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