def __init__(self, signature=FolManySortSignature(), logic="QF_UF"):
self.signature = signature
if not isinstance(signature, FolManySortSignature):
self.signature = FolManySortSignature(signature.funcs,
signature.preds)
self.logic = logic
self.builtin_mappings = {
'R': "Real",
'Z': "Int",
FolWithEquality.Signature.eq: "="
}
def demo2():
V = DeclareSort("V")
R = Function("R", V, V, BoolSort())
u, v, w = Consts("u v w", V)
po = And(
ForAll([u, v, w], Implies(And(R(u, v), R(v, w)),
R(u, w))), # transitive
ForAll([u, v], Not(And(R(u, v), R(v, u)))) # antisymmetric
)
s = Solver()
s.add(po)
s.add(u != v)
s.add(u != w)
s.add(And(Or(R(u, v), R(v, u)), Or(R(u, w), R(w, u))))
print s.check()
m = s.model()
class _:
u = Identifier('u', 'function')
v = Identifier('v', 'function')
w = Identifier('w', 'function')
R = Identifier('R', 'predicate')
sorts = FolSorts({u: u'→V', v: u'→V', w: u'→V', R: u'V×V→'})
sig = FolManySortSignature.from_sorts(_.sorts)
z3m = Z3ModelToFolStructure(sig, {_.u: u, _.v: v, _.w: w, _.R: R})
mm = z3m(m)
print mm
from adt.graph.format import DEFAULT as FORMATTER
mm.domain = mm.domain['V']
print FORMATTER(AlgebraicStructureGraphTool(mm, [_.R])())
def __call__(self, phi):
"""
Makes sure any two identifiers that are not equal (not __eq__(x,y))
have distinct literals. Note: performs modification of formulas in-place.
@param phi a FolFormula instance
it may also be:
* an Identifier, in which case the corresponding unique identifier is returned
* a FolManySortSignature, in which case returns a *new* signature where all
symbols are translated to unique ones
"""
if isinstance(phi, FolFormula):
symset = OrderedSet(node.root for node in phi.nodes
if node.root.ns is not None)
symdict = {sym: self.gen_uniq(sym) for sym in symset}
return TreeNodeRename(symdict).inplace(phi)
elif isinstance(phi, Identifier):
sym = phi
return self.gen_uniq(sym) if sym.ns is not None else sym
elif isinstance(phi, FolManySortSignature):
renamed_sorts = FolSorts(
{self(k): v
for k, v in phi.sorts.sorts.iteritems()})
return FolManySortSignature.from_sorts(renamed_sorts)
else:
raise TypeError, "expected identifier, formula, or signature; found '%s'" % (
type(phi).__name__)
def declare_relations(self, signature):
for f in self.funcs:
Rf = self._rel(f)
if Rf not in signature.preds:
for from_, to_ in signature.sort_of(f):
signature |= FolManySortSignature.from_sorts(
FolSorts({Rf: [(tuple(from_) + (to_, ), 'bool')]}))
def __init__(self):
self.z3_sort_map = OneToOne({
'bool': BoolSort(),
'': BoolSort()
}).of(self._mksort)
self.z3_decls = {}
self.signature = FolManySortSignature()
self.expansion = None
class Signature(FolWithEquality.Signature):
add = Identifier("+", 'function')
cos = Identifier("cos", 'function')
sin = Identifier("sin", 'function')
x = Identifier("x", 'function')
sorts = FolSorts({add: "R×R→R", sin: "R→R", cos: "R→R", x: "→R"})
formal = FolManySortSignature.from_sorts(sorts)
def domain_of(self, symbol, signature):
d = MultiDomain.promote(self.domain)
s = FolManySortSignature.promote(signature)
for (from_, _) in s.sort_of(symbol):
if len(from_) == 1:
l = d[from_[0]]
else:
l = d[from_]
for el in l:
yield el
def blur(self, phi):
reg = self
s = self.symbols
class FormulaWalk(RichTreeWalk.Visitor):
def enter(self, subtree, prune=lambda: None):
r = subtree.root
if isinstance(r, Identifier) and \
r.kind in ['function', 'predicate'] and r not in reg.symbols:
prune()
return reg.UNDEF
else:
return r
def join(self, node, prefix, infix, postfix):
known = [a for a in infix if a.root != reg.UNDEF]
if len(known) == len(infix):
return type(node)(prefix, known)
elif prefix == FolFormula.AND and len(known) == 1:
return known[0]
else:
return type(node)(reg.UNDEF)
def done(self, root, final):
return final
if isinstance(phi, FolFormula):
return RichTreeWalk(FormulaWalk())(phi)
elif isinstance(phi, FolTheory):
sig = self.blur(phi.signature)
t = (y for y in (self.blur(x) for x in phi)
if y.root != self.UNDEF)
return FolTheory(t).with_signature(sig)
elif isinstance(phi, FolManySortSignature):
return FolManySortSignature([(f, a)
for f, a in phi.funcs if f in s],
[(p, a)
for p, a in phi.preds if p in s],
self.blur(phi.sorts))
elif isinstance(phi, FolSignature):
return FolSignature([(f, a) for f, a in phi.funcs if f in s],
[(p, a) for p, a in phi.preds if p in s])
elif isinstance(phi, FolSorts):
return FolSorts(self.blur(phi.sorts))
elif isinstance(phi, FolStructure):
return FolStructure(domain=phi.domain,
interpretation=self.blur(phi.interpretation))
elif isinstance(phi, dict):
return dict(
(sym, v) for sym, v in phi.iteritems() if sym in self.symbols)
else:
raise TypeError, phi
def structure(self, structure, signature, domain=None):
if domain is None: domain = structure.domain
domain = MultiDomain.promote(domain)
s = FolManySortSignature.promote(signature)
i = structure.interpretation
explicit_i = {}
for f, _ in signature.funcs:
explicit_i[f] = self.join(
self.function(i[f], from_, domain)
for from_, _ in s.sort_of(f))
for p, _ in signature.preds:
explicit_i[p] = self.join(
self.predicate(i[p], from_, domain)
for from_, _ in s.sort_of(p))
return FolStructure(domain, explicit_i)
class Signature(FolWithEquality.Signature):
add = Identifier("+", 'function')
cos = Identifier("cos", 'function')
sin = Identifier("sin", 'function')
x = Identifier("[x]", 'function', mnemonic="x")
sorts = FolSorts({add: "R×R→R",
sin: "R→R",
cos: "R→R",
x: "→R"})
formal = FolManySortSignature.from_sorts(sorts)
standard_interpretation = \
{sin: lambda x: math.sin(x*math.pi/180),
cos: lambda x: math.cos(x*math.pi/180)}
class FolIntegerArithmetic(FolNaturalPartialOrder):
"""
Signature for basic arithmetic.
"""
add = Identifier('+', 'function')
sub = Identifier(u'−', 'function')
mul = Identifier(u'⋅', 'function')
div = Identifier('/', 'function')
neg = Identifier('-', 'function')
sorts = FolSorts({add: u"Z×Z→Z", mul: u"Z×Z→Z", sub: u"Z×Z→Z", div: u"Z×Z→Z",
neg: u"Z→Z",
FolNaturalPartialOrder.gt: u"Z×Z→",
FolNaturalPartialOrder.lt: u"Z×Z→",
FolNaturalPartialOrder.ge: u"Z×Z→",
FolNaturalPartialOrder.le: u"Z×Z→"})
formal = FolManySortSignature.from_sorts(sorts)
STANDARD_INTERPRETATION = \
{add : operator.add, sub: lambda a,b=None: -a if b is None else a-b,
u'−': operator.sub,
mul: operator.mul, u'*': operator.mul,
div: operator.div # integer division
}
FolFormula.INFIXES += [mul, sub]
class Sugar:
"""FolSymbolic mix-in"""
def _b(self, root, subs):
return type(self)(root, [self.promote(x) for x in subs])
def __add__(self, other): return self.build(self._.add, [self, other])
def __radd__(self, other): return self.build(self._.add, [other, self])
def __sub__(self, other): return self.build(self._.sub, [self, other])
def __rsub__(self, other): return self.build(self._.sub, [other, self])
def __mul__(self, other): return self.build(self._.mul, [self, other])
def __rmul__(self, other): return self.build(self._.mul, [other, self])
def __neg__(self): return self.build(self._.neg, [self])
# Naturally should be part of FolWithEquality
def __eq__(self, other): return self.build(self._.eq, [self, other])
# Naturally should be part of FolNaturalPartialOrder
def __gt__(self, other): return self.build(self._.gt, [self, other])
def __lt__(self, other): return self.build(self._.lt, [self, other])
def __ge__(self, other): return self.build(self._.ge, [self, other])
def __le__(self, other): return self.build(self._.le, [self, other])
def __call__(self, z3_elements):
sorts = FolSorts()
decls = {}
for z3_symbol in z3_elements:
idf = Identifier(z3_symbol.name(), self._get_kind(z3_symbol))
decls[idf] = z3_symbol
if is_const(z3_symbol):
sorts[idf] = FolSorts.FunctionType((), z3_symbol.sort())
elif is_func_decl(z3_symbol):
z3_domains = [
z3_symbol.domain(i) for i in xrange(z3_symbol.arity())
]
z3_range = z3_symbol.range()
sorts[idf] = FolSorts.FunctionType(tuple(z3_domains), z3_range)
sigma = FolManySortSignature.from_sorts(sorts)
sigma.z3_decls = decls
return sigma
def signature(self):
"""
Extracts a first-order sorted signature describing the type of
the structure.
"""
sorts = FolSorts()
for name, (args, ret, _) in self.funcs.iteritems():
# ~ symbols with '!' are reserved for compiler-generated aux
if not "!" in name:
nameid = Identifier(
name, '?') # 'kind' will be filled by FolManySortSignature
argtypes = [
Identifier(a.subtrees[0].root, 'sort') for a in args
]
rettype = Identifier(ret.root, 'sort')
if rettype == 'bool': rettype = ''
sorts[nameid] = [FolSorts.FunctionType(argtypes, rettype)]
return FolManySortSignature.from_sorts(sorts)
class Signature(FolWithEquality.Signature):
add = Identifier("+", 'function')
mul = Identifier("*", 'function')
gt = Identifier(">", 'predicate', mnemonic="gt")
ge = Identifier("≥", 'predicate', mnemonic="ge")
lt = Identifier("<", 'predicate', mnemonic="lt")
le = Identifier("≤", 'predicate', mnemonic="le")
x = Identifier("[x]", 'function', mnemonic="x")
y = Identifier("[y]", 'function', mnemonic="y")
z = Identifier("[z]", 'function', mnemonic="z")
b = Identifier(":b:", 'function', mnemonic="b")
i = Identifier("i", 'variable')
sorts = FolSorts({add: "Z×Z→Z", mul: "Z×Z→Z", gt: "Z×Z→",
x: "Z→Z", y: "Z→Z", z: "Z→Z", b: "Z×Z→Z", i: "→Z"})
formal = FolManySortSignature.from_sorts(sorts)
def range_of(self, symbol, signature):
d = MultiDomain.promote(self.domain)
s = FolManySortSignature.promote(signature)
for r in set(to_ for _, to_ in s.sort_of(symbol)):
for el in d[r]:
yield el
def __init__(self):
self.declarations = FolSorts()
self.signature = FolManySortSignature(set(), set())
self.special_symbols = {sym.literal: sym for sym in self.SPECIALTIES}
body_prime = self._add_quantifier_prefix(body_prime, r, new_names)
return _(t.root, [s[0], body_prime])
def is_fine(self, atom):
'''Only atoms of the form f(x)=y are allowed to have functions.'''
r, s = atom.root, atom.subtrees
if r == FolWithEquality.eq and len(s) == 2:
return all(not w.subtrees for x in s for w in x.subtrees)
elif r.kind == 'predicate':
return all(not x.subtrees for x in s)
else:
return True
if __name__ == '__main__':
from logic.fol.syntax.parser import FolFormulaParser
from eprv.synopsis.declare import TypeInference
L = FolFormulaParser()
f = Identifier('f', 'function')
g = Identifier('g', 'function')
sigma = FolManySortSignature.from_sorts(FolSorts({f: u'V→V', g: u'V→V'}))
ti = TypeInference()
ti.declarations = sigma.sorts
phi = ti(L * "forall u (ite(u=u, f(u) = u, f(u) = g(u)))")
from eprv.synopsis.proof import ProofSynopsis
ef = EasyFunctionsExtension()
ef.inspect_formula(phi, ProofSynopsis.Context(None, sigma))
print phi
class SmtLibInputFormat(object):
class Attribute(object):
def __init__(self, name, value):
self.name = name
self.value = value
def __str__(self):
return ":%s %s" % (self.name, self.value)
def __init__(self, signature=FolManySortSignature(), logic="QF_UF"):
self.signature = signature
if not isinstance(signature, FolManySortSignature):
self.signature = FolManySortSignature(signature.funcs,
signature.preds)
self.logic = logic
self.builtin_mappings = {
'R': "Real",
'Z': "Int",
FolWithEquality.Signature.eq: "="
}
def predicates(self, preds):
S = SchemeExpression
return S("", [
S(p,
self.sort_of(p, a) or [""])
for p, a in preds if p not in self.builtin_mappings
])
def functions(self, funcs):
S = SchemeExpression
return S("", [
S(f, self.sort_of(f, a))
for f, a in funcs if f not in self.builtin_mappings
])
def quantified_expression(self, variable):
S = SchemeExpression
return S("?%s" % variable, self.sort_of(variable, 0))
def mkid(self, identifier, *a):
x = identifier
return SchemeExpression(self.builtin_mappings.get(x, x), *a)
def sort_of(self, symbol, arity):
S = self.mkid
sr = self.signature.sort_of(symbol, arity)
if not sr:
raise TypeError, "No sort for '%s/%d'" % (symbol, arity)
from_, to_ = sr[0] # TODO support more than one
to_ = [x for x in [to_] if x is not None]
return [S(f) for l in (from_, to_) for f in l]
def benchmark(self, title):
return SchemeExpression("benchmark %s" % title)
def list(self, l): #@ReservedAssignment
S = SchemeExpression
if l:
return S("", [S.promote(x) for x in l])
else:
return S("", [""])
def formula(self, formula):
r = formula.root
quantifiers_and_connectives = {
FolFormula.FORALL: "forall",
FolFormula.EXISTS: "exists",
FolFormula.IMPLIES: "implies",
FolFormula.AND: "and",
FolFormula.OR: "or",
FolFormula.IFF: "iff",
FolFormula.NOT: "not"
}
it = r
kind = None
if isinstance(r, FolFormula.Identifier):
kind = r.kind
if r.kind == 'quantifier' or r.kind == 'connective':
it = quantifiers_and_connectives[r]
if r.kind == 'variable':
it = "?%s" % r
S = self.mkid
if kind == 'quantifier':
expr = S(it, [
self.quantified_expression(s.root)
for s in formula.subtrees[:-1]
] + [self.formula(s) for s in formula.subtrees[-1:]])
else:
expr = S(it, [self.formula(s) for s in formula.subtrees])
return self._tree_fold(expr)
def theory(self, formulas, title="theory"):
bm = self.benchmark(title)
A = self.Attribute
signature = self.signature
all_sorts = set(["V"]) | self.signature.sorts.set()
all_sorts = set(x for x in all_sorts if x not in self.builtin_mappings)
attrs = [
A("logic", SchemeExpression(self.logic)),
A("extrasorts", self.list(all_sorts)),
A("extrafuns", self.functions(signature.funcs)),
A("extrapreds", self.predicates(signature.preds))
]
attrs = [a for a in attrs if a.value]
bm.attributes.extend(attrs)
if formulas:
bm.attributes.extend(
[A("assumption", self.formula(f)) for f in formulas[:-1]])
bm.attributes.append(A("formula", self.formula(formulas[-1])))
return bm
def __call__(self, thing):
if isinstance(thing, list):
thing = self.theory(thing, "theory")
elif isinstance(thing, FolFormula):
thing = self.theory([thing], "formula")
return thing
def _tree_fold(self, tree, foldables=("and", "or")):
r = tree.root
if r in foldables:
subs = []
for s in tree.subtrees:
if s.root == r:
subs.extend(self._tree_fold(s, foldables).subtrees)
else:
subs.append(s)
else:
subs = tree.subtrees
return type(tree)(r, [self._tree_fold(s, foldables) for s in subs])
