def main(known):
s = Solver()
matrix = [[Int(f"m{x}{y}") for x in range(1, 10)] for y in range(1, 10)]
for i in range(9):
for j in range(9):
v = matrix[i][j]
if known[i][j]:
s.add(v == known[i][j])
else:
s.add(v >= 1)
s.add(v <= 9)
for i in range(9):
s.add(Distinct(*[matrix[i][j] for j in range(9)]))
s.add(Distinct(*[matrix[j][i] for j in range(9)]))
for i in range(3):
for j in range(3):
s.add(
Distinct(*[
matrix[3 * i + k][3 * j + l] for k in range(3)
for l in range(3)
]))
s.check()
m = s.model()
print("Solution:")
for i in range(9):
print(*[m[matrix[i][j]] for j in range(9)])
def get_state(doubles, browser):
if browser == "node":
browser = "chrome"
elif browser not in ("chrome", "firefox", "safari"):
raise ValueError(f"invalid browser {browser}")
if browser == "chrome":
doubles = doubles[::-1]
# from the doubles, generate known piece of the original uint64
generated = [from_double(double, browser) for double in doubles]
# setup symbolic state for xorshift128+
ostate0, ostate1 = BitVecs("ostate0 ostate1", 64)
sym_state0 = ostate0
sym_state1 = ostate1
solver = Solver()
conditions = []
# run symbolic xorshift128+ algorithm for three iterations
# using the recovered numbers as constraints
for val in generated:
sym_state0, sym_state1, ret_conditions = sym_xs128p(
solver, sym_state0, sym_state1, val, browser)
conditions += ret_conditions
if solver.check(conditions) == sat:
# get a solved state
m = solver.model()
state0 = m[ostate0].as_long()
state1 = m[ostate1].as_long()
solver.add(Or(ostate0 != m[ostate0], ostate1 != m[ostate1]))
if solver.check(conditions) == sat:
print("WARNING: multiple solutions found! use more doubles!")
return state0, state1
else:
raise ValueError("unsat model")
def test_concrete_calldata_calldatasize():
# Arrange
calldata = ConcreteCalldata(0, [1, 4, 7, 3, 7, 2, 9])
solver = Solver()
# Act
solver.check()
model = solver.model()
result = model.eval(calldata.calldatasize)
# Assert
assert result == 7
def abstract_helper_test(fun: Callable[..., ExprRef], \
params: Tuple[ExprRef, ...],
expected: ExprRef,
xfail=False):
ctx = Context()
solver = Solver(ctx=ctx)
args = [z3_val(p, ctx) for p in params]
solver.add(fun(*args) == z3_val(expected, ctx))
if not xfail:
assert solver.check() == sat
else:
assert solver.check() == unsat
def sat_solve_prev(keystream, next, prev):
"""Find previous keystream by solving boolean satisfiability problem.
Parameters
----------
keystream : list of bool
Current keystream converted to list of boolean representing binary
values
next : list of z3.BoolRef
list containing bit triplet DNF constraints from the original next()
prev : list of z3.BoolRef
list containing bool representation of the previous keystream, to be
used as an index for Solver().model()
Returns
-------
keystream : list of bool
bool representation of keystream input from the previous call of next()
"""
solver = Solver()
for idx in range(N):
# add next keystream bools as constraints
solver.add(next[idx] == keystream[idx])
# check() should be 'sat' because we know we can get back to seed with
# these constraints, but we want to catch semantic errors just in case
if solver.check() == unsat:
raise Exception('Error in SAT DNF constraints')
model = solver.model()
# replace current keystream with solved previous values
for idx in range(N):
keystream[idx] = bool(model[prev[idx]])
return keystream
def collide(target_str, base_str, count=10, size_suffix=6, prefix=False):
'''Generates a string with the following properties:
* strcmp(res, base_str) = 0
* H(res) == H(target_str)'''
solver = Solver()
if prefix:
res = generate_ascii_printable_string(
'res', size_suffix, solver) + str_to_BitVecVals8(base_str)
else:
res = str_to_BitVecVals8(base_str) + generate_ascii_printable_string(
'res', size_suffix, solver)
target_checksum = H(str_to_BitVecVals8(target_str))
res_checksum = H(res)
solver.add(res_checksum == target_checksum)
for i in range(count):
if solver.check() == sat:
model = solver.model()
if prefix:
solution = "".join(
chr(model[x].as_long())
for x in res[:size_suffix]) + base_str
solver.add(
[x != model[x].as_long() for x in res[:size_suffix]])
else:
solution = base_str + "".join(
chr(model[x].as_long()) for x in res[-size_suffix:])
solver.add(
[x != model[x].as_long() for x in res[-size_suffix:]])
yield solution
def xform_correct(x, typing):
# type: (XForm, VarTyping) -> bool
"""
Given an XForm x and a concrete variable typing for x check whether x is
semantically preserving for the concrete typing.
"""
assert x.ti.permits(typing)
# Create copies of the x.src and x.dst with their concrete types
src_m = {v: Var(v.name, typing[v]) for v in x.src.vars()} # type: VarAtomMap # noqa
src = x.src.copy(src_m)
dst = x.apply(src)
dst_m = x.dst.substitution(dst, {})
# Build maps for the inputs/outputs for src->dst
inp_m = {} # type: VarAtomMap
out_m = {} # type: VarAtomMap
for v in x.src.vars():
src_v = src_m[v]
assert isinstance(src_v, Var)
if v.is_input():
inp_m[src_v] = dst_m[v]
elif v.is_output():
out_m[src_v] = dst_m[v]
# Get the primitive semantic Rtls for src and dst
prim_src = elaborate(src)
prim_dst = elaborate(dst)
asserts = equivalent(prim_src, prim_dst, inp_m, out_m)
s = Solver()
s.add(*asserts)
return s.check() == unsat
def solve(formulas):
s = Solver()
s.add(formulas)
status = s.check(); print(status)
if status == sat:
m = s.model(); print(m)
return m
def Password():
from z3 import Int, Solver
Buffer = Int('Buffer')
v9 = Int('v9')
v10 = Int('v10')
v11 = Int('v11')
v12 = Int('v12')
v13 = Int('v13')
v14 = Int('v14')
v15 = Int('v15')
v16 = Int('v16')
v17 = Int('v17')
s = Solver()
s.add(v14 + v15 == 205)
s.add(v13 + v16 == 201)
s.add(v11 + v14 + v15 == 314)
s.add(v13 + v16 + v12 + v17 == 367)
s.add(Buffer + v9 == 194)
s.add(v17 + v16 + v15 + v14 + v13 + v12 + v11 + v10 + v9 + Buffer == 923)
s.add(v16 == 85)
s.add(Buffer + v10 == 128)
s.add(v12 - v15 == -50)
s.add(v14 + v17 == 219)
return "Status : %s \nPassword = %s" % (
s.check(),
chr(s.model()[Buffer].as_long()) + chr(s.model()[v9].as_long()) +
chr(s.model()[v10].as_long()) + chr(s.model()[v11].as_long()) +
chr(s.model()[v12].as_long()) + chr(s.model()[v13].as_long()) +
chr(s.model()[v14].as_long()) + chr(s.model()[v15].as_long()) +
chr(s.model()[v16].as_long()) + chr(s.model()[v17].as_long()))
def _z3_bounded_model_count(solver: z3.Solver, variables: List[z3.ExprRef],
u: int) -> Optional[int]:
"""
If the solver assertions have less than u models that are distinct for the given variables it
returns the exact model count, otherwise it returns None.
:param solver:
:param variables:
:param u:
"""
solver.push()
for i in range(u):
response = solver.check()
if response == z3.unknown:
solver.pop()
raise RuntimeError("Solver responded with unknown")
elif response == z3.unsat:
solver.pop()
return i
# in the last iteration adding the constraint would be unnecessary, thus is skipped
if i != u - 1:
# add assertion that found model cannot be satisfying again
m = solver.model()
solver.add(z3.Or([x != m[x] for x in variables]))
solver.pop()
return None
def find_min(constraints, expr, default_min=0):
if type(expr) == int:
return expr
constraint_strs = [f'{c}' for c in constraints]
min_optimize = Optimize()
min_optimize.set('timeout', 10000)
min_optimize.assert_exprs(*constraints)
min_optimize.minimize(expr)
status = min_optimize.check()
if status != sat:
print(f'Unable to find min ({status}) for:\n' +
'\n'.join(constraint_strs))
return None
min_val = min_optimize.model().eval(expr).as_long()
# Make sure it's actually the min, since z3 has a bug
# https://github.com/Z3Prover/z3/issues/4670
solver = Solver()
solver.set('timeout', 10000)
solver.add(constraints + [expr < min_val])
status = solver.check()
if status != unsat:
print(
f'Z3 bug\nFind min ({expr}) => {min_val} with status ({status}):\n'
+ '\n'.join(constraint_strs))
return None
return min_val
def abstract_sfun_test(fun: Callable[..., ExprRef], \
params: Tuple[Union[str, int], ...],
expected: Union[str, int],
xfail=False):
ctx = Context()
solver = Solver(ctx=ctx)
cs = [Const(k, kind(p), k, ctx) for k, p in enumerate(params)]
solver.add([c == z3_val(p, ctx) for c, p in zip(cs, params)])
solver.add(z3_val(expected, ctx) == fun(*cs))
if not xfail:
assert solver.check() == sat
else:
assert solver.check() == unsat
def collide(target_str, base_str, count = 10, size_suffix = 6, prefix = False):
'''Generates a string with the following properties:
* strcmp(res, base_str) = 0
* H(res) == H(target_str)'''
solver = Solver()
if prefix:
res = generate_ascii_printable_string('res', size_suffix, solver) + str_to_BitVecVals8(base_str)
else:
res = str_to_BitVecVals8(base_str) + generate_ascii_printable_string('res', size_suffix, solver)
target_checksum = H(str_to_BitVecVals8(target_str))
res_checksum = H(res)
solver.add(res_checksum == target_checksum)
for i in range(count):
if solver.check() == sat:
model = solver.model()
if prefix:
solution = "".join(chr(model[x].as_long()) for x in res[:size_suffix]) + base_str
solver.add([x != model[x].as_long() for x in res[:size_suffix]])
else:
solution = base_str + "".join(chr(model[x].as_long()) for x in res[-size_suffix:])
solver.add([x != model[x].as_long() for x in res[-size_suffix:]])
yield solution
def solve():
start = time.time()
s = Solver()
s.reset()
for req in conf.reqs:
target = req[0]
accessConstraint = req[1]
requirementEncoding = encodeRequirement(target, accessConstraint)
s.add(ForAll(template.getAttributeVars(), requirementEncoding))
timeToTranslate = time.time() - start
measurements.addToTranslationTime(timeToTranslate)
start = time.time()
solution = None
if s.check() == sat:
solution = {}
model = s.model()
for PEP in conf.PEPS:
solution[PEP] = template.PEPPolicy(PEP, model)
else:
solution = unsat
timeToSolve = time.time() - start
measurements.addToSMTTime(timeToSolve)
return solution
def xform_correct(x, typing):
# type: (XForm, VarTyping) -> bool
"""
Given an XForm x and a concrete variable typing for x check whether x is
semantically preserving for the concrete typing.
"""
assert x.ti.permits(typing)
# Create copies of the x.src and x.dst with their concrete types
src_m = {v: Var(v.name, typing[v])
for v in x.src.vars()} # type: VarAtomMap # noqa
src = x.src.copy(src_m)
dst = x.apply(src)
dst_m = x.dst.substitution(dst, {})
# Build maps for the inputs/outputs for src->dst
inp_m = {} # type: VarAtomMap
out_m = {} # type: VarAtomMap
for v in x.src.vars():
src_v = src_m[v]
assert isinstance(src_v, Var)
if v.is_input():
inp_m[src_v] = dst_m[v]
elif v.is_output():
out_m[src_v] = dst_m[v]
# Get the primitive semantic Rtls for src and dst
prim_src = elaborate(src)
prim_dst = elaborate(dst)
asserts = equivalent(prim_src, prim_dst, inp_m, out_m)
s = Solver()
s.add(*asserts)
return s.check() == unsat
def visit_type_check(self, tree, symbol_table: SymbolTable):
"""
Attempts to typecheck a program if enabled.
:param tree: the AST of a program.
:param symbol_table: The symbol table of the program.
:return: None
"""
if self.config.typecheck:
combiner = self.config.combine.get_combiner(
self.config.epa_defs, self.config.abstract_interaction)
type_checker = TypeCheckVisitor(symbol_table, combiner,
self.config.types_used)
type_checker.visit(tree)
try:
z3 = Solver()
z3.set(unsat_core=True)
self.log.info(type_checker.smt_string)
z3.from_string(type_checker.smt_string)
if not z3.check():
raise TypeError(
f"The program {self.config.input_file}.bs could not be safely type checked."
)
else:
self.log.info(f"The model: \n{z3.model()}")
except AttributeError as e:
self.log.error(e)
except TypeError as a:
self.log.critical(a)
exit(123)
else:
self.log.debug("Type checking has been disabled.")
def solve(self, board, pieces, sum_requirements=[]):
if len(pieces) == 0:
return []
solver = Solver()
# Create z3 variables for each cell
extended_board = [(row, column, value, Int(self.cell_name(row,
column)))
for (row, column, value) in board]
constraints = \
self.set_prefilled_cell_values(extended_board) + \
self.set_possible_target_cell_values(extended_board, pieces) + \
self.require_unique_row_and_column_cells(extended_board) + \
self.match_sum_requirements(extended_board, sum_requirements) + \
self.target_cells_use_all_available_pieces(extended_board, pieces)
for constraint in constraints:
solver.add(constraint)
if solver.check() == sat:
model = solver.model()
return [(row, column, model[cell].as_long())
for (row, column, value, cell) in extended_board
if self.is_cell_empty(value)]
else:
return False
def find_queens():
from z3 import Int, And, Distinct, Solver
queens = [Int(f"Q{i+1}") for i in range(8)]
columns = [And(1 <= q, q <= 8) for q in queens]
distinct = [Distinct(queens)]
diags = [
And(queens[i] - queens[j] != i - j, queens[i] - queens[j] != j - i)
for i in range(8) for j in range(i)
]
solver = Solver()
solver.add(columns + distinct + diags)
solver.check()
m = solver.model()
return m
def main():
if len(sys.argv) != 2 or sys.argv[1] in ("-h", "help"):
print("Usage:", sys.argv[0], "NONOGRAMM-FILE")
exit(1)
elif not os.path.isfile(sys.argv[1]):
print("'{}' is not a valid file".format(sys.argv[1]))
exit(1)
nonogramm = Nonogramm.from_file(sys.argv[1])
if nonogramm is None:
print("'{}' doesn't contain a valid nonogramm".format(sys.argv[1]))
exit(2)
solver = Solver()
print("Generating constraints ...")
for constraint in nonogramm.gen_constraints():
solver.add(simplify(constraint))
print("Solving...")
if solver.check() == sat:
print("Solved:")
nonogramm.print_grid(solver.model())
else:
print("Unsolvable!")
def testZ3Distinctness():
'''
This test is simply a playground to explore
how z3 handles distinctness and equality checking.
'''
s = Solver()
x, y = Consts('x y', language.PointSort)
s.add(Distinct(x))
s.add(Distinct(y))
print s
# print s.add(Not(eq(x,y)))
# print eq(simplify(x),simplify(y))
# s.add(eq(x,y))
s.add(Not(eq(x,y)))
print s.check()
print s
def trim_unrechable_states(self):
# (parent, trace, child) tuples
pending_parent_trace_child_tuples = [(None, None, self.root_wstate)]
deleted_counter = 0
s = Solver()
while(len(pending_parent_trace_child_tuples)):
s.push()
parent_wstate, trace, curr_wstate = pending_parent_trace_child_tuples.pop()
if curr_wstate.status != WorldStateStatus.REACHABLE:
s.add(curr_wstate.constraints)
res = s.check()
if res == sat:
curr_wstate.status = WorldStateStatus.REACHABLE
elif res == unsat:
curr_wstate.status = WorldStateStatus.UNREACHABLE
elif res == z3.unknown:
print(curr_wstate.get_full_trace())
raise Exception("pdb")
if curr_wstate.status == WorldStateStatus.REACHABLE:
if curr_wstate != self.root_wstate:
parent_wstate.trace_to_children[trace].append(curr_wstate)
for child_trace, children in curr_wstate.trace_to_children.items():
for child_wstate in children:
pending_parent_trace_child_tuples.append((curr_wstate, child_trace, child_wstate))
curr_wstate.trace_to_children.clear()
else:
curr_wstate.status = WorldStateStatus.DELETED
self.gen_to_wstates[curr_wstate.gen].remove(curr_wstate)
deleted_counter += 1
s.pop()
logging.info('%d WorldStates are deleted', deleted_counter)
logging.info('SVM initialized')
def z3_solve(self, n, timeout_amount):
""" Integer factorization using z3 theorem prover implementation:
We can factor composite integers by SAT solving the model N=PQ directly using the clasuse (n==p*q),
wich gives a lot of degree of freedom to z3, so we want to contraint the search space.
Since every composite number n=pq, there always exists some p>sqrt(n) and q<sqrt(n).
We can safely asume the divisor p is in the range n > p >= next_prime(sqrt(n))
if this later clause doesn't hold and sqrt(p) is prime the number is a perfect square.
We can also asume that p and q are alyaws odd otherwise our whole composite is even.
Not all composite numbers generate a valid model that z3 can SAT.
SAT solving is efficient with low bit count set in the factors,
the complexity of the algorithm grows exponential with every bit set.
The problem of SAT solving integer factorization still is NP complete,
making this just a showcase. Don't expect big gains.
"""
s = Solver()
s.set("timeout", timeout_amount * 1000)
p = Int("p")
q = Int("q")
i = int(isqrt(n))
np = int(next_prime(i))
s.add(p * q == n, n > p, n > q, p >= np, q < i, q > 1, p > 1,
q % 2 != 0, p % 2 != 0)
try:
s_check_output = s.check()
if s_check_output == sat:
res = s.model()
P, Q = res[p].as_long(), res[q].as_long()
assert P * Q == n
return P, Q
else:
return None, None
except:
return None, None
def check_adversarial_robustness_z3():
from z3 import Reals, Int, Solver, If, And
sk, sk_1, zk, zk_1 = Reals('sk sk_1 zk zk_1')
i = Int('i')
s = Solver()
s.add(And(i >= 0, i <= 20, sk_1 >= 0, sk >= 0, zk >= 0, zk_1 >= 0))
A = If(sk * 1 >= 0, sk * 1, 0)
B = If(zk * 1 >= 0, zk * 1, 0)
s.add(If(i == 0,
And(sk >= 0, sk <= 3, zk >= 10, zk <= 21, sk_1 == 0, zk_1 == 0),
sk - zk >= sk_1 - zk_1 + 21 / i))
s.add(And(A < B, i == 20))
# # we negate the condition, instead if for all sk condition we check if there exists sk not condition
# s.add(sk_ReLU * w > ylim)
t = s.check()
if t == sat:
print("z3 result:", s.model())
return False
else:
# print("z3 result:", t)
return True
def z3_adversarial_unfold(n=10, to_pass=True):
from z3 import Reals, Solver, sat
s = Solver()
A0, B0 = Reals('A0 B0')
s.add(A0 == 0)
# if to_pass:
s.add(B0 == n)
#
# s.add(B0 == n+1)
A_prev = A0
B_prev = B0
for i in range(1, n + 1):
A_temp, B_temp = Reals('A{} B{}'.format(i, i))
# This is the invariant
if to_pass:
s.add(A_temp == A_prev + 1)
else:
s.add(A_temp == A_prev + 0.9)
s.add(B_temp == B_prev)
A_prev = A_temp
B_prev = B_temp
s.add(A_temp < B_temp)
# print(s)
t = s.check()
if t == sat:
# print("z3 result:", s.model())
return False
else:
# print("z3 result:", t)
return True
def solve():
start = time.time()
s = Solver()
s.reset()
for req in conf.reqs:
target = req[0]
accessConstraint = req[1]
requirementEncoding = encodeRequirement(target, accessConstraint)
s.add(ForAll(template.getAttributeVars(), requirementEncoding))
timeToTranslate = time.time() - start
measurements.addToTranslationTime(timeToTranslate)
start = time.time()
solution = None
if s.check() == sat:
solution = {}
model = s.model()
for PEP in conf.PEPS:
solution[PEP] = template.PEPPolicy(PEP, model)
else:
solution = unsat
timeToSolve = time.time() - start
measurements.addToSMTTime(timeToSolve)
return solution
def find_max(constraints, expr, l = None):
if l is None:
l = logger
if type(expr) == int:
return expr
constraint_strs = [f'{c}' for c in constraints]
max_optimize = Optimize()
max_optimize.set('timeout', 10000)
max_optimize.assert_exprs(*constraints)
max_optimize.maximize(expr)
status = max_optimize.check()
if status != sat:
l.warning(f'Unable to find max ({status}) for:\n' + '\n'.join(constraint_strs))
return None
max_val = max_optimize.model().eval(expr).as_long()
# Make sure it's actually the max, since z3 has a bug
# https://github.com/Z3Prover/z3/issues/4670
solver = Solver()
solver.set('timeout', 10000)
solver.add(constraints + [expr > max_val])
status = solver.check()
if status != unsat:
l.error(f'Z3 bug\nFind max ({expr}) => {max_val} with status ({status}):\n' + '\n'.join(constraint_strs))
return None
return max_val
def check_unique_solution(pins, problem):
s = Solver()
s.add(problem)
print(problem)
if s.check() == z3.sat:
print(s.model())
s.add(Not(model_to_condition(pins, s.model())))
if s.check() == z3.sat:
print(s.model())
print('Solution not unique!')
else:
print('Solution is unique!')
else:
print('Not solvable!')
def main():
dim = int(input("size of the array: "))
i = [Int(f"i_{x}") for x in range(dim + 1)]
j = [Int(f"j_{x}") for x in range(dim + 1)]
A = [Array(f"A_{x}", IntSort(), IntSort()) for x in range(dim + 1)]
tmp = [Int(f"tmp_{x}") for x in range(dim)]
s = Solver()
init_condition = init(i[0], j[0])
s.add(init_condition)
tran_condition = mk_tran_condition(A, i, j, tmp, dim)
s.add(And(*tran_condition))
values = [Int(f"n_{x}") for x in range(dim)]
init_check_condition = check(values, A[-1], dim)
s.add(And(*init_check_condition))
post_condition = mk_post_condition(values)
print("Bubble sort")
print("---------------------")
s.push()
s.add(Not(post_condition))
print("Testing the validity of the algorithm; `valid expected`:")
if s.check() == sat:
print(f"counterexample:\n{s.model()}")
else:
print("valid")
print("---------------------")
s.pop()
s.add(post_condition)
print("Getting a model...")
print("Model:")
if s.check() == sat:
print(s.model())
def __init__(self, rand: random.Random, expr: z3.ExprRef,
solver: z3.Solver):
RandomizedBinaryPathNode.__init__(self, rand)
notexpr = z3.Not(expr)
true_sat, false_sat = solver.check(expr), solver.check(notexpr)
if true_sat == z3.unknown or false_sat == z3.unknown:
raise UnknownSatisfiability
could_be_true = (true_sat == z3.sat)
could_be_false = (false_sat == z3.sat)
if (not could_be_true) and (not could_be_false):
debug(' *** Reached impossible code path *** ', true_sat,
false_sat, expr)
debug('Current solver state:\n', str(solver))
raise CrosshairInternal('Reached impossible code path')
elif not could_be_true:
self.forced_path = False
elif not could_be_false:
self.forced_path = True
def test_knaves_3(self):
"""
It can not be that a person has the role of a knave and tells the truth.
"""
f = knaves_tell_lies()
s = Solver()
s.add(f)
s.add(And(R(C) == Knave, S(C) == True))
self.assertEqual(unsat,s.check(), "Your formula should say that being a knave implies not telling the truth for all persons.")
def check_inductiveness(slv: Solver, inv2pinv: Dict[Clause, Clause]) -> bool:
# assumes trans has already been added to solver
slv.push()
assert_clauses(slv, inv2pinv.keys())
slv.add(Not(And([c._expr for c in inv2pinv.values()])))
# it's inductive if check is unsat
res = str(slv.check()) == "unsat"
slv.pop()
return res
def get_z3_result(query, debug=False) -> bool:
s = Solver()
s.add(query)
if s.check() == sat:
if debug:
print(s, s.model())
return True
return False
def test_knaves_2(self):
"""
The formula returned by `knaves_tell_lies` and a lying knave must be satisfiable.
"""
f = knaves_tell_lies()
s = Solver()
s.add(f)
s.add(And(R(A) == Knave, S(A) == False))
self.assertEqual(sat,s.check(), "Your formula should say that being a knave implies not telling the truth for all persons.")
def myprove(claim):
s = Solver()
s.set(timeout=1*1000)
s.add(claim)
rs = s.check()
if rs == unsat:
return True
else:
return False
class MySolver(object):
def __init__(self):
self._solver = Solver()
# TODO: Initialize datatypes here
# TODO: Port the below functions to here as methods
def push(self):
"""Push solver state."""
self._solver.push()
def pop(self):
"""Pop solver state."""
self._solver.pop()
def add(self, assertion):
"""Add an assertion to the solver state.
Arguments:
assertion : Z3-friendly predicate or boolean
"""
return self._solver.add(assertion)
def model(self):
"""Return a model for the current solver state.
Returns:
: Z3 model. TODO: Modify this all so that it returns sets, etc.
"""
return self._solver.model()
def check(self):
"""Check satisfiability of current satisfiability state.
Returns:
: boolean -- True if sat, False if unsat
"""
# check() returns either unsat or sat
# sat.r is 1, unsat.r is -1
return self._solver.check().r > 0
@contextmanager
def context(self):
"""To do something in between a push and a pop, use a `with context()`."""
self.push()
yield
self.pop()
def quick_check(self, assertion):
"""Add an assertion only temporarily, and check sat."""
with self.context():
self.add(assertion)
return self.check()
def remove_derived_clauses(self):
clauses = sorted(self, key=lambda x: (len(x), x.rank, id(x)))
solver = Solver()
for c in clauses:
if c.must_keep:
solver.add(c.z3_expr)
continue
solver.push()
solver.add(*[(~l).z3_expr for l in c])
if solver.check() != unsat:
solver.pop()
solver.add(c.z3_expr)
else:
solver.pop()
self.clauses.remove(c)
def is_equivalent(self, state):
solver = Solver()
b1, b2 = FreshBool(), FreshBool()
solver.add(b1 == self.z3_expr)
solver.add(b2 == state.z3_expr)
solver.add(Not(And(Implies(b1, b2), Implies(b2, b1))))
# print "comparing"
# print self
# print "-------"
# print state
# if solver.check() != unsat:
# m = solver.model()
# print m
# raw_input("press any key to continue")
# return False
# return True
return solver.check() == unsat
def PEPPolicy(PEP, model):
disjunctions = []
for or_id in range(NUM_ORS):
conjunctions = []
for enum_id in range(NUM_ENUMS):
enumVar = TEMPLATE_ENUM_VARS[PEP][or_id][enum_id]
if model[enumVar] is not None:
synthVal = model[enumVar].as_long()
else:
synthVal = -1
if synthVal >= 0 and synthVal < len(ENUM_INDEX.keys()):
if not isinstance(ENUM_INDEX[synthVal], list):
boolVar = ENUM_INDEX[synthVal]
conjunctions.append(boolVar)
else:
[enumVar, val] = ENUM_INDEX[synthVal]
conjunctions.append(enumVar == val)
elif synthVal >= len(ENUM_INDEX.keys()) and synthVal < 2 * len(ENUM_INDEX.keys()):
if not isinstance(ENUM_INDEX[synthVal - len(ENUM_INDEX.keys())], list):
boolVar = ENUM_INDEX[synthVal - len(ENUM_INDEX.keys())]
conjunctions.append(Not(boolVar))
else:
[enumVar, val] = ENUM_INDEX[synthVal - len(ENUM_INDEX.keys())]
conjunctions.append(enumVar != val)
elif synthVal == 2 * len(ENUM_INDEX.keys()):
conjunctions.append(True)
else:
conjunctions.append(False)
for num_id in range(NUM_NUMERIC):
minVar = TEMPLATE_NUMERIC_VARS[PEP][or_id][num_id]['min']
maxVar = TEMPLATE_NUMERIC_VARS[PEP][or_id][num_id]['max']
if model[minVar] is not None and model[maxVar] is not None:
crosscheck = Solver()
crosscheck.add(model[minVar] <= model[maxVar])
if crosscheck.check() == sat:
conjunctions.append(NUM_VAR >= model[minVar].as_long())
conjunctions.append(NUM_VAR <= model[maxVar].as_long())
elif model[minVar] is not None:
conjunctions.append(NUM_VAR >= model[minVar].as_long())
elif model[maxVar] is not None:
conjunctions.append(NUM_VAR <= model[maxVar].as_long())
disjunctions.append(simplify(And(conjunctions)))
return simplify(Or(disjunctions))
def is_tautology(formula) :
"""Check whether the formula is a tautology, and give a counterexample
if it is not.
Parameters
----------
formula@Formula - The formula to be tested.
Returns
----------
check_res@bool - Whether the formula is a tautology.
counterexample@Model - None if the formula is a tautology, otherwise a
counterexample.
"""
s = Solver()
s.add(Not(formula))
if s.check() == unsat :
return True, None
return False, s.model()
def check_termination(trs, DEBUG=False):
# create the variables we're solving for
W = WeightFunction(trs.functions)
P = Precedence(trs.functions)
# create solver and constraints
solver = Solver()
if DEBUG: print("Solver created...")
# constraint 1: weight constraints
solver.assert_exprs(W[0] > 0)
for f, arity in trs.functions.items():
solver.assert_exprs(P[f] >= 0)
if arity == 0:
solver.assert_exprs(W[f] >= W[0])
elif arity == 1:
local_constraints = []
for g in trs.functions.keys():
local_constraints.append(P[f] >= P[g])
solver.assert_exprs(Implies(W[f] == 0, And(local_constraints)))
if DEBUG: print("Weight constraints asserted...")
# constraint 2: kbo constraint per rule
for lhs, rhs in trs.rules:
solver.assert_exprs(kbo(lhs, rhs, trs, W, P))
if DEBUG: print("Rule constraint asserted...")
if DEBUG:
for c in solver.assertions():
print(c.sexpr())
print("Starting z3...")
start_time = clock()
if solver.check() == unsat:
if DEBUG: print("Done - z3 ran for {} seconds.".format(clock() - start_time))
return None
else:
model = solver.model()
weights, precedence = {}, {}
for f in trs.functions.keys():
weights[f] = model[W[f]].as_long()
precedence[f] = model[P[f]].as_long()
weights[0] = model[W[0]].as_long()
if DEBUG: print("Done - z3 ran for {} seconds.".format(clock() - start_time))
return weights, precedence
def get_models(F):
result = []
s = Solver()
s.add(F)
while True:
if s.check() == sat:
m = s.model()
result.append(m)
# Create a new constraint the blocks the current model
block = []
for d in m:
# d is a declaration
if d.arity() > 0:
raise Z3Exception(
"uninterpreted functions are not suppported")
# create a constant from declaration
c = d()
if is_array(c) or c.sort().kind() == Z3_UNINTERPRETED_SORT:
raise Z3Exception(
"arrays and uninterpreted sorts are not supported")
block.append(c != m[d])
s.add(Or(block))
else:
return result
def Synthesize(spec, specInputPorts, specConn, circuits):
wellFormedConstraints = [Bool(True)]
for circuit in circuits:
wellFormedConstraints.append(circuit.GenerateWellFormednessConstraints())
wellFormedConstraints.append(GenerateCircuitSimilarityConstraints(circuits))
psiWfp = And(wellFormedConstraints)
psiConn = And([circuit.GenerateConnectionConstraints() for circuit in circuits])
#if logger.IsLogging(): print psiConn
examples = []
iterCounter = 0
synthSolver = Solver()
verifSolver = Solver()
synthSolver.assert_exprs(psiWfp)
synthSolver.assert_exprs(GenerateConstraintForExample(spec, specConn, circuits,
psiConn, {}, iterCounter))
while True:
iterCounter+=1
if logger.IsLogging(): print 'Attempting Synthesis...'
if synthSolver.check() == unsat:
if logger.IsLogging(): print 'Synth Failed!'
return False
model = synthSolver.model()
if logger.IsLogging(): print 'Synthesized program:\n'
if logger.IsLogging():
for circuit in circuits:
print circuit.funcName
print circuit.LValToProg(model)
print '\n'
verifConstraint = GenerateVerificationConstraint(spec, specConn,
circuits, psiConn, model)
verifSolver.push()
verifSolver.assert_exprs(verifConstraint)
if logger.IsLogging(): print 'Attempting Verification...'
#if logger.IsLogging(): print verifConstraint
#raw_input("waiting for keypress")
if (verifSolver.check() == unsat):
if logger.IsLogging(): print 'Verification succeeded!\n'
verifSolver.pop()
if logger.IsLogging(): print 'Took %d iterations to '\
'complete synthesis' % (iterCounter)
if logger.IsLogging(): print 'Final Circuits:\n'
printedCircuits = []
for circuit in circuits:
if not circuit.funcName in printedCircuits:
print circuit.GenerateCircuitExpression(model)
printedCircuits.append(circuit.funcName)
return True
if logger.IsLogging(): print 'Verification Failed!\n'
newExample = GetExampleFromInputModel(verifSolver.model(), specInputPorts)
verifSolver.pop()
examples.append(newExample)
synthConstraintForExample = GenerateConstraintForExample(spec, specConn,
circuits, psiConn, newExample, iterCounter)
synthSolver.assert_exprs(synthConstraintForExample)
if logger.IsLogging(): print 'Added Example %d:' % iterCounter
if logger.IsLogging(): print newExample
def synthesize_with_components(self, components, constraint, system):
# components maps unique comp_ids to component objects
# not one-to-one, but def. onto
# step 0: some useful values
card_I, N = len(self.synth_function.parameters), len(components)
# step 1: make some solvers
synth_solver = Solver()
verify_solver = Solver()
# step 2: initialize examples
S = []
# step 2b: create location variables and location constraints
initial_constraints = []
L = create_location_variables(components)
if system:
patterns = create_pattern_constraints(L, components, system)
initial_constraints.append(patterns)
wfp = create_wfp_constraint(L, N)
initial_constraints.append(wfp)
# step 3: looooooooop
while True:
# step 4: assert L constraint
synth_solver.assert_exprs(*initial_constraints)
# step 5: start the looooop
for i, X in enumerate(S):
I, O, T = [], [], []
for w in range(constraint.width):
# step 6: create I, O, T for synth at width i
I_w = create_input_variables(self.synth_function.parameters, i, w)
O_w = create_output_variables(self.synth_function.output, i, w)
T_w = create_temp_variables(components, i, w)
# step 7: assert library and connection constraints
lib = create_lib_constraint(T_w, I_w, components)
# for conn constraint, need map from component_id to l value
locations = create_location_map(O_w, T_w, L, N)
conn = create_conn_constraint(O_w, T_w, locations)
synth_solver.assert_exprs(lib, conn)
I += I_w
O += O_w
T += T_w
# step 8: once we've got all the I, O, we can assert the spec constraint
conn_spec, spec = create_spec_constraint(I, O, X, constraint)
synth_solver.assert_exprs(conn_spec, spec)
# get a model, or just bail
if synth_solver.check() == unsat:
if DEBUG: print("Failed to find a model.")
return None
model = synth_solver.model()
curr_l = [l._replace(value=model[l.value]) for l in L]
# step 9: need to verify the model we just found, so we'll construct verificatio constraint
I, O, T = [], [], []
for w in range(constraint.width):
# same as above, but we only have a single example
I_w = create_input_variables(self.synth_function.parameters, 0, w)
O_w = create_output_variables(self.synth_function.output, 0, w)
T_w = create_temp_variables(components, 0, w)
lib = create_lib_constraint(T_w, I_w, components)
locations = create_location_map(O_w, T_w, curr_l, N)
conn = create_conn_constraint(O_w, T_w, locations)
verify_solver.assert_exprs(lib, conn)
I += I_w
O += O_w
T += T_w
# now we need to create variables for X so we can check our spec
X = create_spec_variables(constraint, self.variables)
conn_spec, spec = create_spec_constraint(I, O, X, constraint)
verify_solver.assert_exprs(conn_spec, Not(spec))
# now we'll try and get a model of our exectution
if verify_solver.check() == unsat:
return curr_l
model = verify_solver.model()
example = [x._replace(value=model[x.value]) for x in X]
S.append(example)
if DEBUG:
print("Found bad solution: ", [l.value.sexpr() for l in curr_l])
# clear synthesizers and start anew
synth_solver.reset()
verify_solver.reset()
return None
def exploit(t):
target = lambda: remote(t['hostname'], t['port'], timeout=3)
try:
with target() as s:
n_inputs = int(s.recvline().strip())
for _ in range(n_inputs):
inp = s.recvline().strip()
c1, c2, c3 = solve_for(inp)
s.sendline('{} {} {}'.format(c1, c2, c3))
s.recvuntil("want?:")
s.send("2\n3\n19\n"+t['flag_id']+"\n16\n"+"A"*16+"\n0\n")
s.recvuntil("GameTime:")
v19_c, v20_c, x_c, v22_c, v23_c, y_c = map(int, s.recv().split(', '))
solv = Solver()
v18 = Int('v18')
v21 = Int('v21')
v19 = Int('v19')
solv.add(v19 == v19_c)
v20 = Int('v20')
solv.add(v20 == v20_c)
v22 = Int('v22')
solv.add(v22 == v22_c)
v23 = Int('v23')
solv.add(v23 == v23_c)
x = Int('x')
solv.add(x == x_c)
solv.add(x == v19*v18 + v20*v21)
y = Int('y')
solv.add(y == y_c)
solv.add(y == v22*v18 + v21*v23)
solv.check()
solv.model()
s.sendline(str(solv.model()[v18].as_long()))
s.sendline(str(solv.model()[v21].as_long()))
s.recvuntil("log:")
s.send("15\n:83xkHFchNObsWf\n")
s.recvuntil("):")
s.sendline("478175")
s.recvuntil("Name:")
magic = s.recvuntil(":")[:-1]
with target() as s:
n_inputs = int(s.recvline().strip())
for _ in range(n_inputs):
inp = s.recvline().strip()
c1, c2, c3 = solve_for(inp)
s.sendline('{} {} {}'.format(c1, c2, c3))
s.recvuntil("want?:")
s.send("2\n5\n19\n"+t['flag_id']+"\n16\n"+magic+"\n")
s.recvline()
flag = s.recvline().strip()
return flag
except:
return
def main():
cppfile = 'Vtop_b12.cpp'
hfile = 'Vtop_b12.h'
CoveragesToTest = range(104 + 1)
cppstring=[]
hstring=[]
check = check_mod()
check.readfi(cppfile, cppstring)
check.readfi(hfile, hstring)
seqfunc=[]
check.get_seqfunc(cppstring, seqfunc)
AllSigList = check.get_all_sig_names(hstring, seqfunc)
parser = CParser()
generator = c_generator.CGenerator()
# cov_no = 1
for cov_no in CoveragesToTest:
#cov_no = 3
#Making global instance of CParser and CCgenerator
#as their they take a long time to run. So, just making it once
print("\n\nCoverage No. to test : {}".format(cov_no))
trueCond, falseCond = check.getAllCondsForCovNo(seqfunc, cov_no)
# Most probably this full search can be simplified to getting first false or true condition after detecting it
CondType, CondNo = check.CondAboveCovNo(seqfunc, cov_no, trueCond, falseCond)
SigOnDecision = []
if CondType == "falseCond":
SigOnDecision = check.getSigFromCond(falseCond[CondNo], AllSigList)
print("Coverage {} requires {} to be false".format(cov_no, falseCond[CondNo]))
else:
SigOnDecision = check.getSigFromCond(trueCond[CondNo], AllSigList)
print("Coverage {} requires {} to be true".format(cov_no, trueCond[CondNo]))
print("It involves Signal : {}".format(SigOnDecision))
SigAssignDict1 = check.GetAllAssignOfSig(SigOnDecision, seqfunc)
SigAssgnStrip1 = check.StripSign(SigAssignDict1)
NewSig1 = check.findNewSig(SigAssgnStrip1, SigOnDecision, AllSigList)
print("Which is assigned values at {}".format(SigAssgnStrip1))
SigAssignDict2 = check.GetAllAssignOfSig(NewSig1, seqfunc)
SigAssgnStrip2 = check.StripSign(SigAssignDict2)
if len(NewSig1)>0:
print("This involves New Signals = {}".format(NewSig1))
print("Which is assigned values at {}".format(SigAssgnStrip2))
if len(SigAssgnStrip2)>0:
NewSig2 = []
SigOnDecision += NewSig1
NewSig2 = check.findNewSig(SigAssgnStrip2, SigOnDecision, AllSigList)
if len(NewSig2)>0:
print("This involves New Signals = {}".format(NewSig2))
SigOnDecision += NewSig2
NewSig3 = "Going 3 levels deep hasn't been implemented, Visit later :)"
print(NewSig3)
print("Final list of signals on which coverage {} depends {}\n".format(cov_no, SigOnDecision))
# Done getting all signals till here.
#z3 signals initialization code
z3Sigs = []
flag = 0
for Sigs in SigOnDecision:
if type(AllSigList[Sigs]) == int:
msb = AllSigList[Sigs]
z3Sigs.append(BitVec('{}'.format(Sigs), msb))
# print("{} = BitVec(\'{}\', {})".format(Sigs, Sigs, msb))
else:
print("""Error: Probably an array, whose definition for z3 hasn't
been written\n\n""")
flag = 1
break
if flag == 1: continue
print("All constraints on final signals are")
s = Solver()
for idx, cond in enumerate(trueCond):
cond = cond.strip().replace(('vlTOPp->'), '').replace('(IData)','')
for Sig in SigOnDecision:
if re.search(r"\b{}\b".format(Sig), cond) is not None:
print("{} should be true".format(trueCond[idx]))
clause1 = check.GetZ3String(1, trueCond[idx], SigOnDecision, z3Sigs, parser, generator, AllSigList)
if clause1 != "":
clauseeval = eval(clause1)
s.add(clauseeval)
break
for idx, cond in enumerate(falseCond):
cond = cond.strip().replace(('vlTOPp->'), '').replace('(IData)','')
for Sig in SigOnDecision:
if re.search(r"\b{}\b".format(Sig), cond) is not None:
#if any new signal is found then don't take this cond
print("{} should be false".format(falseCond[idx]))
clause1 = check.GetZ3String(0, falseCond[idx], SigOnDecision, z3Sigs, parser, generator, AllSigList)
if clause1 != "":
clauseeval = eval(clause1)
s.add(clauseeval)
break
#need to put main condition first
print("Assignments 1 level deep and their coverage no. are {}".format(SigAssgnStrip1))
print("Assignments 2 level deep and their coverage no. are {}".format(SigAssgnStrip2))
satis = []
unsatis = []
for Sig1 in SigAssgnStrip1:
Sig1cl = check.GetZ3String(1, Sig1.replace(';',''), SigOnDecision, z3Sigs, parser, generator, AllSigList)
if Sig1cl != "":
Sig1s = eval(Sig1cl)
s.push()
s.add(Sig1s)
Cover_No = int(SigAssgnStrip1[Sig1])
if s.check() == sat:
if Cover_No not in satis: satis.append(Cover_No)
else:
if Cover_No not in unsatis: unsatis.append(Cover_No)
for Sig2 in SigAssgnStrip2:
sig2ss = check.GetZ3String(1, Sig2.replace(';',''), SigOnDecision, z3Sigs, parser, generator, AllSigList)
if sig2ss != "":
sig2s = eval(sig2ss)
s.push()
s.add(sig2s)
# print s.check()
Cover_No = int(SigAssgnStrip1[Sig1])
if s.check() == sat:
if Cover_No not in satis: satis.append(Cover_No)
else:
if Cover_No not in unsatis: unsatis.append(Cover_No)
s.pop()
if Sig1cl != "":
s.pop()
print("Satisfiable Coverages are {}".format(satis))
print("Unsatisfiable Coverages are {}".format(unsatis))
if ((len(satis) == 0) and (len(unsatis) == 0)):
print("Looks like 'Input' so can be assigned any value")
from z3 import Solver, BitVec
s = Solver()
x = BitVec('x', 32)
s.add(x > 1337)
s.add(x*7 + 4 == 1337)
s.check()
print s.model()
def __call__(self, project, test, dump):
logger.info('inferring specification for test \'{}\''.format(test))
environment = dict(os.environ)
if self.config['klee_max_forks'] is not None:
environment['ANGELIX_KLEE_MAX_FORKS'] = str(self.config['klee_max_forks'])
if self.config['klee_max_depth'] is not None:
environment['ANGELIX_KLEE_MAX_DEPTH'] = str(self.config['klee_max_depth'])
if self.config['klee_search'] is not None:
environment['ANGELIX_KLEE_SEARCH'] = self.config['klee_search']
if self.config['klee_timeout'] is not None:
environment['ANGELIX_KLEE_MAX_TIME'] = str(self.config['klee_timeout'])
if self.config['klee_solver_timeout'] is not None:
environment['ANGELIX_KLEE_MAX_SOLVER_TIME'] = str(self.config['klee_solver_timeout'])
if self.config['klee_debug']:
environment['ANGELIX_KLEE_DEBUG'] = 'YES'
if self.config['klee_ignore_errors']:
environment['KLEE_DISABLE_MEMORY_ERROR'] = 'YES'
if self.config['use_semfix_syn']:
environment['ANGELIX_USE_SEMFIX_SYN'] = 'YES'
environment['ANGELIX_KLEE_WORKDIR'] = project.dir
test_dir = self.get_test_dir(test)
shutil.rmtree(test_dir, ignore_errors='true')
klee_dir = join(test_dir, 'klee')
os.makedirs(klee_dir)
self.run_test(project, test, klee=True, env=environment)
# loading dump
# name -> value list
oracle = dict()
vars = os.listdir(dump)
for var in vars:
instances = os.listdir(join(dump, var))
for i in range(0, len(instances)):
if str(i) not in instances:
logger.error('corrupted dump for test \'{}\''.format(test))
raise InferenceError()
oracle[var] = []
for i in range(0, len(instances)):
file = join(dump, var, str(i))
with open(file) as f:
content = f.read()
oracle[var].append(content)
# solving path constraints
angelic_paths = []
solver = Solver()
smt_glob = join(project.dir, 'klee-out-0', '*.smt2')
smt_files = glob(smt_glob)
for smt in smt_files:
logger.info('solving path {}'.format(relpath(smt)))
try:
path = z3.parse_smt2_file(smt)
except:
logger.warning('failed to parse {}'.format(smt))
continue
variables = [str(var) for var in get_vars(path)
if str(var).startswith('int!')
or str(var).startswith('bool!')
or str(var).startswith('char!')
or str(var).startswith('reachable!')]
outputs, choices, constants, reachable, original_available = parse_variables(variables)
# name -> value list (parsed)
oracle_constraints = dict()
def str_to_int(s):
return int(s)
def str_to_bool(s):
if s == 'false':
return False
if s == 'true':
return True
raise InferenceError()
def str_to_char(s):
if len(s) != 1:
raise InferenceError()
return s[0]
dump_parser_by_type = dict()
dump_parser_by_type['int'] = str_to_int
dump_parser_by_type['bool'] = str_to_bool
dump_parser_by_type['char'] = str_to_char
def bool_to_bv32(b):
if b:
return BitVecVal(1, 32)
else:
return BitVecVal(0, 32)
def int_to_bv32(i):
return BitVecVal(i, 32)
to_bv32_converter_by_type = dict()
to_bv32_converter_by_type['bool'] = bool_to_bv32
to_bv32_converter_by_type['int'] = int_to_bv32
def bv32_to_bool(bv):
return bv.as_long() != 0
def bv32_to_int(bv):
l = bv.as_long()
if l >> 31 == 1: # negative
l -= 4294967296
return l
from_bv32_converter_by_type = dict()
from_bv32_converter_by_type['bool'] = bv32_to_bool
from_bv32_converter_by_type['int'] = bv32_to_int
matching_path = True
for expected_variable, expected_values in oracle.items():
if expected_variable == 'reachable':
expected_reachable = set(expected_values)
if not (expected_reachable == reachable):
logger.info('labels \'{}\' executed while {} required'.format(
list(reachable),
list(expected_reachable)))
matching_path = False
break
continue
if expected_variable not in outputs.keys():
outputs[expected_variable] = (None, 0) # unconstraint does not mean wrong
required_executions = len(expected_values)
actual_executions = outputs[expected_variable][1]
if required_executions != actual_executions:
logger.info('value \'{}\' executed {} times while {} required'.format(
expected_variable,
actual_executions,
required_executions))
matching_path = False
break
oracle_constraints[expected_variable] = []
for i in range(0, required_executions):
type = outputs[expected_variable][0]
try:
value = dump_parser_by_type[type](expected_values[i])
except:
logger.error('variable \'{}\' has incompatible type {}'.format(expected_variable,
type))
raise InferenceError()
oracle_constraints[expected_variable].append(value)
if not matching_path:
continue
solver.reset()
solver.add(path)
def array_to_bv32(array):
return Concat(Select(array, BitVecVal(3, 32)),
Select(array, BitVecVal(2, 32)),
Select(array, BitVecVal(1, 32)),
Select(array, BitVecVal(0, 32)))
def angelic_selector(expr, instance):
s = 'angelic!{}!{}!{}!{}!{}'.format(expr[0], expr[1], expr[2], expr[3], instance)
return BitVec(s, 32)
def original_selector(expr, instance):
s = 'original!{}!{}!{}!{}!{}'.format(expr[0], expr[1], expr[2], expr[3], instance)
return BitVec(s, 32)
def env_selector(expr, instance, name):
s = 'env!{}!{}!{}!{}!{}!{}'.format(name, expr[0], expr[1], expr[2], expr[3], instance)
return BitVec(s, 32)
for name, values in oracle_constraints.items():
type, _ = outputs[name]
for i, value in enumerate(values):
array = self.output_variable(type, name, i)
bv_value = to_bv32_converter_by_type[type](value)
solver.add(bv_value == array_to_bv32(array))
for (expr, item) in choices.items():
type, instances, env = item
for instance in range(0, instances):
selector = angelic_selector(expr, instance)
array = self.angelic_variable(type, expr, instance)
solver.add(selector == array_to_bv32(array))
selector = original_selector(expr, instance)
array = self.original_variable(type, expr, instance)
solver.add(selector == array_to_bv32(array))
for name in env:
selector = env_selector(expr, instance, name)
env_type = 'int' #FIXME
array = self.env_variable(env_type, expr, instance, name)
solver.add(selector == array_to_bv32(array))
result = solver.check()
if result != z3.sat:
logger.info('UNSAT')
continue
model = solver.model()
# store smt2 files
shutil.copy(smt, klee_dir)
# generate IO file
self.generate_IO_file(test, choices, oracle_constraints, outputs)
# expr -> (angelic * original * env) list
angelic_path = dict()
for (expr, item) in choices.items():
angelic_path[expr] = []
type, instances, env = item
for instance in range(0, instances):
bv_angelic = model[angelic_selector(expr, instance)]
angelic = from_bv32_converter_by_type[type](bv_angelic)
bv_original = model[original_selector(expr, instance)]
original = from_bv32_converter_by_type[type](bv_original)
if original_available:
logger.info('expression {}[{}]: angelic = {}, original = {}'.format(expr,
instance,
angelic,
original))
else:
logger.info('expression {}[{}]: angelic = {}'.format(expr,
instance,
angelic))
env_values = dict()
for name in env:
bv_env = model[env_selector(expr, instance, name)]
value = from_bv32_converter_by_type['int'](bv_env)
env_values[name] = value
if original_available:
angelic_path[expr].append((angelic, original, env_values))
else:
angelic_path[expr].append((angelic, None, env_values))
# TODO: add constants to angelic path
angelic_paths.append(angelic_path)
# update IO files
for smt in glob(join(klee_dir, '*.smt2')):
with open(smt) as f_smt:
for line in f_smt.readlines():
if re.search("declare-fun [a-z]+!output!", line):
output_var = line.split(' ')[1]
output_var_type = output_var.split('!')[0]
for io_file in glob(join(test_dir, '*.IO')):
if not output_var in open(io_file).read():
with open(io_file, "a") as f_io:
f_io.write("\n")
f_io.write("@output\n")
f_io.write('name {}\n'.format(output_var))
f_io.write('type {}\n'.format(output_var_type))
if self.config['max_angelic_paths'] is not None and \
len(angelic_paths) > self.config['max_angelic_paths']:
angelic_paths = self._reduce_angelic_forest(angelic_paths)
else:
logger.info('found {} angelic paths for test \'{}\''.format(len(angelic_paths), test))
return angelic_paths
q2x = p1[0]+t2*(p2[0]-p1[0])
q2y = p1[1]+t2*(p2[1]-p1[1])
q3x = p2[0]+t3*(p3[0]-p2[0])
q3y = p2[1]+t3*(p3[1]-p2[1])
q4x = p3[0]+t4*(p4[0]-p3[0])
q4y = p3[1]+t4*(p4[1]-p3[1])
solver = Solver()
solver.add(t1 >= 0, t2 >= 0, t3 >= 0, t4 >= 0)
solver.add(t1 <= 1, t2 <= 1, t3 <= 1, t4 <= 1)
def orthogonal(px, py, qx, qy, rx, ry):
return (px-qx)*(rx-qx)+(py-qy)*(ry-qy) == 0
solver.add(orthogonal(q1x, q1y, q2x, q2y, q3x, q3y))
solver.add(orthogonal(q2x, q2y, q3x, q4y, q4x, q4y))
solver.add(orthogonal(q3x, q3y, q4x, q4y, q1x, q1y))
solver.add(orthogonal(q4x, q4y, q1x, q1y, q2x, q2y))
solver.add((q2x-q1x)**2+(q2y-q1y)**2 == (q4x-q3x)**2+(q4y-q3y)**2)
solver.add((q3x-q2x)**2+(q3y-q2y)**2 == (q4x-q1x)**2+(q4y-q1y)**2)
solver.add()
solver.check()
print(solver.model())
class ACL22SMT(object):
class status:
def __init__(self, value):
self.value = value
def __str__(self):
if self.value is True:
return "QED"
elif self.value.__class__ == "msg".__class__:
return self.value
else:
raise Exception("unknown status?")
def isThm(self):
return self.value is True
class atom: # added my mrg, 21 May 2015
def __init__(self, string):
self.who_am_i = string.lower()
def __eq__(self, other):
return self.who_am_i == other.who_am_i
def __ne__(self, other):
return self.who_am_i != other.who_am_i
def __str__(self):
return self.who_am_i
def __init__(self, solver=0):
if solver != 0:
self.solver = solver
else:
self.solver = Solver()
self.nameNumber = 0
def newVar(self):
varName = "$" + str(self.nameNumber)
self.nameNumber = self.nameNumber + 1
return varName
def isBool(self, who):
return Bool(who)
def isInt(self, who):
return Int(who)
def isReal(self, who):
return Real(who)
def plus(self, *args):
return reduce(lambda x, y: x + y, args)
def times(self, *args):
return reduce(lambda x, y: x * y, args)
def reciprocal(self, x):
if type(x) is int:
return Q(1, x)
elif type(x) is float:
return 1.0 / x
else:
return 1.0 / x
def negate(self, x):
return -x
def lt(self, x, y):
return x < y
def equal(self, x, y):
return x == y
def notx(self, x):
return Not(x)
def implies(self, x, y):
return Implies(x, y)
def Qx(self, x, y):
return Q(x, y)
# type related functions
def integerp(self, x):
return sort(x) == IntSort()
def rationalp(self, x):
return sort(x) == RealSort()
def booleanp(self, x):
return sort(x) == BoolSort()
def ifx(self, condx, thenx, elsex):
return If(condx, thenx, elsex)
# usage prove(claim) or prove(hypotheses, conclusion)
def prove(self, hypotheses, conclusion=0):
if conclusion is 0:
claim = hypotheses
else:
claim = Implies(hypotheses, conclusion)
self.solver.push()
self.solver.add(Not(claim))
res = self.solver.check()
if res == unsat:
print "proved"
return self.status(True) # It's a theorem
elif res == sat:
print "counterexample"
m = self.solver.model()
print m
# return an counterexample??
return self.status(False)
else:
print "failed to prove"
r = self.status(False)
self.solver.pop()
return r
e = ExprId('e', 1)
left = ExprCond(e + ExprOp('parity', a),
ExprMem(a * a, 64),
ExprMem(a, 64))
cond = ExprSlice(ExprSlice(ExprSlice(a, 0, 32) + b, 0, 16) * c, 0, 8) << ExprOp('>>>', d, ExprInt(uint8(0x5L)))
right = ExprCond(cond,
a + ExprInt(uint64(0x64L)),
ExprInt(uint64(0x16L)))
e = ExprAff(left, right)
# initialise translators
t_z3 = TranslatorZ3()
t_smt2 = TranslatorSMT2()
# translate to z3
e_z3 = t_z3.from_expr(e)
# translate to smt2
smt2 = t_smt2.to_smt2([t_smt2.from_expr(e)])
# parse smt2 string with z3
smt2_z3 = parse_smt2_string(smt2)
# initialise SMT solver
s = Solver()
# prove equivalence of z3 and smt2 translation
s.add(e_z3 != smt2_z3)
assert (s.check() == unsat)
def __call__(self, project, test, dump, validation_project):
logger.info('inferring specification for test \'{}\''.format(test))
environment = dict(os.environ)
if self.config['klee_max_forks'] is not None:
environment['ANGELIX_KLEE_MAX_FORKS'] = str(self.config['klee_max_forks'])
if self.config['klee_max_depth'] is not None:
environment['ANGELIX_KLEE_MAX_DEPTH'] = str(self.config['klee_max_depth'])
if self.config['klee_search'] is not None:
environment['ANGELIX_KLEE_SEARCH'] = self.config['klee_search']
if self.config['klee_timeout'] is not None:
environment['ANGELIX_KLEE_MAX_TIME'] = str(self.config['klee_timeout'])
if self.config['klee_solver_timeout'] is not None:
environment['ANGELIX_KLEE_MAX_SOLVER_TIME'] = str(self.config['klee_solver_timeout'])
if self.config['klee_debug']:
environment['ANGELIX_KLEE_DEBUG'] = 'YES'
if self.config['klee_ignore_errors']:
environment['KLEE_DISABLE_MEMORY_ERROR'] = 'YES'
if self.config['use_semfix_syn']:
environment['ANGELIX_USE_SEMFIX_SYN'] = 'YES'
environment['ANGELIX_KLEE_WORKDIR'] = project.dir
klee_start_time = time.time()
self.run_test(project, test, klee=True, env=environment)
klee_end_time = time.time()
klee_elapsed = klee_end_time - klee_start_time
statistics.data['time']['klee'] += klee_elapsed
statistics.save()
logger.info('sleeping for 1 second...')
time.sleep(1)
smt_glob = join(project.dir, 'klee-out-0', '*.smt2')
smt_files = glob(smt_glob)
err_glob = join(project.dir, 'klee-out-0', '*.err')
err_files = glob(err_glob)
err_list = []
for err in err_files:
err_list.append(os.path.basename(err).split('.')[0])
non_error_smt_files = []
for smt in smt_files:
smt_id = os.path.basename(smt).split('.')[0]
if not smt_id in err_list:
non_error_smt_files.append(smt)
if not self.config['ignore_infer_errors']:
smt_files = non_error_smt_files
if len(smt_files) == 0 and len(err_list) == 0:
logger.warning('No paths explored')
raise NoSmtError()
if len(smt_files) == 0:
logger.warning('No non-error paths explored')
raise NoSmtError()
# loading dump
# name -> value list
oracle = dict()
vars = os.listdir(dump)
for var in vars:
instances = os.listdir(join(dump, var))
for i in range(0, len(instances)):
if str(i) not in instances:
logger.error('corrupted dump for test \'{}\''.format(test))
raise InferenceError()
oracle[var] = []
for i in range(0, len(instances)):
file = join(dump, var, str(i))
with open(file) as f:
content = f.read()
oracle[var].append(content)
# solving path constraints
inference_start_time = time.time()
angelic_paths = []
z3.set_param("timeout", self.config['path_solving_timeout'])
solver = Solver()
for smt in smt_files:
logger.info('solving path {}'.format(relpath(smt)))
try:
path = z3.parse_smt2_file(smt)
except:
logger.warning('failed to parse {}'.format(smt))
continue
variables = [str(var) for var in get_vars(path)
if str(var).startswith('int!')
or str(var).startswith('long!')
or str(var).startswith('bool!')
or str(var).startswith('char!')
or str(var).startswith('reachable!')]
try:
outputs, choices, constants, reachable, original_available = parse_variables(variables)
except:
continue
# name -> value list (parsed)
oracle_constraints = dict()
def str_to_int(s):
return int(s)
def str_to_long(s):
return int(s)
def str_to_bool(s):
if s == 'false':
return False
if s == 'true':
return True
raise InferenceError()
def str_to_char(s):
if len(s) != 1:
raise InferenceError()
return s[0]
dump_parser_by_type = dict()
dump_parser_by_type['int'] = str_to_int
dump_parser_by_type['long'] = str_to_long
dump_parser_by_type['bool'] = str_to_bool
dump_parser_by_type['char'] = str_to_char
def bool_to_bv(b):
if b:
return BitVecVal(1, 32)
else:
return BitVecVal(0, 32)
def int_to_bv(i):
return BitVecVal(i, 32)
def long_to_bv(i):
return BitVecVal(i, 64)
def char_to_bv(c):
return BitVecVal(ord(c), 32)
to_bv_converter_by_type = dict()
to_bv_converter_by_type['bool'] = bool_to_bv
to_bv_converter_by_type['int'] = int_to_bv
to_bv_converter_by_type['long'] = long_to_bv
to_bv_converter_by_type['char'] = char_to_bv
def bv_to_bool(bv):
return bv.as_long() != 0
def bv_to_int(bv):
l = bv.as_long()
if l >> 31 == 1: # negative
l -= pow(2, 32)
return l
def bv_to_long(bv):
l = bv.as_long()
if l >> 63 == 1: # negative
l -= pow(2, 64)
return l
def bv_to_char(bv):
l = bv.as_long()
return chr(l)
from_bv_converter_by_type = dict()
from_bv_converter_by_type['bool'] = bv_to_bool
from_bv_converter_by_type['int'] = bv_to_int
from_bv_converter_by_type['long'] = bv_to_long
from_bv_converter_by_type['char'] = bv_to_char
matching_path = True
for expected_variable, expected_values in oracle.items():
if expected_variable == 'reachable':
expected_reachable = set(expected_values)
if not (expected_reachable == reachable):
logger.info('labels \'{}\' executed while {} required'.format(
list(reachable),
list(expected_reachable)))
matching_path = False
break
continue
if expected_variable not in outputs.keys():
outputs[expected_variable] = (None, 0) # unconstraint does not mean wrong
required_executions = len(expected_values)
actual_executions = outputs[expected_variable][1]
if required_executions != actual_executions:
logger.info('value \'{}\' executed {} times while {} required'.format(
expected_variable,
actual_executions,
required_executions))
matching_path = False
break
oracle_constraints[expected_variable] = []
for i in range(0, required_executions):
type = outputs[expected_variable][0]
try:
value = dump_parser_by_type[type](expected_values[i])
except:
logger.error('variable \'{}\' has incompatible type {}'.format(expected_variable,
type))
raise InferenceError()
oracle_constraints[expected_variable].append(value)
if not matching_path:
continue
solver.reset()
solver.add(path)
def array_to_bv32(array):
return Concat(Select(array, BitVecVal(3, 32)),
Select(array, BitVecVal(2, 32)),
Select(array, BitVecVal(1, 32)),
Select(array, BitVecVal(0, 32)))
def array_to_bv64(array):
return Concat(Select(array, BitVecVal(7, 32)),
Select(array, BitVecVal(6, 32)),
Select(array, BitVecVal(5, 32)),
Select(array, BitVecVal(4, 32)),
Select(array, BitVecVal(3, 32)),
Select(array, BitVecVal(2, 32)),
Select(array, BitVecVal(1, 32)),
Select(array, BitVecVal(0, 32)))
def angelic_variable(type, expr, instance):
pattern = '{}!choice!{}!{}!{}!{}!{}!angelic'
s = pattern.format(type, expr[0], expr[1], expr[2], expr[3], instance)
return Array(s, BitVecSort(32), BitVecSort(8))
def original_variable(type, expr, instance):
pattern = '{}!choice!{}!{}!{}!{}!{}!original'
s = pattern.format(type, expr[0], expr[1], expr[2], expr[3], instance)
return Array(s, BitVecSort(32), BitVecSort(8))
def env_variable(expr, instance, name):
pattern = 'int!choice!{}!{}!{}!{}!{}!env!{}'
s = pattern.format(expr[0], expr[1], expr[2], expr[3], instance, name)
return Array(s, BitVecSort(32), BitVecSort(8))
def output_variable(type, name, instance):
s = '{}!output!{}!{}'.format(type, name, instance)
if type == 'long':
return Array(s, BitVecSort(32), BitVecSort(8))
else:
return Array(s, BitVecSort(32), BitVecSort(8))
def angelic_selector(expr, instance):
s = 'angelic!{}!{}!{}!{}!{}'.format(expr[0], expr[1], expr[2], expr[3], instance)
return BitVec(s, 32)
def original_selector(expr, instance):
s = 'original!{}!{}!{}!{}!{}'.format(expr[0], expr[1], expr[2], expr[3], instance)
return BitVec(s, 32)
def env_selector(expr, instance, name):
s = 'env!{}!{}!{}!{}!{}!{}'.format(name, expr[0], expr[1], expr[2], expr[3], instance)
return BitVec(s, 32)
for name, values in oracle_constraints.items():
type, _ = outputs[name]
for i, value in enumerate(values):
array = output_variable(type, name, i)
bv_value = to_bv_converter_by_type[type](value)
if type == 'long':
solver.add(bv_value == array_to_bv64(array))
else:
solver.add(bv_value == array_to_bv32(array))
for (expr, item) in choices.items():
type, instances, env = item
for instance in range(0, instances):
selector = angelic_selector(expr, instance)
array = angelic_variable(type, expr, instance)
solver.add(selector == array_to_bv32(array))
selector = original_selector(expr, instance)
array = original_variable(type, expr, instance)
solver.add(selector == array_to_bv32(array))
for name in env:
selector = env_selector(expr, instance, name)
array = env_variable(expr, instance, name)
solver.add(selector == array_to_bv32(array))
result = solver.check()
if result != z3.sat:
logger.info('UNSAT') # TODO: can be timeout
continue
model = solver.model()
# expr -> (angelic * original * env) list
angelic_path = dict()
if os.path.exists(self.load[test]):
shutil.rmtree(self.load[test])
os.mkdir(self.load[test])
for (expr, item) in choices.items():
angelic_path[expr] = []
type, instances, env = item
expr_str = '{}-{}-{}-{}'.format(expr[0], expr[1], expr[2], expr[3])
expression_dir = join(self.load[test], expr_str)
if not os.path.exists(expression_dir):
os.mkdir(expression_dir)
for instance in range(0, instances):
bv_angelic = model[angelic_selector(expr, instance)]
angelic = from_bv_converter_by_type[type](bv_angelic)
bv_original = model[original_selector(expr, instance)]
original = from_bv_converter_by_type[type](bv_original)
if original_available:
logger.info('expression {}[{}]: angelic = {}, original = {}'.format(expr,
instance,
angelic,
original))
else:
logger.info('expression {}[{}]: angelic = {}'.format(expr,
instance,
angelic))
env_values = dict()
for name in env:
bv_env = model[env_selector(expr, instance, name)]
value = from_bv_converter_by_type['int'](bv_env)
env_values[name] = value
if original_available:
angelic_path[expr].append((angelic, original, env_values))
else:
angelic_path[expr].append((angelic, None, env_values))
# Dump angelic path to dump folder
instance_file = join(expression_dir, str(instance))
with open(instance_file, 'w') as file:
if isinstance(angelic, bool):
if angelic:
file.write('1')
else:
file.write('0')
else:
file.write(str(angelic))
# Run Tester to validate the dumped values
validated = self.run_test(validation_project, test, load=self.load[test])
if validated:
angelic_paths.append(angelic_path)
else:
logger.info('spurious angelic path')
if self.config['synthesis_bool_only']:
angelic_paths = self._boolean_angelic_forest(angelic_paths)
if self.config['max_angelic_paths'] is not None and \
len(angelic_paths) > self.config['max_angelic_paths']:
angelic_paths = self._reduce_angelic_forest(angelic_paths)
else:
logger.info('found {} angelic paths for test \'{}\''.format(len(angelic_paths), test))
inference_end_time = time.time()
inference_elapsed = inference_end_time - inference_start_time
statistics.data['time']['inference'] += inference_elapsed
iter_stat = dict()
iter_stat['time'] = dict()
iter_stat['time']['klee'] = klee_elapsed
iter_stat['time']['inference'] = inference_elapsed
iter_stat['paths'] = dict()
iter_stat['paths']['explored'] = len(smt_files)
iter_stat['paths']['angelic'] = len(angelic_paths)
statistics.data['iterations']['klee'].append(iter_stat)
statistics.save()
return angelic_paths
def test1():
print "=== Loading Core ==="
solver = Solver()
solver.push()
solver.add(language.axioms)
print "=== Starting tests ==="
print ">> Let p q r s t u v be distinct points"
p, q, r, s, t, u, v = Consts('p q r s t u v', language.PointSort)
solver.add(simplify(Distinct(p,q,r,s,t,u,v), blast_distinct=True))
print ">> Let L M N O be distinct lines"
K, L, M, N, O = Consts('K L M N O', language.LineSort)
solver.add(simplify(Distinct(K,L,M,N,O), blast_distinct=True))
## Diagram description
assumptions = []
assumptions.append(language.OnLine(p,L))
assumptions.append(language.OnLine(q,L))
assumptions.append(language.OnLine(p,N))
assumptions.append(language.OnLine(s,N))
assumptions.append(language.OnLine(t,N))
assumptions.append(language.OnLine(p,M))
assumptions.append(language.OnLine(r,M))
assumptions.append(language.OnLine(q,O))
assumptions.append(language.OnLine(s,O))
assumptions.append(language.OnLine(r,O))
assumptions.append(language.OnLine(q,K))
assumptions.append(language.OnLine(t,K))
assumptions.append(Not(language.OnLine(r,L)))
assumptions.append(language.Between(p,s,t))
assumptions.append(language.Between(q,s,r))
assumptions.append(language.Between(s,u,t))
assumptions.append(Not(p == q))
assumptions.append(language.Between(p,q,v))
print ">> Assume " + str(assumptions)
solver.add(assumptions)
print " << z3: " + str(solver.check())
## Satisfied
solver.push()
solver.add(True)
print ">> Hence True"
print " << z3: " + str(solver.check())
solver.pop()
## Satisfied
solver.push()
solver.add(Not(language.SameSide(s,t,O)))
print ">> Hence s and t are on opposite sides of O"
print " << z3: " + str(solver.check())
solver.pop()
## Satisfied
solver.push()
solver.add(language.SameSide(u,t,M))
print ">> Hence u and t are on same side of M"
print " << z3: " + str(solver.check())
solver.pop()
## unsatisfied
solver.push()
solver.add(language.SameSide(p,t,O))
print ">> Hence p and t are on same side of O"
print " << z3: " + str(solver.check())
solver.pop()
## unsatisfied
solver.push()
solver.add(language.SameSide(s,t,O))
print ">> Hence s and t are on same side of O"
print " << z3: " + str(solver.check())
solver.pop()
## unsatisfied
solver.push()
solver.add(Not(language.SameSide(s,t,M)))
print ">> Hence s and t are on opposite sides of M"
print " << z3: " + str(solver.check())
solver.pop()
## unsatisfied
solver.push()
solver.add(Not(language.SameSide(u,t,M)))
print ">> Hence u and t are on opposite sides of M"
print " << z3: " + str(solver.check())
solver.pop()
## unsatisfied
solver.push()
solver.add(language.Between(s,p,t))
print ">> Hence p is between s and t"
print " << z3: " + str(solver.check())
solver.pop()
## unsatisfied
solver.push()
solver.add(M == N)
print ">> Hence p is between s and t"
print " << z3: " + str(solver.check())
solver.pop()
## unsatisfied
solver.push()
solver.add(language.Between(q,s,u))
print ">> Hence s is between q and u"
print " << z3: " + str(solver.check())
solver.pop()
## unsatisfied
solver.push()
solver.add(Not( language.Segment(s,u) < language.Segment(s,t)))
print ">> Hence seg su is less than seg st"
print " << z3: " + str(solver.check())
solver.pop()
## unsatisfied
solver.push()
solver.add(L==K)
print ">> Hence L = K"
print " << z3: " + str(solver.check())
solver.pop()
def check(self, smt2_formula, context=None):
s = Solver()
s.add(parse_smt2_string((context if context else self.context) + smt2_formula))
return str(s.check())
s.add(q[7] ^ q[3] == q[11])
s.add(q[3] ^ q[8] == 0)
s.add(q[15] & q[3] == q[9] & q[8])
s.add(q[15] >> 3 == 6)
s.add(q[15] % 5 == 3)
s.add(q[4] ^ q[7] == q[5] - q[8])
s.add(q[2] & q[8] == 32)
s.add(q[2] << 2 == q[0] + q[6])
s.add(q[14] % 29 == 0)
s.add(q[14] % 9 == 8)
s.add(q[10] == 77) # We know the ratio between q[13] and q[10] is 7/11, and that q[13] is likely 49. Thus, q[10] has to be 77.
flag = ""
if s.check() == sat:
m = s.model()
print m
chars = [m.evaluate(q[i]) for i in range(len(q))]
for char in chars:
try:
flag += chr(int(str(char)))
except:
flag += "."
print flag
else:
print "failed"
# {n0-fAEr-eM_b1t5}
class LanguageE(object):
'''
@author: krojas
==============
The language of E
==============
The language of E is six sorted, with sorts for
(diagrammatic sorts) points, lines, circles,
(metric assertions) segments, angles, and areas.
There are variables ranging over the first three sorts;
a, b, c, ... to range over points
L, M, N, ... to range over lines
alpha, beta, gamma, ... to range over circles.
There is =, <,
--------------
Constants
--------------
right-angle of angle-sort, 0
--------------
Basic relations
--------------
on(a, L): point * line --> bool
REQUIRES:
ENSURES: true iff point a is on line L
same-side(a,b,L): point * point * line --> bool
REQUIRES: true
ENSURES: true iff points a,b are on the same side of line L
between(a,b,c): point * point * point --> bool
REQUIRES: a,b, and c are distinct and collinear
ENSURES: true iff b is between a and c
on(a,alpha): point * circle --> bool
REQUIRES: true
ENSURES: true iff point a is on circle alpha
inside(a, alpha): point * circle --> bool
REQUIRES: true
ENSURES: point a is inside circle alpha
center(a, alpha): point * circle --> bool
REQUIRES: true
ENSURES: true iff point a is in the center of circle alpha
--------------
Additional relations
--------------
intersects(L, M): line * line --> bool
REQUIRES: true
ENSURES: two lines intersect when they have exactly one point in common
intersects(L, alpha): line * circle --> bool
REQUIRES: true
ENSURES: line and a circle intersect when they have exactly two points in common
intersects(alpha, beta): circle * circle --> bool
REQUIRES: true
ENSURES: two circles intersect when they have exactly two points in common.
'''
def __init__(self):
'''
Constructor
'''
print "=== Initializing language EuclidZ3 ==="
# # make sorts
self.PointSort = DeclareSort("Point")
self.LineSort = DeclareSort("Line")
self.CircleSort = DeclareSort("Circle")
# # make basic relations between diagrammatic sorts
self.Between = Function("Between", self.PointSort, self.PointSort, self.PointSort, BoolSort())
self.OnLine = Function("On", self.PointSort, self.LineSort, BoolSort())
self.OnCircle = Function("Onc", self.PointSort, self.CircleSort, BoolSort())
self.Inside = Function ("Inside" , self.PointSort, self.CircleSort, BoolSort())
self.Center = Function ("Center" , self.PointSort, self.CircleSort, BoolSort())
self.SameSide = Function("SameSide", self.PointSort, self.PointSort, self.LineSort, BoolSort())
self.Intersectsll = Function("Intersectsll", self.LineSort, self.LineSort, BoolSort())
self.Intersectslc = Function("Intersectslc", self.LineSort, self.CircleSort, BoolSort())
self.Intersectscc = Function("Intersectscc", self.CircleSort, self.CircleSort, BoolSort())
# # make the magnitude sorts
self.Segment = Function("Segment", self.PointSort, self.PointSort, RealSort())
self.Angle = Function("Angle", self.PointSort, self.PointSort, self.PointSort, RealSort())
self.Area = Function("Area", self.PointSort, self.PointSort, self.PointSort, RealSort())
# # make constants/terms
self.RightAngle = Const("RightAngle", RealSort())
a, b, c, d, e = Consts('a b c d e', self.PointSort)
L, M, N = Consts('L M N' , self.LineSort)
alpha, beta = Consts('alpha beta', self.CircleSort)
# # assert self.axioms for language E
self.axioms = [ ]
"""
---------- DIAGRAMMATIC AXIOMS ----------
"""
"""
Two points determine a line
1. If a != b, a is on L, and b is L, a is on M and b is on M,
then L = M
"""
self.axioms.append(ForAll([a, b, L, M], \
Implies(And(\
Not(a == b), self.OnLine(a, L), self.OnLine(b, L), \
self.OnLine(a, M), self.OnLine(b, M)), \
L == M)))
"""
self.Center of circle is unique
2. if a and b are both centers of alpha then a=b
3. if a is the center of alpha then a is inside alpha
"""
self.axioms.append(ForAll([a, b, alpha], \
Implies((And (self.Center(a, alpha), self.Center(b, alpha))), a == b)))
self.axioms.append(ForAll([a, alpha], \
Implies(self.Center(a, alpha), self.Inside(a, alpha))))
"""
No degenerate circles
4. if a is inside alpha, then a is not on alpha
"""
self.axioms.append(ForAll([a, alpha] , \
Implies(self.Inside(a, alpha), Not(self.OnCircle(a, alpha)))))
"""
Strict betweeness
1. If b is between a and c then b is between c and a,
a != b, a != c, and a is not between b and c
2. If b is between a and c, a is on L, and b is on L, then c is on L.
3. If b is between a and c, a is on L, and c is on L, then b is on L.
4. If b is between a and c and d is between a and b then d is between a and c.
5. If b is between a and c and c is between b and d then b is between a and d.
6. if a, b, and c are distinct points on a line L, then either b is between a and c, or a is
between b and c, or c is between a and b.
7. if b is between a and c and b is between a and d then b is not between c and d.
"""
self.axioms.append(ForAll([a, b, c], \
Implies(self.Between(a, b, c), \
And(self.Between(c, b, a), Not(a == c), Not(a == b), Not(self.Between(b, a, c))))))
self.axioms.append(ForAll([a, b, c], \
Implies(And(\
self.Between(a, b, c), self.OnLine(a, L), self.OnLine(b, L)), \
self.OnLine(c, L))))
self.axioms.append(ForAll([a, b, c], \
Implies(And(\
self.Between(a, b, c), self.OnLine(a, L), self.OnLine(c, L)), \
self.OnLine(b, L))))
self.axioms.append(ForAll([a, b, c], \
Implies(And(\
self.Between(a, b, c), self.Between(a, d, b)), \
self.Between(a, d, c))))
self.axioms.append(ForAll([a, b, c, d], \
Implies(And(\
self.Between(a, b, c), self.Between(b, c, d)), \
self.Between(a, b, d))))
self.axioms.append(ForAll([a, b, c, L], \
Implies(And(\
self.OnLine(a, L), self.OnLine(b, L), self.OnLine(c, L), \
Not(a == b), Not(a == c), Not(b == c)), \
Or(self.Between(a, b, c), self.Between(b, a, c), self.Between(a, c, b)))))
self.axioms.append(ForAll([a, b, c, d], \
Implies(And(\
self.Between(a, b, c), self.Between(a, b, d)), \
Not(self.Between(d, b, c)))))
"""
Same-side self.axioms
1. if a is not on L, then a and a are on the same side of L.
2. if a and b are on the same side of L, then b and a are on teh same side of L.
3. if a and b are on the same side of L, then a is not on L.
4. if a and b are on the same side of L, and a and c are on the same side of L, then b
and c are on the same side L.
5. if a,b, and c are not on L, and a and b are not on the same side of L,
then either a
and c are on the same side of L, or b
"""
self.axioms.append(ForAll([a, L], \
Implies(Not(self.OnLine(a, L)), \
self.SameSide(a, a, L))))
self.axioms.append(ForAll([a, b, L], \
Implies(self.SameSide(a, b, L), \
And(Not(self.OnLine(a, L), self.SameSide(b, a, L))))))
self.axioms.append(ForAll([a, b, c, L], \
Implies(And(\
self.SameSide(a, b, L), self.SameSide(a, c, L)), \
self.SameSide(b, c, L))))
self.axioms.append(ForAll([ a, b, c, L], \
Implies(And(\
Not(self.OnLine(a, L)), Not(self.OnLine(b, L)), Not(self.OnLine(c, L))), \
Or(self.SameSide(a, b, L), self.SameSide(a, c, L), self.SameSide(b, c, L)))))
# # TODO: check this axiom below with avigad,
# # "either a and c are sameside L, or b"
# # is that sameside(a,c,L) or sameside(b,c,L) and NOT both?
self.axioms.append(ForAll([a, b, c, L], \
Implies(And(\
Not(self.OnLine(a, L)), Not(self.OnLine(b, L)), Not(self.OnLine(c, L)), \
Not(self.SameSide(a, b, L))), \
Or(self.SameSide(a, c, L), self.SameSide(b, c, L)))))
"""
Pasch self.axioms
1. if b is between a and c, and a and c are on the same side of L,
then a and b are on teh same side of L.
2. if b is between a and c, and a is on L and b is not on L,
then b and c are on the same side of L.
3. if b is between a and c and b is on L
then a and c are not on the same side of L
4. if b is the intersection of distinct lines L and M, a and c are distinct points on m,
a != b, c !=b, and a and c are not on teh same side of L,
then b is between a and c.
"""
self.axioms.append(ForAll([a, b, c, L], \
Implies(And(\
self.Between(a, b, c), self.SameSide(a, c, L)), \
self.SameSide(a, b, L))))
self.axioms.append(ForAll([a, b, c, L], \
Implies(And(\
self.Between(a, b, c), self.OnLine(a, L), Not(self.OnLine(b, L))), \
self.SameSide(b, c, L))))
self.axioms.append(ForAll([a, b, c, L], \
Implies(And(\
self.Between(a, b, c), self.OnLine(b, L)), \
Not(self.SameSide(a, c, L)))))
self.axioms.append(ForAll([a, b, c, L, M], \
Implies(And(\
Not(a == b), Not(b == c), Not(L == M), \
self.OnLine(a, M), self.OnLine(b, M), self.OnLine(c, M), Not(self.SameSide(a, c, L)), self.OnLine(b, L)), \
self.Between(a, b, c))))
"""
Triple incidence self.axioms
1. if L, M, and N are lines meeting at a point a, and b, c, and d are points on L, M,and N resctively,
and if c and d are on the same side of L, and b and c are on
the same side of N, then b and d are not on the same side of M.
2. if L,M,N are lines meeting at a point a, and b, c, and d are points on L M and N respectively, and
if c and d are on the same side of L, and b and d are not on the same side of M, and d is not on M
and b != a,
then b and c are on the same side of N.
3. If L, M and N are lines meeting at a point a, and b, c, and d are points on L,M,
and N respectively, and if c and d are on the same side of L, and b and c are on the
same side of N, and d and e are on the sameside of M, and c and e are on the same side of N,
then c and e are on the same side of L.
"""
self.axioms.append(ForAll([a, b, c, d, L, M, N], \
Implies(And(\
self.OnLine(a, L), self.OnLine(a, M), self.OnLine(a, N), self.OnLine(b, L), self.OnLine(c, M), self.OnLine(d, N), \
self.SameSide(c, d, L), self.SameSide(b, c, N)), \
Not(self.SameSide(b, d, M)))))
self.axioms.append(ForAll([a, b, c, d], \
Implies(And(\
self.OnLine(a, L), self.OnLine(a, M), self.OnLine(a, N), \
self.OnLine(b, L), self.OnLine(c, M), self.OnLine(d, N), \
self.SameSide(c, d, L), Not(self.SameSide(d, b, M)), Not(self.OnLine(d, M)), Not(b == a)), \
self.SameSide(b, c, N))))
self.axioms.append(ForAll([a, b, c, d, e, L, M, N], \
Implies(And(\
self.OnLine(a, L), self.OnLine(c, M), self.OnLine(a, N), self.OnLine(b, L), \
self.OnLine(c, M), self.OnLine(d, N), self.SameSide(b, c, N), self.SameSide(d, c, L), \
self.SameSide(d, e, M), self.SameSide(c, e, N)), \
self.SameSide(c, e, L))))
"""
Circle self.axioms
"""
self.axioms.append(ForAll([a, b, c, alpha, L], \
Implies(And(\
self.Inside(a, alpha), self.OnCircle(b, alpha), self.OnCircle(c, alpha), \
self.OnLine(a, L), self.OnLine(b, L), self.OnLine(c, L), Not(b == c)), \
self.Between(b, a, c))))
self.axioms.append(ForAll([a, b, c, alpha], \
Implies(And(\
Or(self.Inside(a, alpha), self.OnCircle(a, alpha)), Or(self.Inside(b, alpha), self.OnCircle(b, alpha)), \
self.Between(a, c, b)), \
And(Not(self.Inside(b, alpha)), Not(self.OnCircle(b, alpha))))))
self.axioms.append(ForAll([a, b, c, alpha, L], \
Implies(And(\
Or(self.Inside(a, alpha), self.OnCircle(a, alpha)), Not(self.Inside(c, alpha)), self.Between(a, c, b)), \
And(Not(self.Inside(b, alpha)), Not(self.OnCircle(b, alpha))))))
self.axioms.append(ForAll([a, b, c, d, alpha, beta], \
Implies(And(\
self.OnCircle(c, alpha), self.OnCircle(c, beta), self.OnCircle(d, alpha), self.OnCircle(d, beta), \
Not(alpha == beta), Not(c == d), self.OnLine(a, L), self.OnLine(b, L), \
self.Center(a, alpha), self.Center(a, beta)), \
Not(self.SameSide(c, d, L)))))
"""
Intersection
"""
self.axioms.append(ForAll([L, M, a, b], \
Implies(And(\
self.OnLine(a, M), self.OnLine(b, M), Not(self.SameSide(a, b, L))), \
self.Intersectsll(L, M))))
self.axioms.append(ForAll([alpha, L, a, b], \
Implies(And(\
Or(self.Inside(a, alpha), self.OnCircle(a, alpha)), \
Or(self.Inside(b, alpha), self.OnCircle(b, alpha)), \
Not(self.OnLine(a, L)), Not(self.OnLine(b, L)), Not(self.SameSide(a, b, L))), \
self.Intersectslc(L, alpha))))
self.axioms.append(ForAll([L, alpha, a], \
Implies(And(\
self.Inside(a, alpha), self.OnLine(a, L)), \
self.Intersectslc(L, alpha))))
self.axioms.append(ForAll([alpha, beta, a, b], \
Implies(And(\
self.OnCircle(a, alpha), Or(self.Inside(b, alpha), self.OnCircle(b, alpha)), \
self.Inside(a, beta), Not(self.Inside(b, beta)), Not(self.OnCircle(b, beta))), \
self.Intersectscc(alpha, beta))))
self.axioms.append(ForAll([alpha, beta, a, b], \
Implies(And(\
self.OnCircle(a, alpha), self.Inside(b, beta), self.Inside(a, beta), self.OnCircle(b, beta)), \
self.Intersectscc(alpha, beta))))
"""
---------- METRIC AXIOMS ----------
"""
"""
Segments
"""
self.axioms.append(ForAll([a, b], \
Implies(self.Segment(a, b) == RealVal(0.0), a == b)))
self.axioms.append(ForAll([a], \
self.Segment(a, a) == RealVal(0.0)))
self.axioms.append(ForAll([a, b], \
(self.Segment(a, b) >= RealVal(0.0))))
self.axioms.append(ForAll([a, b], \
self.Segment(a, b) == self.Segment(b, a)))
"""
Angles
"""
self.axioms.append(ForAll ([a, b, c], \
Implies(And(\
Not(a == b), Not(b == c)), \
self.Angle(a, b, c) == self.Angle(c, b, a))))
self.axioms.append(ForAll ([a, b, c], \
Implies(And(\
Not((a == b)), Not((b == c))), \
And(\
self.Angle(a, b, c) >= RealVal(0.0), \
self.Angle(a, b, c) <= (self.RightAngle + self.RightAngle)))))
"""
Areas
"""
self.axioms.append(ForAll([a, b], \
self.Area(a, a, b) == RealVal(0.0)))
self.axioms.append(ForAll([a, b, c], \
self.Area(a, b, c) >= RealVal(0.0)))
self.axioms.append(ForAll([a, b, c], \
And(self.Area(a, b, c) == self.Area(c, a, b), self.Area(a, b, c) == self.Area(b, a, c))))
"""
---------- Transfer AXIOMS ----------
"""
"""
Diagram-segment transfer self.axioms
"""
self.axioms.append(ForAll([a, b, c], \
Implies(self.Between(a, b, c), ((self.Segment(a, b) + self.Segment(b, c)) == self.Segment(a, c)))))
# # center and radius determine circle
self.axioms.append(ForAll([a, b, c, alpha, beta], \
Implies(And(\
self.Center(a, alpha), self.Center(a, beta), self.OnCircle(b, alpha), self.OnCircle(c, beta), \
self.Segment(a, b) == self.Segment(a, c)), \
(alpha == beta))))
self.axioms.append(ForAll([a, b, c, alpha], \
Implies(And(self.Center(a, alpha), self.OnCircle(b, beta), self.Segment(a, c) == self.Segment(a, b)), \
self.OnCircle(c, alpha))))
self.axioms.append(ForAll([a, b, c, alpha], \
Implies(And(\
self.Center(a, alpha), self.OnCircle(b, alpha)), \
((self.Segment(a, c) == self.Segment(a, b)) == self.Inside(c, alpha)))))
"""
Diagram-angle transfer self.axioms
"""
self.axioms.append(ForAll([a, b, c, L], \
Implies(And(\
Not((a == b)), Not((a == c)), self.OnLine(a, L), self.OnLine(b, L)), \
(And(self.OnLine(c, L), Not(self.Between(c, a, b))) == (self.Angle(b, a, c) == RealVal(0.0))))))
# # Possibly this is superfluous
self.axioms.append(ForAll([a, b], \
Implies(Not((a == b)), self.Angle(a, b, a) == RealVal(0.0))))
# # Point inside angle iff angles sum
self.axioms.append(ForAll([a, b, c, d, L, M], \
Implies(And(\
self.OnLine(a, L), self.OnLine(b, L), self.OnLine(a, M), self.OnLine(c, M), \
Not((a == b)), Not((a == c)), Not(self.OnLine(d, L)), Not(self.OnLine(d, M)), \
Not((L == M))), \
((self.Angle(b, a, c) == (self.Angle(b, a, d) + self.Angle(d, a, c))) == \
And(self.SameSide(b, d, M), self.SameSide(d, c, L))))))
# # Define right angle (and all right angles are equal)
self.axioms.append(ForAll([a, b, c, d, L], \
Implies(And(\
self.OnLine(a, L), self.OnLine(b, L), self.Between(a, c, b), Not(self.OnLine(d, L))), \
((self.Angle(a, c, d) == self.Angle(d, c, b)) == (self.Angle(a, c, d) == self.RightAngle)))))
"""
Diagram-area transfer self.axioms
"""
self.axioms.append(ForAll([a, b, c, L], \
Implies(And(\
self.OnLine(a, L), self.OnLine(b, L), Not((a == b))), \
((self.Area(a, b, c) == RealVal(0.0)) == self.OnLine(c, L)))))
# # Sum of Areas
self.axioms.append(ForAll([a, b, c, d, L], \
Implies(And(\
self.OnLine(a, L), self.OnLine(b, L), self.OnLine(c, L), Not(self.OnLine(d, L)), \
Not((a == b)), Not((c == a)), Not((c == b))), \
(self.Between(a, c, b) == ((self.Area(a, c, d) + self.Area(d, c, b)) == self.Area(a, d, b))))))
self.solver = Solver()
self.solver.add(self.axioms)
print "Axiom set : " + str(self.solver.check())
from z3 import Real, Solver
s = Solver()
x = Real('x')
s.add((x * 3.0 + 18.0) == (x * 12.56637061435917))
print s.check()
print s.model()
print s.model()[x].as_decimal(10)