309 lines
9.0 KiB
Python
309 lines
9.0 KiB
Python
"""Implementation of DPLL algorithm
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Further improvements: eliminate calls to pl_true, implement branching rules,
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efficient unit propagation.
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References:
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- https://en.wikipedia.org/wiki/DPLL_algorithm
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- https://www.researchgate.net/publication/242384772_Implementations_of_the_DPLL_Algorithm
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"""
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from sympy.core.sorting import default_sort_key
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from sympy.logic.boolalg import Or, Not, conjuncts, disjuncts, to_cnf, \
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to_int_repr, _find_predicates
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from sympy.assumptions.cnf import CNF
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from sympy.logic.inference import pl_true, literal_symbol
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def dpll_satisfiable(expr):
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"""
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Check satisfiability of a propositional sentence.
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It returns a model rather than True when it succeeds
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>>> from sympy.abc import A, B
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>>> from sympy.logic.algorithms.dpll import dpll_satisfiable
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>>> dpll_satisfiable(A & ~B)
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{A: True, B: False}
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>>> dpll_satisfiable(A & ~A)
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False
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"""
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if not isinstance(expr, CNF):
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clauses = conjuncts(to_cnf(expr))
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else:
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clauses = expr.clauses
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if False in clauses:
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return False
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symbols = sorted(_find_predicates(expr), key=default_sort_key)
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symbols_int_repr = set(range(1, len(symbols) + 1))
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clauses_int_repr = to_int_repr(clauses, symbols)
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result = dpll_int_repr(clauses_int_repr, symbols_int_repr, {})
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if not result:
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return result
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output = {}
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for key in result:
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output.update({symbols[key - 1]: result[key]})
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return output
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def dpll(clauses, symbols, model):
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"""
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Compute satisfiability in a partial model.
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Clauses is an array of conjuncts.
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>>> from sympy.abc import A, B, D
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>>> from sympy.logic.algorithms.dpll import dpll
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>>> dpll([A, B, D], [A, B], {D: False})
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False
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"""
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# compute DP kernel
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P, value = find_unit_clause(clauses, model)
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while P:
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model.update({P: value})
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symbols.remove(P)
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if not value:
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P = ~P
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clauses = unit_propagate(clauses, P)
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P, value = find_unit_clause(clauses, model)
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P, value = find_pure_symbol(symbols, clauses)
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while P:
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model.update({P: value})
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symbols.remove(P)
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if not value:
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P = ~P
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clauses = unit_propagate(clauses, P)
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P, value = find_pure_symbol(symbols, clauses)
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# end DP kernel
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unknown_clauses = []
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for c in clauses:
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val = pl_true(c, model)
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if val is False:
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return False
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if val is not True:
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unknown_clauses.append(c)
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if not unknown_clauses:
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return model
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if not clauses:
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return model
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P = symbols.pop()
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model_copy = model.copy()
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model.update({P: True})
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model_copy.update({P: False})
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symbols_copy = symbols[:]
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return (dpll(unit_propagate(unknown_clauses, P), symbols, model) or
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dpll(unit_propagate(unknown_clauses, Not(P)), symbols_copy, model_copy))
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def dpll_int_repr(clauses, symbols, model):
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"""
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Compute satisfiability in a partial model.
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Arguments are expected to be in integer representation
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>>> from sympy.logic.algorithms.dpll import dpll_int_repr
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>>> dpll_int_repr([{1}, {2}, {3}], {1, 2}, {3: False})
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False
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"""
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# compute DP kernel
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P, value = find_unit_clause_int_repr(clauses, model)
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while P:
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model.update({P: value})
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symbols.remove(P)
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if not value:
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P = -P
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clauses = unit_propagate_int_repr(clauses, P)
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P, value = find_unit_clause_int_repr(clauses, model)
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P, value = find_pure_symbol_int_repr(symbols, clauses)
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while P:
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model.update({P: value})
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symbols.remove(P)
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if not value:
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P = -P
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clauses = unit_propagate_int_repr(clauses, P)
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P, value = find_pure_symbol_int_repr(symbols, clauses)
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# end DP kernel
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unknown_clauses = []
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for c in clauses:
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val = pl_true_int_repr(c, model)
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if val is False:
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return False
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if val is not True:
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unknown_clauses.append(c)
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if not unknown_clauses:
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return model
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P = symbols.pop()
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model_copy = model.copy()
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model.update({P: True})
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model_copy.update({P: False})
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symbols_copy = symbols.copy()
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return (dpll_int_repr(unit_propagate_int_repr(unknown_clauses, P), symbols, model) or
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dpll_int_repr(unit_propagate_int_repr(unknown_clauses, -P), symbols_copy, model_copy))
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### helper methods for DPLL
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def pl_true_int_repr(clause, model={}):
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"""
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Lightweight version of pl_true.
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Argument clause represents the set of args of an Or clause. This is used
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inside dpll_int_repr, it is not meant to be used directly.
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>>> from sympy.logic.algorithms.dpll import pl_true_int_repr
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>>> pl_true_int_repr({1, 2}, {1: False})
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>>> pl_true_int_repr({1, 2}, {1: False, 2: False})
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False
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"""
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result = False
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for lit in clause:
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if lit < 0:
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p = model.get(-lit)
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if p is not None:
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p = not p
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else:
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p = model.get(lit)
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if p is True:
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return True
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elif p is None:
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result = None
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return result
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def unit_propagate(clauses, symbol):
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"""
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Returns an equivalent set of clauses
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If a set of clauses contains the unit clause l, the other clauses are
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simplified by the application of the two following rules:
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1. every clause containing l is removed
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2. in every clause that contains ~l this literal is deleted
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Arguments are expected to be in CNF.
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>>> from sympy.abc import A, B, D
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>>> from sympy.logic.algorithms.dpll import unit_propagate
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>>> unit_propagate([A | B, D | ~B, B], B)
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[D, B]
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"""
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output = []
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for c in clauses:
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if c.func != Or:
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output.append(c)
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continue
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for arg in c.args:
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if arg == ~symbol:
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output.append(Or(*[x for x in c.args if x != ~symbol]))
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break
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if arg == symbol:
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break
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else:
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output.append(c)
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return output
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def unit_propagate_int_repr(clauses, s):
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"""
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Same as unit_propagate, but arguments are expected to be in integer
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representation
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>>> from sympy.logic.algorithms.dpll import unit_propagate_int_repr
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>>> unit_propagate_int_repr([{1, 2}, {3, -2}, {2}], 2)
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[{3}]
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"""
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negated = {-s}
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return [clause - negated for clause in clauses if s not in clause]
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def find_pure_symbol(symbols, unknown_clauses):
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"""
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Find a symbol and its value if it appears only as a positive literal
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(or only as a negative) in clauses.
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>>> from sympy.abc import A, B, D
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>>> from sympy.logic.algorithms.dpll import find_pure_symbol
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>>> find_pure_symbol([A, B, D], [A|~B,~B|~D,D|A])
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(A, True)
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"""
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for sym in symbols:
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found_pos, found_neg = False, False
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for c in unknown_clauses:
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if not found_pos and sym in disjuncts(c):
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found_pos = True
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if not found_neg and Not(sym) in disjuncts(c):
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found_neg = True
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if found_pos != found_neg:
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return sym, found_pos
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return None, None
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def find_pure_symbol_int_repr(symbols, unknown_clauses):
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"""
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Same as find_pure_symbol, but arguments are expected
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to be in integer representation
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>>> from sympy.logic.algorithms.dpll import find_pure_symbol_int_repr
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>>> find_pure_symbol_int_repr({1,2,3},
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... [{1, -2}, {-2, -3}, {3, 1}])
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(1, True)
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"""
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all_symbols = set().union(*unknown_clauses)
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found_pos = all_symbols.intersection(symbols)
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found_neg = all_symbols.intersection([-s for s in symbols])
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for p in found_pos:
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if -p not in found_neg:
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return p, True
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for p in found_neg:
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if -p not in found_pos:
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return -p, False
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return None, None
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def find_unit_clause(clauses, model):
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"""
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A unit clause has only 1 variable that is not bound in the model.
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>>> from sympy.abc import A, B, D
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>>> from sympy.logic.algorithms.dpll import find_unit_clause
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>>> find_unit_clause([A | B | D, B | ~D, A | ~B], {A:True})
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(B, False)
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"""
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for clause in clauses:
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num_not_in_model = 0
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for literal in disjuncts(clause):
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sym = literal_symbol(literal)
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if sym not in model:
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num_not_in_model += 1
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P, value = sym, not isinstance(literal, Not)
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if num_not_in_model == 1:
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return P, value
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return None, None
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def find_unit_clause_int_repr(clauses, model):
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"""
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Same as find_unit_clause, but arguments are expected to be in
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integer representation.
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>>> from sympy.logic.algorithms.dpll import find_unit_clause_int_repr
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>>> find_unit_clause_int_repr([{1, 2, 3},
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... {2, -3}, {1, -2}], {1: True})
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(2, False)
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"""
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bound = set(model) | {-sym for sym in model}
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for clause in clauses:
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unbound = clause - bound
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if len(unbound) == 1:
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p = unbound.pop()
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if p < 0:
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return -p, False
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else:
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return p, True
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return None, None
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