def formulaDistributeDisjunctions(f):
"""
Convert a Skolemized formula in prefix-NNF form into Conjunctive
Normal Form.
"""
arg1 = None
arg2 = None
if f.isQuantified():
arg1 = f.child1
arg2 = formulaDistributeDisjunctions(f.child2)
f = Formula(f.op, arg1, arg2)
elif f.isLiteral():
pass
else:
if f.hasSubform1():
arg1 = formulaDistributeDisjunctions(f.child1)
if f.hasSubform2():
arg2 = formulaDistributeDisjunctions(f.child2)
f = Formula(f.op, arg1, arg2)
if f.op == "|":
if f.child1.op == "&":
# (P&Q)|R -> (P|R) & (Q|R)
arg1 = Formula("|", f.child1.child1, f.child2)
arg2 = Formula("|", f.child1.child2, f.child2)
f = Formula("&", arg1, arg2)
f = formulaDistributeDisjunctions(f)
elif f.child2.op == "&":
# (R|(P&Q) -> (R|P) & (R|Q)
arg1 = Formula("|", f.child1, f.child2.child1)
arg2 = Formula("|", f.child1, f.child2.child2)
f = Formula("&", arg1, arg2)
f = formulaDistributeDisjunctions(f)
return f
def formulaRekSkolemize(f, variables, subst):
"""
Perform Skolemization of f, which is assumed to be in the scope of
the list of variables provided.
"""
if f.isLiteral():
child = f.child1.instantiate(subst)
f = Formula("", child)
elif f.op == "?":
var = f.child1
skTerm = skolemGenerator(variables)
oldbinding = subst.modifyBinding((var, skTerm))
f = formulaRekSkolemize(f.child2, variables, subst)
subst.modifyBinding((var, oldbinding))
elif f.op == "!":
var = f.child1
variables.append(var)
handle = formulaRekSkolemize(f.child2, variables, subst)
f = Formula("!", var, handle)
variables.pop()
else:
arg1 = None
arg2 = None
if f.hasSubform1():
arg1 = formulaRekSkolemize(f.child1, variables, subst)
if f.hasSubform2():
arg2 = formulaRekSkolemize(f.child2, variables, subst)
f = Formula(f.op, arg1, arg2)
return f
def formulaMiniScope(f):
"""
Perform miniscoping, i.e. move quantors in as far as possible, so
that their scope is only the smallest subformula in which the
variable occurs.
"""
res = False
if f.isQuantified():
op = f.child2.op
quant = f.op
var = f.child1
subf = f.child2
if op == "&" or op == "|":
if not var in subf.child1.collectFreeVars():
# q[X]:(P op Q) -> P op (q[X]:Q) if X not free in P
arg2 = Formula(quant, var, subf.child2)
arg1 = subf.child1
f = Formula(op, arg1, arg2)
res = True
elif not var in subf.child2.collectFreeVars():
# q[X]:(P op Q) -> (q[X]:P) op Q if X not free in Q
arg1 = Formula(quant, var, subf.child1)
arg2 = subf.child2
f = Formula(op, arg1, arg2)
res = True
else:
if op == "&" and quant == "!":
# ![X]:(P&Q) -> ![X]:P & ![X]:Q
arg1 = Formula("!", var, subf.child1)
arg2 = Formula("!", var, subf.child2)
f = Formula("&", arg1, arg2)
res = True
elif op == "|" and quant == "?":
# ?[X]:(P|Q) -> ?[X]:P | ?[X]:Q
arg1 = Formula("?", var, subf.child1)
arg2 = Formula("?", var, subf.child2)
f = Formula("|", arg1, arg2)
res = True
arg1 = f.child1
arg2 = f.child2
modified = False
if f.hasSubform1():
arg1, m = formulaMiniScope(f.child1)
modified |= m
if f.hasSubform2():
arg2, m = formulaMiniScope(f.child2)
modified |= m
if modified:
f = Formula(f.op, arg1, arg2)
f, m = formulaMiniScope(f)
res = True
return f, res
def formulaVarRename(f, subst=None):
"""
Rename variables in f so that all bound variables are unique.
"""
if subst == None:
subst = Substitution()
if f.isQuantified():
# New scope of a variable -> add a new binding to a new
# variable. Store potential old binding to restore when
# leaving the scope later
var = f.child1
newvar = freshVar()
oldbinding = subst.modifyBinding((var, newvar))
if f.isLiteral():
# Create copy with the new variables recorded in subst
child = f.child1.instantiate(subst)
f = Formula("", child)
else:
# This is a composite formula. Rename it...
arg1 = None
arg2 = None
if f.isQuantified():
# Apply new renaming locally to the bound variable and
# recusively to the subformula
arg1 = newvar
arg2 = formulaVarRename(f.child2, subst)
else:
# Apply renaming to all subformulas
if f.hasSubform1():
arg1 = formulaVarRename(f.child1, subst)
if f.hasSubform2():
arg2 = formulaVarRename(f.child2, subst)
f = Formula(f.op, arg1, arg2)
if f.isQuantified():
# We are leaving the scope of the quantifier, so restore
# substitution.
subst.modifyBinding((var, oldbinding))
return f