def remove_conditionals(f):
""" Return a copy of f where conditionals are replaced by & and |. """
if f.op == OP_IMPLIES:
p, q = f.args[0], f.args[1]
return (remove_conditionals(~p) | remove_conditionals(q))
elif f.op == OP_IMPLIED_BY:
p, q = f.args[0], f.args[1]
return (remove_conditionals(p) | remove_conditionals(~q))
elif f.op == OP_EQUIVALENT:
p, q = f.args[0], f.args[1]
return (remove_conditionals(p) & remove_conditionals(q) |
remove_conditionals(~p) & remove_conditionals(~q))
elif is_quantified(f):
return Quantifier(f.op, f.vars, remove_conditionals(f.args[0]))
else:
return Expr(f.op, *[remove_conditionals(x) for x in f.args])
def push_neg(f):
""" Return a copy of f that has negation as close to terms as possible.
Unlike nnf(), push_neg() leaves conditionals as they are.
"""
if f.op == OP_NOT:
arg = f.args[0]
if arg.op == OP_NOT:
return push_neg(arg.args[0])
elif arg.op == OP_AND:
return Expr(OP_OR, *[push_neg(Expr(OP_NOT, x)) for x in arg.args])
elif arg.op == OP_OR:
return Expr(OP_AND, *[push_neg(Expr(OP_NOT, x)) for x in arg.args])
elif arg.op == OP_IMPLIES: # p -> q
p, q = arg.args[0], arg.args[1]
return ~(push_neg(p) >> push_neg(q))
elif arg.op == OP_IMPLIED_BY: # p <- q
p, q = arg.args[0], arg.args[1]
return ~(push_neg(p) << push_neg(q))
elif arg.op == OP_EQUIVALENT: # p <-> q
p, q = arg.args[0], arg.args[1]
return ~(push_neg(p) ** push_neg(q))
elif arg.op == OP_FORALL:
return Quantifier(OP_EXISTS, arg.vars,
push_neg(Expr(OP_NOT, arg.args[0])))
elif arg.op == OP_EXISTS:
return Quantifier(OP_FORALL, arg.vars,
push_neg(Expr(OP_NOT, arg.args[0])))
else:
return f
elif f.op == OP_AND or f.op == OP_OR:
return Expr(f.op, *[push_neg(x) for x in f.args])
elif f.op == OP_IMPLIES:
p, q = f.args[0], f.args[1]
return (push_neg(p) >> push_neg(q))
elif f.op == OP_IMPLIED_BY:
p, q = f.args[0], f.args[1]
return (push_neg(p) << push_neg(q))
elif f.op == OP_EQUIVALENT:
p, q = f.args[0], f.args[1]
return (push_neg(p) ** push_neg(q))
elif is_quantified(f):
return Quantifier(f.op, f.vars, push_neg(f.args[0]))
else:
return f
def helper(f, used, substs):
if is_variable(f):
if f in substs: return substs[f]
else: return f
if is_quantified(f):
# does this quantifier use a variable that is already used?
clashing = (used & set(f.vars)) # set intersection
if len(clashing) > 0:
for var in clashing:
existing = used.union(*list(map(vars, substs.values())))
# rename any clashing variable
var2 = variant(var, existing)
substs[var] = var2
used.update(f.vars)
arg = helper(f.args[0], used, substs)
return Quantifier(f.op,
[subst(substs, x) for x in f.vars],
*[subst(substs, x) for x in f.args])
else:
used.update(f.vars)
arg = helper(f.args[0], used, substs)
return Quantifier(f.op, f.vars, arg)
else:
return Expr(f.op, *[helper(x, used, substs) for x in f.args])
def drop(f):
if is_quantified(f):
return drop(f.args[0])
else:
return f
def pushquants(f):
# print('pushquant({0})'.format(f))
if is_quantified(f):
arg = f.args[0]
if arg.op == OP_NOT:
return ~pushquants(Quantifier(opposite(f.op), f.vars, arg.args[0]))
elif is_quantified(arg):
arg1 = pushquants(arg)
# did pushquants have any effect?
if arg1 == arg:
# # if no, see if changing the order of the quants has an effect
# arg2 = Quantifier(f.op, f.vars, arg.args[0])
# arg2_ = pushquants(arg2)
# if arg2 == arg2_:
# return Quantifier(f.op, f.vars, arg1)
# else:
# return Quantifier(arg.op, arg.vars, arg2_)
return Quantifier(f.op, f.vars, arg1)
else:
return pushquants(Quantifier(f.op, f.vars, arg1))
elif arg.op in [OP_AND, OP_OR]:
arg1 = arg.args[0]
arg2 = arg.args[1]
variable = f.vars[0]
if variable in fvars(arg1) and variable in fvars(arg2):
if ((arg.op == OP_AND and f.op == OP_FORALL) or
(arg.op == OP_OR and f.op == OP_EXISTS)):
return pushquants(Expr(arg.op,
Quantifier(f.op, f.vars, arg1),
Quantifier(f.op, f.vars, arg2)))
else:
return Quantifier(f.op, f.vars, pushquants(arg))
if variable in fvars(arg1):
return pushquants(Expr(arg.op,
Quantifier(f.op, f.vars, arg1), arg2))
if variable in fvars(arg2):
return pushquants(Expr(arg.op, arg1,
Quantifier(f.op, f.vars, arg2)))
else:
get_log().warning('Dropped quantifier "{0} : {1}" from "{2}"'
.format(f.op, f.vars, f))
return arg
elif arg.op == OP_IMPLIES:
arg1 = arg.args[0]
arg2 = arg.args[1]
variable = f.vars[0]
if variable in fvars(arg1) and variable in fvars(arg2):
return Quantifier(f.op, f.vars, pushquants(arg))
elif variable in fvars(arg1):
quant = opposite(f.op)
return (pushquants(Quantifier(quant, variable, arg1)) >>
pushquants(arg2))
elif variable in fvars(arg2):
return (pushquants(arg1) >>
pushquants(Quantifier(f.op, variable, arg2)))
else:
get_log().warning('Dropped quantifier "{0} : {1}" from "{2}"'
.format(f.op, f.vars, f))
return (pushquants(arg1) >> pushquants(arg2))
elif arg.op == OP_IMPLIED_BY:
arg1 = arg.args[0]
arg2 = arg.args[1]
variable = f.vars[0]
if variable in fvars(arg1) and variable in fvars(arg2):
return Quantifier(f.op, f.vars, pushquants(arg))
elif variable in fvars(arg1):
return (pushquants(Quantifier(f.op, variable, arg1)) <<
pushquants(arg2))
elif variable in fvars(arg2):
quant = opposite(f.op)
return (pushquants(arg1) <<
pushquants(Quantifier(quant, variable, arg2)))
else:
get_log().warning('Dropped quantifier "{0} : {1}" from "{2}"'
.format(f.op, f.vars, f))
return (pushquants(arg1) << pushquants(arg2))
else:
return Quantifier(f.op, f.vars[0], pushquants(f.args[0]))
elif f.op == OP_NOT:
return ~pushquants(f.args[0])
elif f.op in BINARY_LOGIC_OPS:
return Expr(f.op, *[pushquants(x) for x in f.args])
else:
return f
def pullquants(f):
# print('pullquants({0})'.format(str(f)))
if f.op == OP_AND:
arg1 = pullquants(f.args[0])
arg2 = pullquants(f.args[1])
if arg1.op == OP_FORALL and arg2.op == OP_FORALL:
return rename_and_pull(f, OP_FORALL,
arg1.vars[0], arg2.vars[0],
arg1.args[0], arg2.args[0])
elif is_quantified(arg1):
return rename_and_pull(f, arg1.op,
arg1.vars[0], None,
arg1.args[0], arg2)
elif is_quantified(arg2):
return rename_and_pull(f, arg2.op,
None, arg2.vars[0],
arg1, arg2.args[0])
else:
return (arg1 & arg2)
elif f.op == OP_OR:
arg1 = pullquants(f.args[0])
arg2 = pullquants(f.args[1])
if arg1.op == OP_EXISTS and arg2.op == OP_EXISTS:
return rename_and_pull(f, OP_EXISTS,
arg1.vars[0], arg2.vars[0],
arg1.args[0], arg2.args[0])
elif is_quantified(arg1):
return rename_and_pull(f, arg1.op,
arg1.vars[0], None,
arg1.args[0], arg2)
elif is_quantified(arg2):
return rename_and_pull(f, arg2.op,
None, arg2.vars[0],
arg1, arg2.args[0])
else:
return (arg1 | arg2)
elif f.op == OP_IMPLIES:
arg1 = pullquants(f.args[0])
arg2 = pullquants(f.args[1])
if is_quantified(arg1):
return rename_and_pull(f, opposite(arg1.op),
arg1.vars[0], None,
arg1.args[0], arg2)
elif is_quantified(arg2):
return rename_and_pull(f, arg2.op,
None, arg2.vars[0],
arg1, arg2.args[0])
else:
return (arg1 >> arg2)
elif f.op == OP_IMPLIED_BY:
arg1 = pullquants(f.args[0])
arg2 = pullquants(f.args[1])
if is_quantified(arg1):
return rename_and_pull(f, arg1.op,
arg1.vars[0], None,
arg1.args[0], arg2)
elif is_quantified(arg2):
return rename_and_pull(f, opposite(arg2.op),
None, arg2.vars[0],
arg1, arg2.args[0])
else:
return (arg1 << arg2)
elif f.op == OP_EQUIVALENT:
arg1 = pullquants(f.args[0])
arg2 = pullquants(f.args[1])
return pullquants((arg1 >> arg2) & (arg1 << arg2))
elif f.op == OP_NOT:
arg = pullquants(f.args[0])
if is_quantified(arg):
return Quantifier(opposite(arg.op), f.vars, ~(arg.args[0]))
else:
return (~arg)
else:
return f
def kleene(f):
""" Take a FOL formula and try to simplify it. """
# print('\nSimplifying op "{0}" args "{1}"'.format(f.op, f.args))
if is_quantified(f): # remove variable that are not in f
vars = set(f.vars)
fv = fvars(f.args[0])
needed = vars & fv
if needed == set():
return kleene(f.args[0])
else:
arg = kleene(f.args[0])
variables = [v for v in f.vars if v in needed] # keep the same order
if (arg != f.args[0]) or (len(needed) < len(f.vars)):
return kleene(Quantifier(f.op, variables, arg))
else:
return f
elif f.op == OP_NOT:
arg = f.args[0]
if arg.op == OP_NOT: # double neg
return kleene(arg.args[0])
elif is_true(arg): # -TRUE --> FALSE
return Expr(OP_FALSE)
elif is_false(arg): # -FALSE --> TRUE
return Expr(OP_TRUE)
elif arg.op == OP_EQUALS:
return Expr(OP_NOTEQUALS,
kleene(arg.args[0]),
kleene(arg.args[1]))
elif arg.op == OP_NOTEQUALS:
return Expr(OP_EQUALS,
kleene(arg.args[0]),
kleene(arg.args[1]))
else:
arg2 = kleene(arg)
if arg2 != arg:
return kleene(Expr(OP_NOT, arg2))
return f
elif f.op == OP_AND:
if any(map(is_false, f.args)): # if one conjuct is FALSE, expr is FALSE
return Expr(OP_FALSE)
elif len(f.args) == 1:
return kleene(f.args[0])
else: # remove conjuncts that are TRUE and simplify args
args = list(map(kleene, filter(lambda x: not is_true(x),f.args)))
used = set()
unique = []
for arg in args:
if arg not in used:
used.add(arg)
unique.append(arg)
if args != unique or f.args != unique:
return kleene(Expr(OP_AND, *unique))
else:
return f
elif f.op == OP_OR:
if any(map(is_true, f.args)): # if one conjuct is TRUE, expr is TRUE
return Expr(OP_TRUE)
elif len(f.args) == 1:
return kleene(f.args[0])
else: # remove conjuncts that are FALSE and simplify args
args = list(map(kleene, filter(lambda x: not is_false(x),f.args)))
used = set()
unique = []
for arg in args:
if arg not in used:
used.add(arg)
unique.append(arg)
if args != unique or f.args != unique:
return kleene(Expr(OP_OR, *unique))
else:
return f
elif f.op == OP_IMPLIES:
if is_false(f.args[0]) or is_true(f.args[1]):
return Expr(OP_TRUE)
elif is_true(f.args[0]):
return kleene(f.args[1])
elif is_false(f.args[1]):
return kleene(Expr(OP_NOT, kleene(f.args[0])))
else:
arg1 = kleene(f.args[0])
arg2 = kleene(f.args[1])
if arg1 != f.args[0] or arg2 != f.args[1]:
return kleene(Expr(OP_IMPLIES, arg1, arg2))
else:
return f
elif f.op == OP_IMPLIED_BY:
if is_false(f.args[1]) or is_true(f.args[0]):
return Expr(OP_TRUE)
elif is_true(f.args[1]):
return kleene(f.args[0])
elif is_false(f.args[0]):
return kleene(Expr(OP_NOT, kleene(f.args[1])))
else:
arg1 = kleene(f.args[0])
arg2 = kleene(f.args[1])
if arg1 != f.args[0] or arg2 != f.args[1]:
return kleene(Expr(OP_IMPLIES, arg1, arg2))
else:
return f
elif f.op == OP_EQUIVALENT:
if is_true(f.args[0]):
return kleene(f.args[1])
elif is_true(f.args[1]):
return kleene(f.args[0])
elif is_false(f.args[0]):
return kleene(Expr(OP_NOT, kleene(f.args[1])))
elif is_false(f.args[1]):
return kleene(Expr(OP_NOT, kleene(f.args[0])))
else:
arg1 = kleene(f.args[0])
arg2 = kleene(f.args[1])
if arg1 != f.args[0] or arg2 != f.args[1]:
return kleene(Expr(OP_EQUIVALENT, arg1, arg2))
else:
return f
else:
return f