def relativize_quantifiers(self, phi):
"""
forall x (P(x)) ---> forall x (exists cx (R[f](x,cx)) -> P(x))
exists x (P(x)) ---> exists x (exists cx (R[f](x,cx)) & P(x))
"""
def gen_tot(t, r, s, connective):
for x in s[:-1]:
if x.subtrees:
raise ValueError, "nonterminal quantified variable '%s'" % (
x, )
imp = self.construct_totality([x.root for x in s[:-1]])
T = type(t)
return T(r, s[:-1] + [T(connective, [imp, s[-1]])])
def xform(t):
r, s = t.root, t.subtrees
if r == 'disable-relativization':
return FolFormula.conjunction(s)
elif r.kind == 'quantifier' and t.context_sign == '--':
return None # skip relativization for this quantifier (disable-relativization)
elif r.kind == 'quantifier' and (u'β' in s[0].root.literal
or u'γ' in s[0].root.literal):
return None
elif r == FolFormula.FORALL and t.context_sign == '+':
return gen_tot(t, r, s, FolFormula.IMPLIES)
elif r == FolFormula.EXISTS and t.context_sign == '-':
return gen_tot(t, r, s, FolFormula.AND)
elif r in [FolFormula.FORALL, FolFormula.EXISTS
] and t.context_sign == '+-':
raise NotImplementedError, "ambiguous context for quantifier: '%s'" % (
t, )
self._mark_context_sign(phi)
return TreeTransform([xform], dir=TreeTransform.BOTTOM_UP).inplace(phi)
def __call__(self, text):
t, = self.forestify(self.b(self.l(text)))
t = TreeTransform(
[functools.partial(AuxTransformers.unfold, symbols={';': 2})],
dir=TreeTransform.BOTTOM_UP,
recurse=True)(t)
return t
def normalize(self):
def transformer(self):
if isinstance(self.root, Identifier) and \
isinstance(self.root.literal, type(self)) and \
not self.subtrees:
return self.root.literal
return TreeTransform([transformer])(self)
def _rename_terms_over(self, phi, variable):
ns = AlphaRenaming.NS()
new_names = BidirectionalDict(
{FolFormula(variable): FolFormula(variable)})
def _scan(t):
if t.root != '=':
return type(t)(t.root, [_collect(x) or x for x in t.subtrees])
def _collect(t):
if t.root.kind == 'function':
try:
t_prime = new_names.get_backwards(t)
return t_prime
except KeyError:
if t.subtrees and all(
new_names.has_key(x) for x in t.subtrees):
c = FolFormula(
Identifier(u"β%d" % len(new_names),
'variable',
ns=ns))
new_names[c] = t
return c
return TreeTransform([_scan],
dir=TreeTransform.BOTTOM_UP)(phi), new_names
def __init__(self, naming_convention=NamingConvention()):
_ = AuxTransformers
fold_f = TreeTransform([_.fold], dir=TreeTransform.BOTTOM_UP)
#fold_f.IS_DESCENDING = True
self.naming = naming_convention
regroup_f = TreeTransform([partial(_.regroup, head='forall', tag='')],
dir=TreeTransform.BOTTOM_UP)
synonyms = {
FolFormula.FORALL: Identifier('forall', 'quantifier'),
FolFormula.EXISTS: Identifier('exists', 'quantifier'),
FolFormula.AND: Identifier('and', 'connective'),
FolFormula.OR: Identifier('or', 'connective'),
FolFormula.IFF: Identifier('iff', 'connective'),
FolFormula.IMPLIES: Identifier('=>', 'connective'),
FolFormula.NOT: Identifier('not', 'connective'),
True: 'true',
False: 'false'
}
synonyms_f = TreeSubstitution(synonyms)
#e = Expansion() # must get rid of it; only needed for the parser
L = DeltaReduction.Transformer._mkparser(
) # suggested optimization: use singleton for parser. Also, only parse macros once
macros = [
"x != y := ~(x=y)", "forall x (psi) := [forall]([]([](x,V)), psi)",
"exists x (psi) := [exists]([]([](x,V)), psi)"
]
macros_f = DeltaReduction(dir=TreeTransform.BOTTOM_UP)
macros_f.IS_DESCENDING = False
macros_f.transformers += \
[DeltaReduction.Transformer(macros_f, L(m)) for m in macros]
from adt.tree.transform.apply import ApplyTo # @Reimport
escape_f = ApplyTo(nodes=naming_convention.escape_safe).inplace
self.phases = ProcessingPhases([(macros_f, "Operators/quantifiers"),
(SExpression.reconstruct,
"S-expression"), (fold_f, "Fold"),
(synonyms_f, "Rename keywords"),
(regroup_f, "Regroup"),
(escape_f, "Escape identifiers")])
self.preface = ''
def __call__(self, t):
if t.root == self.head:
if not self.params and not self.body.subtrees: # e.g. "r := q"
return TreeTransform.Scalar(self.body.root)
elif len(t.subtrees) == len(
self.params): # e.g. "r := p(a,b)" or "r(a) := p(a,b)"
tun = self.o.tuning
subst = dict(
map(self._single_out, zip(self.params, t.subtrees)))
body = tun.local_.before(self.body)
return tun.local_.after(TreeSubstitution(subst)(body))
def e_absapp(self, expr):
is_value = self.is_value
beta_reduce = self.beta_reduce
class Transformer(object):
def __init__(self):
self.stop = False
def __call__(self, t):
if not self.stop and t.root == 'App' and \
all(is_value(x) for x in t.subtrees):
assert len(t.subtrees) == 2
self.stop = True
func, arg = t.subtrees
return beta_reduce(func, arg)
return TreeTransform([Transformer()])(expr)
def replace_equalities_with_atomic(self, phi, quantifier_type):
'''f(x) = y ---> R[f](x, y)
y = f(x) ---> R[f](x, y)
f(x) = g(y) ---> forall z (R[f](x,z) -> R[g](y,z)) /
exists z (R[f](x,z) & R[g](y,z))
The quantifier is determined according to quantifier_type (either FolFormula.FORALL or FolFormula.EXISTS)
when the atomic formula occurs in a positive context. In a negative
context, the other quantifier is used.
'''
EQ = FolWithEquality.eq
def xform(t):
if not self.unnest.is_fine(t):
raise ValueError, "nested term not allowed; in '%s'" % t
r, s = t.root, t.subtrees
s = sorted(s, key=lambda x: len(x.subtrees), reverse=True)
if r == EQ and len(s) == 2 and s[0].root in self.funcs:
if s[1].root in self.funcs:
_ = FolFormula
gamma = Identifier(u'γ', 'variable', ns=AlphaRenaming.NS())
if (quantifier_type, t.context_sign) in [(_.FORALL, "+"),
(_.EXISTS, "-")]:
quantifier, connective = _.FORALL, _.IMPLIES
elif (quantifier_type, t.context_sign) in [(_.FORALL, "-"),
(_.EXISTS, "+")
]:
quantifier, connective = _.EXISTS, _.AND
else:
raise ValueError, "invalid quantifier context %s" % (
(quantifier_type, t.context_sign), )
return _(quantifier, [
_(gamma),
_(connective, [
_(self._rel(s[0].root),
s[0].subtrees + [_(gamma)]),
_(self._rel(s[1].root), s[1].subtrees + [_(gamma)])
])
])
else:
return FolFormula(self._rel(s[0].root),
s[0].subtrees + s[1:])
self._mark_context_sign(phi)
return TreeTransform([xform]).inplace(phi)
def __call__(self, tree):
w = self.replace_what
if isinstance(w, type(tree)):
compare_with = w
else:
compare_with = type(tree)(w)
if tree.root == w and isinstance(self.replace_with, list):
return type(tree)([], self.replace_with)
if tree.root == w and not isinstance(self.replace_with,
type(tree)):
return TreeTransform.Scalar(self.replace_with)
elif tree == compare_with:
if isinstance(self.replace_with, list):
return type(tree)([], self.replace_with)
elif isinstance(self.replace_with, type(tree)):
return self.replace_with
else:
return type(tree)(self.replace_with)
def to_quantified(cls, phi, quantifier_context):
'''
@param quantifier_context: either FolFormula.FORALL or FolFormula.EXISTS
'''
def xform(t):
r, s = t.root, t.subtrees
if r == cls.LET:
_ = FolFormula
if t.quantifier_context == _.FORALL:
return _(_.FORALL, [s[0].subtrees[0], _(_.IMPLIES, s)])
elif t.quantifier_context == _.EXISTS:
return _(_.EXISTS, [s[0].subtrees[0], _(_.AND, s)])
else:
raise ValueError, "invalid quantifier context '%s' for '%s'" % (
t.quantifier_context, t)
cls._mark_quantifier_context(phi, quantifier_context)
return TreeTransform([xform])(phi)
def __call__(self, formula):
q = []
def pull_out(t, op_prefix=(), q=q):
""" Extracts quantifiers nested in (sub-)formula t. """
r, s = t.root, t.subtrees
if r in self.quantifiers:
q += [self._apply_ops(op_prefix, type(t)(r, s[:-1]))]
if r in self.instructions:
ops = self.instructions[r]
for x, op in zip(s, ops):
pull_out(x, op_prefix + (op,))
def xform(t):
r, s = t.root, t.subtrees
if r in self.quantifiers:
return s[-1]
pull_out(formula)
phi = TreeTransform([xform], recurse=True)(formula)
for prefix in reversed(q):
prefix.subtrees += [phi]
phi = prefix
return phi
def simplify(self, t):
def transform(t):
r, arity, s = t.root, len(t.subtrees), t.subtrees
h = None
if r == 'S' and arity == 1:
h = s[0]
#elif r == 'id' and arity == 1:
# h = s[0]
elif r == 'Abs' and arity == 3:
h = type(t)(r, s[1:])
elif r == 'S' and arity == 3 and s[0].root == '(' and s[
2].root == ')':
h = s[1]
elif isinstance(r, Tagged) and hasattr(r, 'token'):
h = type(t)(r.token)
if h is not None:
return recurse(h)
recurse = TreeTransform([transform])
return recurse(t)
def inspect_formula(self, phi, ctx):
'''Collects constants occurring in assertions.'''
# TODO: remove 'used_fragment', take all the functions ever
s = ctx.signature #.used_fragment([phi])
self.constants = [
c for c, _ in s.funcs if s.sorts.is_const(c, of_sort="V")
]
self.funcs = [
c for c, a in s.funcs if a == 1 and s.sorts.returns(c, "V")
]
#self.rel_funcs = self.funcs
#self.rel_funcs = [c for c in self.funcs if (c,1) in s.used_fragment([phi]).funcs or c in ["k0", "km0"]]
_ = Tree
#rel_funcs = _('', [_('k10', [_('k20'), _('km20')]),
# _('km10', [_('k20'), _('km20')])
# ] + [_(x) for x in ["f", "g", "l0", "k0", "km0", "k20", "km20"]])
rel_funcs = _('', [
_(x) for x in
["f", "g", "l0", "k0", "km0", "k10", "km10", "k20", "km20", "en0"]
])
def xform(t):
r, s = t.root, t.subtrees
if r != '':
f = [c for c in self.funcs if c.literal == r]
if f:
new_r = f[0]
else:
return Tree('')
else:
new_r = r
return Tree(new_r, [x for x in s if x.root != ''])
self.rel_funcs = TreeTransform([xform],
dir=TreeTransform.BOTTOM_UP)(rel_funcs)
#[c for c in self.funcs if c in ["f", "g", "l0", "k0", "km0", "k10", "km10", "k20", "km20", "k2k10", "km2k10", "k2km10", "km2km10"]]
self.declare_relations(ctx.signature)
self.inplace(phi)
def __call__(self, phi, ctx=None):
occurrences = ListWithAdd()
self._collect_ternies(phi, occurrences)
def xform(phi):
_ = FolFormula
for (atom, ter) in occurrences:
if phi is atom:
if len(ter.subtrees) != 3:
raise ValueError, "invalid ternary expression: '%s'" % ter
cond, then_, else_ = ter.subtrees
phi = _(self.Signature.ite, [
cond,
self._choose(atom, ter, then_),
self._choose(atom, ter, else_)
])
return self(phi)
if occurrences:
return TreeTransform([xform])(phi)
else:
return phi
def _collect(self, t, parts_f):
parts = []
xform = TreeTransform([partial(parts_f, collect=parts)],
dir=TreeTransform.BOTTOM_UP)
parts += [xform(t)]
return parts
def __call__(self, phi, ctx=None):
return TreeTransform([self.xform], dir=TreeTransform.BOTTOM_UP)(phi)
astf = fe("k:=i.n ; i.n:=j; j:=i; i:=k", 'wp ')
astf = fe(
"( if $C( i )$ then ( t := i.n ; j.n := null ; j.n := t ) else j := i ) ; i := i.n",
'wp ')
astf.subtrees += [FolFormula.promote("Q")]
def compose_xform(t):
import copy
r, s = t.root, t.subtrees
a = len(s)
if r == 'wp ;' and a == 3:
c1 = copy.deepcopy(s[0])
c2 = copy.deepcopy(s[1])
c2.subtrees += [s[2]]
c1.subtrees += [c2]
return c1
if r == 'wp if' and a == 4:
ct, ce = copy.deepcopy(s[1]), copy.deepcopy(s[2])
ct.subtrees += [s[3]]
ce.subtrees += [s[3]]
return type(t)(r, [s[0], ct, ce])
print astf
xf = TreeTransform([compose_xform],
dir=TreeTransform.TOP_DOWN,
recurse=True)
xf.IS_DESCENDING = True
print xf(astf)
def unprotect(self):
return TreeTransform([self._xform_unprotect])
#print unicode(FOL_GRAMMAR)
for rule in FOL_GRAMMAR.rules:
if rule.head == 'fml': print rule
#raise SystemExit
class TX(ParserXforms):
PRIM = {'forall': FolFormula.FORALL, '|': FolFormula.OR, '&': FolFormula.AND, '~': FolFormula.NOT, '->': FolFormula.IMPLIES, '<->': FolFormula.IFF}
for v in PRIM.values(): PRIM[v.literal] = v
def singl(self, t): return FolFormula(Identifier((t.subtrees[0] if t.subtrees else t).root.token.value, '?'))
def qua(self, t): return FolFormula(self.PRIM[t.subtrees[0].root.token.value], [self.singl(t.subtrees[1]), t.subtrees[3]])
def cal(self, t): return FolFormula(self.singl(t).root, self.split(t.subtrees[2], ('term*', 'term+', ',')))
def bin(self, t): return FolFormula(self.PRIM[t.subtrees[1].root.token.value], [t.subtrees[0], t.subtrees[2]])
def un(self, t): return FolFormula(self.PRIM[t.subtrees[0].root.token.value], [t.subtrees[1]])
def sam(self, t): return t.subtrees[0]
def sam1(self, t): return t.subtrees[1]
lG = set((Tagged(tok).with_(annotation='sam'), tok) for tok in tokens)
ast = Earley(FOL_GRAMMAR.rules, lG, tokens, lexer=lambda x: x)
PrintTrees(ast)
assert len(ast) == 1
ast = FolFormula.reconstruct(ast[0])
print ast
phi = TreeTransform([TX()], dir=TreeTransform.BOTTOM_UP)(ast)
print phi
def _choose(self, atom, point_of_choice, choice):
def xform(phi):
if phi is point_of_choice: return choice
return TreeTransform([xform])(atom)
def __call__(self, phi, constants, quantifier_context):
phi = TreeTransform([self._xform], dir=TreeTransform.BOTTOM_UP)(phi)
return self._xform_toplevel(phi, constants, quantifier_context)