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 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 separateQuantors(f, varlist=None):
"""
Remove all quantors from f, returning the quantor-free core of the
formula and a list of quanified variables. This will only be
applied to Skolemized formulas, thus finding an existential
quantor is an error. To be useful, the inpt formula also has to be
variable-normalized.
"""
if varlist == None:
varlist = list()
if f.isQuantified():
assert f.op == "!"
varlist.append(f.child1)
f, dummy = separateQuantors(f.child2, varlist)
elif f.isLiteral():
pass
else:
arg1 = None
arg2 = None
if f.hasSubform1():
arg1, dummy = separateQuantors(f.child1, varlist)
if f.hasSubform2():
arg2, dummy = separateQuantors(f.child2, varlist)
f = Formula(f.op, arg1, arg2)
return f, varlist
def formulaSimplify(f):
"""
Exhaustively apply simplification to f. See formulaTopSimplify()
above for the
Returns (f', True) if f!=f, (f', False) otherwise.
"""
if f.isLiteral():
return f, False
modified = False
# First simplify the subformulas
if f.hasSubform1():
child1, m = formulaSimplify(f.child1)
modified |= m
else:
child1 = f.child1
if f.hasSubform2():
child2, m = formulaSimplify(f.child2)
modified |= m
else:
child2 = None
if modified:
f = Formula(f.op, child1, child2)
topmod = True
while topmod:
f, topmod = formulaTopSimplify(f)
modified |= topmod
return f, modified
def formulaShiftQuantorsOut(f):
"""
Shift all (universal) quantor to the outermost level.
"""
f, varlist = separateQuantors(f)
while varlist:
f = Formula("!", varlist.pop(), f)
return f
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
def formulaOpSimplify(f):
"""
Simplify the formula by eliminating the <=, ~|, ~& and <~>. This
is not strictly necessary, but means fewer cases to consider
later. The following rules are applied exhaustively:
F~|G -> ~(F|G)
F~&G -> ~(F&G)
F<=G -> G=>F
F<~>G -> ~(F<=>G)
Returns f', modified
"""
if f.isLiteral():
return f, False
modified = False
# First simplify the subformulas
if f.hasSubform1():
child1, m = formulaOpSimplify(f.child1)
modified |= m
else:
child1 = f.child1
if f.hasSubform2():
child2, m = formulaOpSimplify(f.child2)
modified |= m
else:
child2 = None
if modified:
f = Formula(f.op, child1, child2)
if f.op == "<~>":
handle = Formula("<=>", f.child1, f.child2)
newform = Formula("~", handle)
return newform, True
elif f.op == "<=":
newform = Formula("=>", f.child2, f.child1)
return newform, True
elif f.op == "~|":
handle = Formula("|", f.child1, f.child2)
newform = Formula("~", handle)
return newform, True
elif f.op == "~&":
handle = Formula("&", f.child1, f.child2)
newform = Formula("~", handle)
return newform, True
return f, modified
def formulaNNF(f, polarity):
"""
Convert f into a NNF. Equivalences (<=>) are eliminated
polarity-dependend, top to bottom. Returns (f', m), where f' is a
NNF of f, and m indicates if f!=f'
"""
normalform = False
modified = False
while not normalform:
normalform = True
f, m = rootFormulaNNF(f, polarity)
modified |= m
if f.op == "~":
handle, m = formulaNNF(f.child1, -polarity)
if m:
normalform = False
f = Formula("~", handle)
elif f.op in ["!", "?"]:
handle, m = formulaNNF(f.child2, polarity)
if m:
normalform = False
f = Formula(f.op, f.child1, handle)
elif f.op in ["|", "&"]:
handle1, m1 = formulaNNF(f.child1, polarity)
handle2, m2 = formulaNNF(f.child2, polarity)
m = m1 or m2
if m:
normalform = False
f = Formula(f.op, handle1, handle2)
else:
assert f.isLiteral()
modified |= m
return f, modified
def formulaSimplify(f):
"""
Exhaustively apply simplification to f, creating the simplified
version f'. See formulaTopSimplify()
above for the individual rules.
Returns (f', True) if f'!=f, (f', False) otherwise (i.e. if no
simplification happened because the formula already was completely
simplified.
"""
if f.isLiteral():
return f, False
modified = False
# First simplify the subformulas
if f.hasSubform1():
child1, m = formulaSimplify(f.child1)
modified |= m
else:
child1 = f.child1
if f.hasSubform2():
child2, m = formulaSimplify(f.child2)
modified |= m
else:
child2 = None
if modified:
f = Formula(f.op, child1, child2)
topmod = True
while topmod:
f, topmod = formulaTopSimplify(f)
modified |= topmod
return f, modified
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 rootFormulaNNF(f, polarity):
"""
Apply all NNF transformation rules that can be applied at top
level. Return result and modification flag.
"""
normalform = False
modified = False
while not normalform:
normalform = True
m = False
if f.op == "~":
if f.child1.isLiteral():
# Move negations into literals
f = Formula("", f.child1.child1.negate())
m = True
elif f.child1.op == "|":
# De Morgan: ~(P|Q) -> ~P & ~Q
f = Formula("&", Formula("~", f.child1.child1),
Formula("~", f.child1.child2))
m = True
elif f.child1.op == "&":
# De Morgan: ~(P&Q) -> ~P | ~Q
f = Formula("|", Formula("~", f.child1.child1),
Formula("~", f.child1.child2))
m = True
elif f.child1.op == "!":
# ~(![X]:P) -> ?[X]:~P
f = Formula("?", f.child1.child1,
Formula("~", f.child1.child2))
m = True
elif f.child1.op == "?":
# ~(?[X]:P) -> ![X]:~P
f = Formula("!", f.child1.child1,
Formula("~", f.child1.child2))
m = True
elif f.op == "=>":
# Expand P=>Q into ~P|Q
f = Formula("|", Formula("~", f.child1), f.child2)
m = True
elif f.op == "<=>":
if polarity == 1:
# P<=>Q -> (P=>Q)&(Q=>P)
f = Formula("&", Formula("=>", f.child1, f.child2),
Formula("=>", f.child2, f.child1))
m = True
else:
assert polarity == -1
# P<=>Q -> (P & Q) | (~P & ~Q)
f = Formula(
"|", Formula("&", f.child1, f.child2),
Formula("&", Formula("~", f.child1),
Formula("~", f.child2)))
m = True
normalform = not m
modified |= m
return f, modified
def formulaTopSimplify(f):
"""
Try to apply the following simplification rules to f at the top
level. Return (f',m), where f' is the result of simplification,
and m indicates if f'!=f.
"""
if f.op == "~":
if f.child1.isLiteral():
# Push ~ into literals if possible. This covers the case
# of ~~P -> P if one of the negations is in the literal.
return Formula("", f.child1.child1.negate()), True
elif f.op == "|":
if f.child1.isPropConst(True):
# T | P -> T. Note that child1 is $true or
# equivalent. This applies to several other cases where we
# can reuse a $true or $false child instead of creating a
# new formula.
return f.child1, True
elif f.child2.isPropConst(True):
# P | T -> T
return f.child2, True
elif f.child1.isPropConst(False):
# F | P -> P
return f.child2, True
elif f.child2.isPropConst(False):
# P | F -> P
return f.child1, True
elif f.child1.isEqual(f.child2):
# P | P -> P
return f.child2, True
elif f.op == "&":
if f.child1.isPropConst(True):
# T & P -> P
return f.child2, True
elif f.child2.isPropConst(True):
# P & T -> P
return f.child1, True
elif f.child1.isPropConst(False):
# F & P -> F
return f.child1, True
elif f.child2.isPropConst(False):
# P & F -> F
return f.child2, True
elif f.child1.isEqual(f.child2):
# P & P -> P
return f.child2, True
elif f.op == "<=>":
if f.child1.isPropConst(True):
# T <=> P -> P
return f.child2, True
elif f.child2.isPropConst(True):
# P <=> T -> P
return f.child1, True
elif f.child1.isPropConst(False):
# P <=> F -> ~P
newform = Formula("~", f.child2)
newform, m = formulaSimplify(newform)
return newform, True
elif f.child2.isPropConst(False):
# F <=> P -> ~P
newform = Formula("~", f.child1)
newform, m = formulaSimplify(newform)
return newform, True
elif f.child1.isEqual(f.child2):
# P <=> P -> T
return Formula("", Literal(["$true"])), True
elif f.op == "=>":
if f.child1.isPropConst(True):
# T => P -> P
return f.child2, True
elif f.child1.isPropConst(False):
# F => P -> T
return Formula("", Literal(["$true"])), True
elif f.child2.isPropConst(True):
# P => T -> T
return Formula("", Literal(["$true"])), True
elif f.child2.isPropConst(False):
# P => F -> ~P
newform = Formula("~", f.child1)
newform, m = formulaSimplify(newform)
return newform, True
elif f.child1.isEqual(f.child2):
# P => P -> T
return Formula("", Literal(["$true"])), True
elif f.op in ["!", "?"]:
# ![X] F -> F if X is not free in F
# ?[X] F -> F if X is not free in F
vars = f.child2.collectFreeVars()
if not f.child1 in vars:
return f.child2, True
else:
assert f.op == "" or "Unexpected op"
return f, False