Source code for diofant.logic.inference

"""Inference in propositional logic"""

from ..core import sympify
from ..core.compatibility import ordered
from .boolalg import And, Not, conjuncts, to_cnf


def literal_symbol(literal):
    """
    The symbol in this literal (without the negation).

    Examples
    ========

    >>> literal_symbol(a)
    a
    >>> literal_symbol(~a)
    a
    """

    if literal is True or literal is False:
        return literal
    try:
        if literal.is_Symbol:
            return literal
        if literal.is_Not:
            return literal_symbol(literal.args[0])
        else:
            raise ValueError
    except (AttributeError, ValueError):
        raise ValueError("Argument must be a boolean literal.")


[docs]def satisfiable(expr, algorithm="dpll2", all_models=False): """ Check satisfiability of a propositional sentence. Returns a model when it succeeds. Returns {true: true} for trivially true expressions. On setting all_models to True, if given expr is satisfiable then returns a generator of models. However, if expr is unsatisfiable then returns a generator containing the single element False. Examples ======== >>> satisfiable(a & ~b) {a: True, b: False} >>> satisfiable(a & ~a) False >>> satisfiable(True) {true: True} >>> next(satisfiable(a & ~a, all_models=True)) False >>> models = satisfiable((a >> b) & b, all_models=True) >>> next(models) {a: False, b: True} >>> next(models) {a: True, b: True} >>> def use_models(models): ... for model in models: ... if model: ... # Do something with the model. ... return model ... else: ... # Given expr is unsatisfiable. ... print("UNSAT") >>> use_models(satisfiable(a >> ~a, all_models=True)) {a: False} >>> use_models(satisfiable(a ^ a, all_models=True)) UNSAT """ expr = to_cnf(expr) if algorithm == "dpll": from .algorithms.dpll import dpll_satisfiable return dpll_satisfiable(expr) elif algorithm == "dpll2": from .algorithms.dpll2 import dpll_satisfiable return dpll_satisfiable(expr, all_models) else: raise NotImplementedError
def valid(expr): """ Check validity of a propositional sentence. A valid propositional sentence is True under every assignment. Examples ======== >>> valid(a | ~a) True >>> valid(a | b) False References ========== * https//en.wikipedia.org/wiki/Validity """ return not satisfiable(Not(expr)) def pl_true(expr, model={}, deep=False): """ Returns whether the given assignment is a model or not. If the assignment does not specify the value for every proposition, this may return None to indicate 'not obvious'. Parameters ========== model : dict, optional, default: {} Mapping of symbols to boolean values to indicate assignment. deep: boolean, optional, default: False Gives the value of the expression under partial assignments correctly. May still return None to indicate 'not obvious'. Examples ======== >>> pl_true(a & b, {a: True, b: True}) True >>> pl_true(a & b, {a: False}) False >>> pl_true(a & b, {a: True}) >>> pl_true(a & b, {a: True}, deep=True) >>> pl_true(a >> (b >> a)) >>> pl_true(a >> (b >> a), deep=True) True >>> pl_true(a & ~a) >>> pl_true(a & ~a, deep=True) False >>> pl_true(a & b & (~a | ~b), {a: True}) >>> pl_true(a & b & (~a | ~b), {a: True}, deep=True) False """ from ..core import Symbol from .boolalg import BooleanFunction boolean = (True, False) def _validate(expr): if isinstance(expr, Symbol) or expr in boolean: return True if not isinstance(expr, BooleanFunction): return False return all(_validate(arg) for arg in expr.args) if expr in boolean: return expr expr = sympify(expr) if not _validate(expr): raise ValueError("%s is not a valid boolean expression" % expr) model = {k: v for k, v in model.items() if v in boolean} result = expr.subs(model) if result in boolean: return bool(result) if deep: model = {k: True for k in result.atoms()} if pl_true(result, model): if valid(result): return True else: if not satisfiable(result): return False return def entails(expr, formula_set={}): """ Check whether the given expr_set entail an expr. If formula_set is empty then it returns the validity of expr. Examples ======== >>> entails(a, [a >> b, b >> c]) False >>> entails(c, [a >> b, b >> c, a]) True >>> entails(a >> b) False >>> entails(a >> (b >> a)) True References ========== * https//en.wikipedia.org/wiki/Logical_consequence """ formula_set = list(formula_set) formula_set.append(Not(expr)) return not satisfiable(And(*formula_set)) class KB: """Base class for all knowledge bases""" def __init__(self, sentence=None): self.clauses_ = set() if sentence: self.tell(sentence) def tell(self, sentence): raise NotImplementedError def ask(self, query): raise NotImplementedError def retract(self, sentence): raise NotImplementedError @property def clauses(self): return list(ordered(self.clauses_)) class PropKB(KB): """A KB for Propositional Logic. Inefficient, with no indexing.""" def tell(self, sentence): """Add the sentence's clauses to the KB Examples ======== >>> l = PropKB() >>> l.clauses [] >>> l.tell(x | y) >>> l.clauses [Or(x, y)] >>> l.tell(y) >>> l.clauses [y, Or(x, y)] """ for c in conjuncts(to_cnf(sentence)): self.clauses_.add(c) def ask(self, query): """Checks if the query is true given the set of clauses. Examples ======== >>> l = PropKB() >>> l.tell(x & ~y) >>> l.ask(x) True >>> l.ask(y) False """ return entails(query, self.clauses_) def retract(self, sentence): """Remove the sentence's clauses from the KB Examples ======== >>> l = PropKB() >>> l.clauses [] >>> l.tell(x | y) >>> l.clauses [Or(x, y)] >>> l.retract(x | y) >>> l.clauses [] """ for c in conjuncts(to_cnf(sentence)): self.clauses_.discard(c)