def get_all_relevant_facts(
proposition,
assumptions=True,
context=global_assumptions,
use_known_facts=True,
iterations=oo,
):
# 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()
exprs = None
all_exprs = set()
while exprs != set():
exprs, relevant_facts = get_relevant_facts(
proposition,
assumptions,
context,
exprs=exprs,
relevant_facts=relevant_facts,
)
all_exprs |= exprs
i += 1
if i >= iterations:
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
def minisat22_satisfiable(expr, all_models=False, minimal=False):
if not isinstance(expr, EncodedCNF):
exprs = EncodedCNF()
exprs.add_prop(expr)
expr = exprs
from pysat.solvers import Minisat22
# Return UNSAT when False (encoded as 0) is present in the CNF
if {0} in expr.data:
if all_models:
return (f for f in [False])
return False
r = Minisat22(expr.data)
if minimal:
r.set_phases([-(i + 1) for i in range(r.nof_vars())])
if not r.solve():
return False
if not all_models:
return {expr.symbols[abs(lit) - 1]: lit > 0 for lit in r.get_model()}
else:
# Make solutions sympy compatible by creating a generator
def _gen(results):
satisfiable = False
while results.solve():
sol = results.get_model()
yield {expr.symbols[abs(lit) - 1]: lit > 0 for lit in sol}
if minimal:
results.add_clause([-i for i in sol if i > 0])
else:
results.add_clause([-i for i in sol])
satisfiable = True
if not satisfiable:
yield False
raise StopIteration
return _gen(r)
def pycosat_satisfiable(expr, all_models=False):
import pycosat
if not isinstance(expr, EncodedCNF):
exprs = EncodedCNF()
exprs.add_prop(expr)
expr = exprs
# Return UNSAT when False (encoded as 0) is present in the CNF
if {0} in expr.data:
if all_models:
return (f for f in [False])
return False
if not all_models:
r = pycosat.solve(expr.data)
result = r != "UNSAT"
if not result:
return result
return dict((expr.symbols[abs(lit) - 1], lit > 0) for lit in r)
else:
r = pycosat.itersolve(expr.data)
result = r != "UNSAT"
if not result:
return result
# Make solutions sympy compatible by creating a generator
def _gen(results):
satisfiable = False
try:
while True:
sol = next(results)
yield dict(
(expr.symbols[abs(lit) - 1], lit > 0) for lit in sol)
satisfiable = True
except StopIteration:
if not satisfiable:
yield False
return _gen(r)
def dpll_satisfiable(expr, all_models=False):
"""
Check satisfiability of a propositional sentence.
It returns a model rather than True when it succeeds.
Returns a generator of all models if all_models is True.
Examples
========
>>> from sympy.abc import A, B
>>> from sympy.logic.algorithms.dpll2 import dpll_satisfiable
>>> dpll_satisfiable(A & ~B)
{A: True, B: False}
>>> dpll_satisfiable(A & ~A)
False
"""
if not isinstance(expr, EncodedCNF):
exprs = EncodedCNF()
exprs.add_prop(expr)
expr = exprs
# Return UNSAT when False (encoded as 0) is present in the CNF
if {0} in expr.data:
if all_models:
return (f for f in [False])
return False
solver = SATSolver(expr.data, expr.variables, set(), expr.symbols)
models = solver._find_model()
if all_models:
return _all_models(models)
try:
return next(models)
except StopIteration:
return False
def ask(proposition, assumptions=True, context=global_assumptions):
"""
Function to evaluate the proposition with assumptions.
**Syntax**
* ask(proposition)
Evaluate the *proposition* in global assumption context.
* ask(proposition, assumptions)
Evaluate the *proposition* with respect to *assumptions* in
global assumption context.
This function evaluates the proposition to ``True`` or ``False`` if
the truth value can be determined. If not, it returns ``None``.
It should be discerned from :func:`~.refine()` which, when applied to a
proposition, simplifies the argument to symbolic ``Boolean`` instead of
Python built-in ``True``, ``False`` or ``None``.
Parameters
==========
proposition : any boolean expression
Proposition which will be evaluated to boolean value. If this is
not ``AppliedPredicate``, it will be wrapped by ``Q.is_true``.
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.
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(4*x), Q.integer(x))
False
If the truth value cannot be determined, ``None`` will be returned.
>>> print(ask(Q.odd(3*x))) # cannot determine unless we know x
None
**Remarks**
Relations in assumptions are not implemented (yet), so the following
will not give a meaningful result.
>>> ask(Q.positive(x), x > 0)
It is however a work in progress.
See Also
========
sympy.assumptions.refine.refine : Simplification using assumptions.
Proposition is not reduced to ``None`` if the truth value cannot
be determined.
"""
from sympy.assumptions.satask import satask
proposition = sympify(proposition)
assumptions = sympify(assumptions)
if isinstance(proposition,
Predicate) or proposition.kind is not BooleanKind:
raise TypeError("proposition must be a valid logical expression")
if isinstance(assumptions,
Predicate) or assumptions.kind is not BooleanKind:
raise TypeError("assumptions must be a valid logical expression")
binrelpreds = {Eq: Q.eq, Ne: Q.ne, Gt: Q.gt, Lt: Q.lt, Ge: Q.ge, Le: Q.le}
if isinstance(proposition, AppliedPredicate):
key, args = proposition.function, proposition.arguments
elif proposition.func in binrelpreds:
key, args = binrelpreds[proposition.func], proposition.args
else:
key, args = Q.is_true, (proposition, )
# convert local and global assumptions to CNF
assump = CNF.from_prop(assumptions)
assump.extend(context)
# extract the relevant facts from assumptions with respect to args
local_facts = _extract_all_facts(assump, args)
known_facts_cnf = get_all_known_facts()
known_facts_dict = get_known_facts_dict()
# convert default facts and assumed facts to encoded CNF
enc_cnf = EncodedCNF()
enc_cnf.from_cnf(CNF(known_facts_cnf))
enc_cnf.add_from_cnf(local_facts)
# check the satisfiability of given assumptions
if local_facts.clauses and satisfiable(enc_cnf) is False:
raise ValueError("inconsistent assumptions %s" % assumptions)
if local_facts.clauses:
# quick exit if the prerequisite of proposition is not true
# e.g. proposition = Q.odd(x), assumptions = ~Q.integer(x)
if len(local_facts.clauses) == 1:
cl, = local_facts.clauses
if len(cl) == 1:
f, = cl
if f.is_Not and f.arg in known_facts_dict.get(key, []):
return False
for clause in local_facts.clauses:
if len(clause) == 1:
f, = clause
fdict = known_facts_dict.get(f.arg,
None) if not f.is_Not else None
if fdict is None:
pass
elif key in fdict:
# quick exit if proposition is directly satisfied by assumption
# e.g. proposition = Q.integer(x), assumptions = Q.odd(x)
return True
elif Not(key) in fdict:
# quick exit if proposition is directly rejected by assumption
# example might be proposition = Q.even(x), assumptions = Q.odd(x)
# but known_facts_dict does not have such information yet and
# such example is computed by satask.
return False
# direct resolution method, no logic
res = key(*args)._eval_ask(assumptions)
if res is not None:
return bool(res)
# using satask (still costly)
res = satask(proposition, assumptions=assumptions, context=context)
return res
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(4*x), Q.integer(x))
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))
It is however a work in progress.
"""
from sympy.assumptions.satask import satask
if not isinstance(proposition, (BooleanFunction, AppliedPredicate, bool, BooleanAtom)):
raise TypeError("proposition must be a valid logical expression")
if not isinstance(assumptions, (BooleanFunction, AppliedPredicate, bool, BooleanAtom)):
raise TypeError("assumptions must be a valid logical expression")
if isinstance(proposition, AppliedPredicate):
key, expr = proposition.func, sympify(proposition.arg)
else:
key, expr = Q.is_true, sympify(proposition)
assump = CNF.from_prop(assumptions)
assump.extend(context)
local_facts = _extract_all_facts(assump, expr)
known_facts_cnf = get_all_known_facts()
known_facts_dict = get_known_facts_dict()
enc_cnf = EncodedCNF()
enc_cnf.from_cnf(CNF(known_facts_cnf))
enc_cnf.add_from_cnf(local_facts)
if local_facts.clauses and satisfiable(enc_cnf) is False:
raise ValueError("inconsistent assumptions %s" % assumptions)
if local_facts.clauses:
local_facts_ = CNF.CNF_to_cnf(local_facts)
# 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 (isinstance(local_facts_, 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 isinstance(assum, 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
isinstance(local_facts_, Not) and local_facts_.args[0].is_Atom):
if local_facts_.args[0] in known_facts_dict[key]:
return False
# direct resolution method, no logic
res = key(expr)._eval_ask(assumptions)
if res is not None:
return bool(res)
# using satask (still costly)
res = satask(proposition, assumptions=assumptions, context=context)
return res
def ask(proposition, assumptions=True, context=global_assumptions):
"""
Function to evaluate the proposition with assumptions.
**Syntax**
* ask(proposition)
Evaluate the *proposition* in global assumption context.
* ask(proposition, assumptions)
Evaluate the *proposition* with respect to *assumptions* in
global assumption context.
This function evaluates the proposition to ``True`` or ``False`` if
the truth value can be determined. If not, it returns ``None``.
It should be discerned from :func:`~.refine()` which does not reduce
the expression to ``None``.
Parameters
==========
proposition : any boolean expression
Proposition which will be evaluated to boolean value. If this is
not ``AppliedPredicate``, it will be wrapped by ``Q.is_true``.
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.
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(4*x), Q.integer(x))
False
If the truth value cannot be determined, ``None`` will be returned.
>>> print(ask(Q.odd(3*x))) # cannot determine unless we know x
None
**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))
It is however a work in progress.
See Also
========
sympy.assumptions.refine.refine : Simplification using assumptions.
Proposition is not reduced to ``None`` if the truth value cannot
be determined.
"""
from sympy.assumptions.satask import satask
proposition = sympify(proposition)
assumptions = sympify(assumptions)
if isinstance(proposition,
Predicate) or proposition.kind is not BooleanKind:
raise TypeError("proposition must be a valid logical expression")
if isinstance(assumptions,
Predicate) or assumptions.kind is not BooleanKind:
raise TypeError("assumptions must be a valid logical expression")
if isinstance(proposition, AppliedPredicate):
key, args = proposition.function, proposition.arguments
else:
key, args = Q.is_true, (proposition, )
assump = CNF.from_prop(assumptions)
assump.extend(context)
local_facts = _extract_all_facts(assump, args)
known_facts_cnf = get_all_known_facts()
known_facts_dict = get_known_facts_dict()
enc_cnf = EncodedCNF()
enc_cnf.from_cnf(CNF(known_facts_cnf))
enc_cnf.add_from_cnf(local_facts)
if local_facts.clauses and satisfiable(enc_cnf) is False:
raise ValueError("inconsistent assumptions %s" % assumptions)
if local_facts.clauses:
if len(local_facts.clauses) == 1:
cl, = local_facts.clauses
f, = cl if len(cl) == 1 else [None]
if f and f.is_Not and f.arg in known_facts_dict.get(key, []):
return False
for clause in local_facts.clauses:
if len(clause) == 1:
f, = clause
fdict = known_facts_dict.get(f.arg,
None) if not f.is_Not else None
if fdict and key in fdict:
return True
if fdict and Not(key) in known_facts_dict[f.arg]:
return False
# direct resolution method, no logic
res = key(*args)._eval_ask(assumptions)
if res is not None:
return bool(res)
# using satask (still costly)
res = satask(proposition, assumptions=assumptions, context=context)
return res
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(4*x), Q.integer(x))
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))
It is however a work in progress.
"""
from sympy.assumptions.satask import satask
if not isinstance(proposition,
(BooleanFunction, AppliedPredicate, bool, BooleanAtom)):
raise TypeError("proposition must be a valid logical expression")
if not isinstance(assumptions,
(BooleanFunction, AppliedPredicate, bool, BooleanAtom)):
raise TypeError("assumptions must be a valid logical expression")
if isinstance(proposition, AppliedPredicate):
key, expr = proposition.func, sympify(proposition.arg)
else:
key, expr = Q.is_true, sympify(proposition)
assump = CNF.from_prop(assumptions)
assump.extend(context)
local_facts = _extract_all_facts(assump, expr)
known_facts_cnf = get_all_known_facts()
known_facts_dict = get_known_facts_dict()
enc_cnf = EncodedCNF()
enc_cnf.from_cnf(CNF(known_facts_cnf))
enc_cnf.add_from_cnf(local_facts)
if local_facts.clauses and satisfiable(enc_cnf) is False:
raise ValueError("inconsistent assumptions %s" % assumptions)
if local_facts.clauses:
if len(local_facts.clauses) == 1:
cl, = local_facts.clauses
f, = cl if len(cl) == 1 else [None]
if f and f.is_Not and f.arg in known_facts_dict.get(key, []):
return False
for clause in local_facts.clauses:
if len(clause) == 1:
f, = clause
fdict = known_facts_dict.get(f.arg,
None) if not f.is_Not else None
if fdict and key in fdict:
return True
if fdict and Not(key) in known_facts_dict[f.arg]:
return False
# direct resolution method, no logic
res = key(expr)._eval_ask(assumptions)
if res is not None:
return bool(res)
# using satask (still costly)
res = satask(proposition, assumptions=assumptions, context=context)
return res
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_facts()``. 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()
exprs = None
all_exprs = set()
while True:
if i == 0:
exprs = extract_predargs(proposition, assumptions, context)
all_exprs |= exprs
exprs, relevant_facts = get_relevant_facts(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
def ask(proposition, assumptions=True, context=global_assumptions):
"""
Function to evaluate the proposition with assumptions.
Explanation
===========
This function evaluates the proposition to ``True`` or ``False`` if
the truth value can be determined. If not, it returns ``None``.
It should be discerned from :func:`~.refine()` which, when applied to a
proposition, simplifies the argument to symbolic ``Boolean`` instead of
Python built-in ``True``, ``False`` or ``None``.
**Syntax**
* ask(proposition)
Evaluate the *proposition* in global assumption context.
* ask(proposition, assumptions)
Evaluate the *proposition* with respect to *assumptions* in
global assumption context.
Parameters
==========
proposition : Any boolean expression.
Proposition which will be evaluated to boolean value. If this is
not ``AppliedPredicate``, it will be wrapped by ``Q.is_true``.
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.
Returns
=======
``True``, ``False``, or ``None``
Raises
======
TypeError : *proposition* or *assumptions* is not valid logical expression.
ValueError : assumptions are inconsistent.
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(4*x), Q.integer(x))
False
If the truth value cannot be determined, ``None`` will be returned.
>>> print(ask(Q.odd(3*x))) # cannot determine unless we know x
None
``ValueError`` is raised if assumptions are inconsistent.
>>> ask(Q.integer(x), Q.even(x) & Q.odd(x))
Traceback (most recent call last):
...
ValueError: inconsistent assumptions Q.even(x) & Q.odd(x)
Notes
=====
Relations in assumptions are not implemented (yet), so the following
will not give a meaningful result.
>>> ask(Q.positive(x), x > 0)
It is however a work in progress.
See Also
========
sympy.assumptions.refine.refine : Simplification using assumptions.
Proposition is not reduced to ``None`` if the truth value cannot
be determined.
"""
from sympy.assumptions.satask import satask
proposition = sympify(proposition)
assumptions = sympify(assumptions)
if isinstance(proposition,
Predicate) or proposition.kind is not BooleanKind:
raise TypeError("proposition must be a valid logical expression")
if isinstance(assumptions,
Predicate) or assumptions.kind is not BooleanKind:
raise TypeError("assumptions must be a valid logical expression")
binrelpreds = {Eq: Q.eq, Ne: Q.ne, Gt: Q.gt, Lt: Q.lt, Ge: Q.ge, Le: Q.le}
if isinstance(proposition, AppliedPredicate):
key, args = proposition.function, proposition.arguments
elif proposition.func in binrelpreds:
key, args = binrelpreds[proposition.func], proposition.args
else:
key, args = Q.is_true, (proposition, )
# convert local and global assumptions to CNF
assump_cnf = CNF.from_prop(assumptions)
assump_cnf.extend(context)
# extract the relevant facts from assumptions with respect to args
local_facts = _extract_all_facts(assump_cnf, args)
# convert default facts and assumed facts to encoded CNF
known_facts_cnf = get_all_known_facts()
enc_cnf = EncodedCNF()
enc_cnf.from_cnf(CNF(known_facts_cnf))
enc_cnf.add_from_cnf(local_facts)
# check the satisfiability of given assumptions
if local_facts.clauses and satisfiable(enc_cnf) is False:
raise ValueError("inconsistent assumptions %s" % assumptions)
# quick computation for single fact
res = _ask_single_fact(key, local_facts)
if res is not None:
return res
# direct resolution method, no logic
res = key(*args)._eval_ask(assumptions)
if res is not None:
return bool(res)
# using satask (still costly)
res = satask(proposition, assumptions=assumptions, context=context)
return res