
|
|
|
||
|
|
return ctx
|