def solve_eqts(avs, bvs, c, d):
"""
solve for c,d in a[i] = c+d*b[i]
#[5 = c+d*0, 137 = c + d*44, 59 = c+d*18] => c=5,d=3
sage: c,d = var('c d')
sage: rs = solve_eqts([5,137,59],[0,44,18],c,d)
sage: rs[c], rs[d]
(3, 5)
"""
if __debug__:
assert (isinstance(avs, (tuple, list))
and all(is_scalar(v) and CM.isnum(v) for v in avs)), avs
assert (isinstance(bvs, (tuple, list))
and all(is_scalar(v) and CM.isnum(v) for v in bvs)), bvs
assert len(avs) == len(bvs)
assert is_sage_expr(c), c
assert is_sage_expr(d), d
eqts = [a == d + c * b for a, b in zip(avs, bvs)]
try:
sol = solve(eqts, c, d, solution_dict=True)
if len(sol) >= 2:
print '{} results {} sols {}'.format(eqts, len(sol), sol)
return sol[0]
except Exception:
return None
def strOfExp(p):
"""
-p^3 => -(p*p*p)
n*p^4 => n*(p*p*p*p)
ab^3 => (ab*ab*ab)
x*y*z^3 => x*y*(z*z*z)
"""
assert is_sage_expr(p), p
def getPow(p):
try:
oprs = p.operands()
except Exception:
return []
if p.operator() == sage.all.operator.pow:
x, y = oprs
pow_s = '*'.join(
[str(x) if x.is_symbol() else "({})".format(x)] * int(y))
return [(str(p), '({})'.format(pow_s))]
else:
return [xy for o in oprs for xy in getPow(o)]
s = str(p)
if '^' not in s:
return s
rs = getPow(p)
for (x, y) in rs:
s = s.replace(x, y)
return s
def getConstraints(m, resultAsDict=False):
"""
Input a model m, returns its set of constraints in either
1) sage dict {x:7,y:10}
1) z3 expr [x==7,y==0]
sage: S = z3.Solver()
sage: S.add(z3.Int('x') + z3.Int('y') == z3.IntVal('7'))
sage: S.check()
sat
sage: M = S.model()
sage: d = getConstraints(M, resultAsDict=True)
sage: sorted(d.items(), key=lambda(k,_): str(k))
[(x, 7), (y, 0)]
sage: getConstraints(M)
[y == 0, x == 7]
sage: S.reset()
"""
assert m is not None, m
if resultAsDict: #sage format
rs = [(var(str(v())),sage_eval(str(m[v]))) for v in m]
rs = dict(rs)
assert all(is_sage_expr(x) for x in rs.keys())
assert all(is_sage_real(x) or is_sage_int(x) for x in rs.values())
else: #z3 format
rs = [v()==m[v] for v in m]
assert all(z3.is_expr(x) for x in rs)
return rs
def __str__(self, printStat=False):
if is_sage_expr(self.inv):
s = Inv.strOfExp(self.inv)
else:
s = str(self.inv)
if printStat: s = "{} {}".format(s, self.stat)
return s
def __init__(self, p):
"""
Ex:
x + y >= 0
x^2 + y^2 + 3 == 0
"""
if __debug__:
assert is_sage_expr(p), p
super(InvExp, self).__init__(p)
def __str__(self, printStat=False):
if is_sage_expr(self.inv):
inv = miscs.elim_denom(self.inv)
s = miscs.strOfExp(inv)
else:
s = str(self.inv)
if printStat: s = "{} {}".format(s, self.stat)
return s
def infer(self, loc, template, uks, exprs, dtraces, inps):
assert isinstance(loc, str), loc
assert sageutil.is_sage_expr(template), template
assert isinstance(uks, list), uks
assert isinstance(exprs, set) and exprs, exprs
assert isinstance(dtraces, DTraces) and dtraces, dtraces
assert isinstance(inps, Inps) and inps, inps
vs = tuple(self.invdecls[loc])
cache = set()
eqts = set() #results
exprs = list(exprs)
newInps = []
curIter = 0
while True:
curIter += 1
logger.debug("{}: iter {} infer using {} exprs".format(
loc, curIter, len(exprs)))
newEqts = Miscs.solveEqts(exprs, uks, template)
unchecks = [eqt for eqt in newEqts if eqt not in cache]
if not unchecks:
logger.debug("{}: no new results -- break".format(loc))
break
logger.debug('{}: {} candidates:\n{}'.format(
loc, len(newEqts), '\n'.join(map(str, newEqts))))
logger.debug("{}: check {} unchecked ({} candidates)".format(
loc, len(unchecks), len(newEqts)))
dinvs = DInvs.mk(loc, Invs.mk(map(Inv, unchecks)))
dInps, dCexs, dinvs = self.prover.checkRange(dinvs, inps=None)
if dInps: newInps.append(dInps)
_ = [eqts.add(inv) for inv in dinvs[loc] if not inv.isDisproved]
_ = [
cache.add(inv.inv) for inv in dinvs[loc]
if inv.stat is not None
]
if loc not in dCexs:
logger.debug(
"{}: no disproved candidates -- break".format(loc))
break
cexs = Traces.extract(dCexs[loc], vs)
exprs_ = cexs.instantiate(template, None)
logger.debug("{}: {} new cex exprs".format(loc, len(exprs_)))
exprs.extend(exprs_)
return eqts, newInps
def instantiateTemplate(template, sols):
"""
Instantiate a template with solved coefficient values
sage: var('uk_0,uk_1,uk_2,uk_3,uk_4,r14,r15,a,b,y')
(uk_0, uk_1, uk_2, uk_3, uk_4, r14, r15, a, b, y)
#when sols are in dict form
sage: sols = [{uk_0: -2*r14 + 7/3*r15, uk_1: -1/3*r15, uk_4: r14, uk_2: r15, uk_3: -2*r14}]
sage: Template(uk_1*a + uk_2*b + uk_3*x + uk_4*y + uk_0 == 0).instantiateSols(sols)
[-2*x + y - 2 == 0, -1/3*a + b + 7/3 == 0]
# #when sols are not in dict form
sage: sols = [[uk_0== -2*r14 + 7/3*r15, uk_1== -1/3*r15, uk_4== r14, uk_2== r15, uk_3== -2*r14]]
sage: Template(uk_1*a + uk_2*b + uk_3*x + uk_4*y + uk_0 == 0).instantiateSols(sols)
[-2*x + y - 2 == 0, -1/3*a + b + 7/3 == 0]
sage: Template(uk_1*a + uk_2*b + uk_3*x + uk_4*y + uk_0 == 0).instantiateSols([])
[]
"""
assert sageutil.is_sage_expr(template), template
if not sols: return []
if len(sols) > 1:
logger.warn('instantiateTemplateWithSols: len(sols) = {}'.format(
len(sols)))
logger.warn(str(sols))
def f_eq(d):
if isinstance(d, list):
f_ = template
for d_ in d:
f_ = f_.subs(d_)
rhsVals = CM.vset([d_.rhs() for d_ in d])
uk_vars = sageutil.get_vars(rhsVals)
else:
f_ = template(d)
uk_vars = sageutil.get_vars(d.values()) #e.g., r15,r16 ...
if not uk_vars: return f_
iM = sage.all.identity_matrix(len(uk_vars)) #standard basis
rs = [dict(zip(uk_vars, l)) for l in iM.rows()]
rs = [f_(r) for r in rs]
return rs
sols = sage.all.flatten([f_eq(s) for s in sols])
#remove trivial (tautology) str(x) <=> str(x)
sols = [
s for s in sols
if not (s.is_relational() and str(s.lhs()) == str(s.rhs()))
]
return sols
def __init__(self, p):
"""
Ex:
x + y >= 0
x^2 + y^2 + 3 == 0
"""
if __debug__:
assert is_sage_expr(p), p
super(InvExp,self).__init__(p)
def toZ3(p, is_real):
"""
Convert a Sage expression to a Z3 expression
Initially implements with a dictionary containing variables
e.g. {x:Real('x')} but the code is longer and more complicated.
This implemention does not require a dictionary pass in.
Todo: cache this function
sage: toZ3(x*x*x, False)
x*x*x
"""
assert is_sage_expr(p), p
def retval(p):
if p.is_symbol():
_f = z3.Real if is_real else z3.Int
else:
_f = z3.RealVal if is_real else z3.IntVal
return _f(str(p))
try:
oprs = p.operands()
except Exception:
return retval(p)
if is_empty(oprs):
return retval(p)
else:
op = p.operator()
#z3 has problem w/ t^c , so use t*t*t..
if op == operator.pow:
assert len(oprs) == 2, oprs
t, c = oprs
t = toZ3(t, is_real)
vs = [t] * c
z3exp = reduce(operator.mul, vs)
else:
oprs = [toZ3(o, is_real) for o in oprs]
z3exp = reduce(op, oprs)
assert z3.is_expr(z3exp), z3exp
return z3exp
def instantiate(self, term, nTraces):
assert is_sage_expr(term), term
assert nTraces is None or nTraces >= 1, nTraces
if nTraces is None:
exprs = set(term.subs(t) for t in self.mydicts)
else:
nTracesExtra = nTraces * 5
exprs = set()
for i, t in enumerate(self.mydicts):
expr = term.subs(t)
if expr not in exprs:
exprs.add(expr)
if len(exprs) >= nTracesExtra:
break
#instead of doing this, can find out the # 0's in traces
#the more 0's , the better
exprs = sorted(exprs, key=lambda expr: len(get_vars(expr)))
exprs = set(exprs[:nTraces])
return exprs
def get_constraints(m, result_as_dict=False):
"""
Input a model m, returns its set of constraints in either
1) sage dict {x:7,y:10}
1) z3 expr [x==7,y==0]
sage: S = z3.Solver()
sage: S.add(z3.Int('x') + z3.Int('y') == z3.IntVal('7'))
sage: S.check()
sat
sage: M = S.model()
sage: d = SMT_Z3.get_constraints(M, result_as_dict=True)
sage: sorted(d.items(), key=lambda(k,_): str(k))
[(x, 7), (y, 0)]
sage: SMT_Z3.get_constraints(M)
[y == 0, x == 7]
sage: S.reset()
"""
assert m is not None, m
if result_as_dict: #sage format
rs = [(var(str(v())),sage_eval(str(m[v]))) for v in m]
rs = dict(rs)
if __debug__:
assert all(is_sage_expr(x) for x in rs.keys())
assert all(is_sage_real(x) or is_sage_int(x) for x in rs.values())
else: #z3 format
rs = [v()==m[v] for v in m]
if __debug__:
assert all(z3.is_expr(x) for x in rs)
return rs
def to_z3exp(p, is_real):
"""
Convert a Sage expression to a Z3 expression
Initially implements with a dictionary containing variables
e.g. {x:Real('x')} but the code is longer and more complicated.
This implemention does not require a dictionary pass in.
Todo: cache this function
"""
assert is_sage_expr(p), p
def retval(p):
if p.is_symbol():
z3exp = (z3.Real if is_real else z3.Int)(str(p))
else:
z3exp = (z3.RealVal if is_real else z3.IntVal)(str(p))
if __debug__:
assert z3.is_expr(z3exp), z3exp
return z3exp
try:
oprs = p.operands()
except Exception:
return retval(p)
if is_empty(oprs):
return retval(p)
else:
oprs = [SMT_Z3.to_z3exp(o, is_real) for o in oprs]
z3exp = SMT_Z3._reduce(p.operator(),oprs)
assert z3.is_expr(z3exp), z3exp
return z3exp
def __init__(self, p):
if __debug__:
assert (is_sage_expr(p)
and p.operator() in [operator.le, operator.ge])
super(InvIeq, self).__init__(p)
def __init__(self, p):
if __debug__:
assert is_sage_expr(p) and p.operator() == operator.eq
super(InvEqt, self).__init__(p)
def __init__(self, p):
if __debug__:
assert (is_sage_expr(p) and
p.operator() in [operator.le, operator.ge])
super(InvIeq,self).__init__(p)
def __init__(self, p):
if __debug__:
assert is_sage_expr(p) and p.operator() == operator.eq
super(InvEqt,self).__init__(p)
def myeval(self, term):
assert is_sage_expr(term), term
return [trace.myeval(term, self.vs) for trace in self]
def myeval(self, term, vs):
assert is_sage_expr(term), term
return term.subs(self.mydict(vs))
def __init__(self, template):
assert sageutil.is_sage_expr(template), template
self.template = template
