def over_post_bdd(self, src_states_bdd, sys_strat=None):
""" Over-approximated version of concrete post which can be done even
without the transition relation """
strat = BDD.true()
if sys_strat is not None:
strat &= sys_strat
# to do this, we use an over-simplified transition relation, EXu,Xc
b = BDD.true()
for x in self.iterate_latches():
temp = BDD.make_eq(BDD(self.get_primed_var(x.lit)),
self.lit2bdd(x.next))
b &= temp.and_abstract(
strat,
BDD.make_cube(
imap(funcomp(BDD, symbol_lit),
self.iterate_controllable_inputs())))
b = b.restrict(src_states_bdd)
b &= src_states_bdd
b = b.exist_abstract(
BDD.make_cube(
imap(
funcomp(BDD, symbol_lit),
chain(self.iterate_latches(),
self.iterate_uncontrollable_inputs()))))
return self.unprime_latches_in_bdd(b)
def upost(self, q):
assert isinstance(q, BDD)
if q in self.succ_cache:
return iter(self.succ_cache[q])
A = BDD.true()
M = set()
while A != BDD.false():
a = A.get_one_minterm(self.uinputs)
trans = BDD.make_cube(
imap(lambda x: BDD.make_eq(BDD(self.aig.get_primed_var(x.lit)),
self.aig.lit2bdd(x.next)
.and_abstract(q, self.latch_cube)),
self.aig.iterate_latches()))
lhs = trans & a
rhs = self.aig.prime_all_inputs_in_bdd(trans)
simd = BDD.make_impl(lhs, rhs).univ_abstract(self.platch_cube)\
.exist_abstract(self.pcinputs_cube)\
.univ_abstract(self.cinputs_cube)
simd = self.aig.unprime_all_inputs_in_bdd(simd)
A &= ~simd
Mp = set()
for m in M:
if not (BDD.make_impl(m, simd) == BDD.true()):
Mp.add(m)
M = Mp
M.add(a)
log.DBG_MSG("Upost |M| = " + str(len(M)))
self.succ_cache[q] = map(lambda x: (q, x), M)
return iter(self.succ_cache[q])
def upost(self, q):
assert isinstance(q, BDD)
if q in self.succ_cache:
return iter(self.succ_cache[q])
A = BDD.true()
M = set()
while A != BDD.false():
a = A.get_one_minterm(self.uinputs)
trans = BDD.make_cube(
imap(
lambda x: BDD.make_eq(
BDD(self.aig.get_primed_var(x.lit)),
self.aig.lit2bdd(x.next).and_abstract(
q, self.latch_cube)), self.aig.iterate_latches()))
lhs = trans & a
rhs = self.aig.prime_all_inputs_in_bdd(trans)
simd = BDD.make_impl(lhs, rhs).univ_abstract(self.platch_cube)\
.exist_abstract(self.pcinputs_cube)\
.univ_abstract(self.cinputs_cube)
simd = self.aig.unprime_all_inputs_in_bdd(simd)
A &= ~simd
Mp = set()
for m in M:
if not (BDD.make_impl(m, simd) == BDD.true()):
Mp.add(m)
M = Mp
M.add(a)
log.DBG_MSG("Upost |M| = " + str(len(M)))
self.succ_cache[q] = map(lambda x: (q, x), M)
return iter(self.succ_cache[q])
def strat_is_inductive(self, strat, use_trans=False):
strat_dom = strat.exist_abstract(
BDD.make_cube(imap(funcomp(BDD, symbol_lit),
chain(self.iterate_controllable_inputs(),
self.iterate_uncontrollable_inputs()))))
p_bdd = self.substitute_latches_next(strat_dom, use_trans=use_trans)
return BDD.make_impl(strat, p_bdd) == BDD.true()
def extract_output_funs(self, strategy, care_set=None):
"""
Calculate BDDs for output functions given non-deterministic winning
strategy.
"""
if care_set is None:
care_set = BDD.true()
output_models = dict()
all_outputs = [BDD(x.lit) for x in self.iterate_controllable_inputs()]
for c_symb in self.iterate_controllable_inputs():
c = BDD(c_symb.lit)
others = set(set(all_outputs) - set([c]))
if others:
others_cube = BDD.make_cube(others)
c_arena = strategy.exist_abstract(others_cube)
else:
c_arena = strategy
# pairs (x,u) in which c can be true
can_be_true = c_arena.cofactor(c)
# pairs (x,u) in which c can be false
can_be_false = c_arena.cofactor(~c)
must_be_true = (~can_be_false) & can_be_true
must_be_false = (~can_be_true) & can_be_false
local_care_set = care_set & (must_be_true | must_be_false)
# Restrict operation:
# on care_set: must_be_true.restrict(care_set) <-> must_be_true
c_model = min([must_be_true.safe_restrict(local_care_set),
(~must_be_false).safe_restrict(local_care_set)],
key=lambda x: x.dag_size())
output_models[c_symb.lit] = c_model
log.DBG_MSG("Size of function for " + str(c.get_index()) + " = " +
str(c_model.dag_size()))
strategy &= BDD.make_eq(c, c_model)
return output_models
def strat_is_inductive(self, strat, use_trans=False):
strat_dom = strat.exist_abstract(
BDD.make_cube(
imap(
funcomp(BDD, symbol_lit),
chain(self.iterate_controllable_inputs(),
self.iterate_uncontrollable_inputs()))))
p_bdd = self.substitute_latches_next(strat_dom, use_trans=use_trans)
return BDD.make_impl(strat, p_bdd) == BDD.true()
def trans_rel_bdd(self):
# check cache
if self._cached_transition is not None:
return self._cached_transition
b = BDD.true()
for x in self.iterate_latches():
b &= BDD.make_eq(BDD(self.get_primed_var(x.lit)),
self.lit2bdd(x.next))
self._cached_transition = b
log.BDD_DMP(b, "Composed and cached the concrete transition relation.")
return b
def trans_rel_bdd(self):
# check cache
if self._cached_transition is not None:
return self._cached_transition
b = BDD.true()
for x in self.iterate_latches():
b &= BDD.make_eq(BDD(self.get_primed_var(x.lit)),
self.lit2bdd(x.next))
self._cached_transition = b
log.BDD_DMP(b, "Composed and cached the concrete transition relation.")
return b
def comp_synth(games):
s = BDD.true()
cum_w = BDD.true()
cnt = 0
for game in games:
assert isinstance(game, BackwardGame)
w = backward_safety_synth(game)
cnt += 1
# short-circuit a negative response
if w is None:
log.DBG_MSG("Short-circuit exit after sub-game #" + str(cnt))
return (None, None)
if s is None:
s = game.cpre(w, get_strat=True)
cum_w = w
else:
s &= game.cpre(w, get_strat=True)
cum_w &= w
# sanity check before moving forward
if (not s or not game.init() & s):
return (None, None)
# we must aggregate everything now
log.DBG_MSG("Solved " + str(cnt) + " sub games.")
return (cum_w, s)
def over_post_bdd(self, src_states_bdd, sys_strat=None):
""" Over-approximated version of concrete post which can be done even
without the transition relation """
strat = BDD.true()
if sys_strat is not None:
strat &= sys_strat
# to do this, we use an over-simplified transition relation, EXu,Xc
b = BDD.true()
for x in self.iterate_latches():
temp = BDD.make_eq(BDD(self.get_primed_var(x.lit)),
self.lit2bdd(x.next))
b &= temp.and_abstract(
strat,
BDD.make_cube(imap(
funcomp(BDD, symbol_lit),
self.iterate_controllable_inputs()
)))
b = b.restrict(src_states_bdd)
b &= src_states_bdd
b = b.exist_abstract(
BDD.make_cube(imap(funcomp(BDD, symbol_lit),
chain(self.iterate_latches(),
self.iterate_uncontrollable_inputs()))))
return self.unprime_latches_in_bdd(b)
def comp_synth(games):
s = BDD.true()
cum_w = BDD.true()
cnt = 0
for game in games:
assert isinstance(game, BackwardGame)
w = backward_safety_synth(game)
cnt += 1
# short-circuit a negative response
if w is None:
log.DBG_MSG("Short-circuit exit after sub-game #" + str(cnt))
return (None, None)
if s is None:
s = game.cpre(w, get_strat=True)
cum_w = w
else:
s &= game.cpre(w, get_strat=True)
cum_w &= w
# sanity check before moving forward
if (not s or not game.init() & s):
return (None, None)
# we must aggregate everything now
log.DBG_MSG("Solved " + str(cnt) + " sub games.")
return (cum_w, s)
def bdd2aig(self, a_bdd):
"""
Walk given BDD node (recursively). If given input BDD requires
intermediate AND gates for its representation, the function adds them.
Literal representing given input BDD is `not` added to the spec.
"""
if a_bdd in self.bdd_gate_cache:
return self.bdd_gate_cache[a_bdd]
if a_bdd.is_constant():
res = int(a_bdd == BDD.true()) # in aiger 0/1 = False/True
return res
# get an index of variable,
# all variables used in bdds also introduced in aiger,
# except fake error latch literal,
# but fake error latch will not be used in output functions (at least
# we don't need this..)
a_lit = a_bdd.get_index()
assert (a_lit != self.error_fake_latch.lit),\
("using error latch in the " +
"definition of output " +
"function is not allowed")
t_bdd = a_bdd.then_child()
e_bdd = a_bdd.else_child()
t_lit = self.bdd2aig(t_bdd)
e_lit = self.bdd2aig(e_bdd)
# ite(a_bdd, then_bdd, else_bdd)
# = a*then + !a*else
# = !(!(a*then) * !(!a*else))
# -> in general case we need 3 more ANDs
a_t_lit = self.get_optimized_and_lit(a_lit, t_lit)
na_e_lit = self.get_optimized_and_lit(negate_lit(a_lit), e_lit)
n_a_t_lit = negate_lit(a_t_lit)
n_na_e_lit = negate_lit(na_e_lit)
ite_lit = self.get_optimized_and_lit(n_a_t_lit, n_na_e_lit)
res = negate_lit(ite_lit)
if a_bdd.is_complement():
res = negate_lit(res)
# cache result
self.bdd_gate_cache[a_bdd] = res
return res
def decompose(aig, argv):
if argv.decomp == 1:
if lit_is_negated(aig.error_fake_latch.next):
log.DBG_MSG("Decomposition opt possible (BIG OR case)")
(A, B) = aig.get_1l_land(strip_lit(aig.error_fake_latch.next))
return imap(
lambda a: ConcGame(BDDAIG(aig).short_error(a),
use_trans=argv.use_trans),
merge_some_signals(BDD.true(), A, aig, argv))
else:
(A, B) = aig.get_1l_land(aig.error_fake_latch.next)
if not B:
log.DBG_MSG("No decomposition opt possible")
return None
else:
log.DBG_MSG("Decomposition opt possible (A ^ [C v D] case)")
log.DBG_MSG(str(len(A)) + " AND leaves: " + str(A))
# critical heuristic: which OR leaf do we distribute?
# here I propose to choose the one with the most children
b = B.pop()
(C, D) = aig.get_1l_land(b)
for bp in B:
(Cp, Dp) = aig.get_1l_land(bp)
if len(Cp) > len(C):
b = bp
C = Cp
log.DBG_MSG("Chosen OR: " + str(b))
rem_AND_leaves = filter(lambda x: strip_lit(x) != b, A)
rdeps = set()
for r in rem_AND_leaves:
rdeps |= aig.get_lit_deps(strip_lit(r))
log.DBG_MSG("Rem. AND leaves' deps: " + str(rdeps))
cube = BDD.make_cube(map(aig.lit2bdd, rem_AND_leaves))
log.DBG_MSG(
str(len(C)) + " OR leaves: " + str(map(aig.get_lit_name, C)))
return imap(
lambda a: ConcGame(BDDAIG(aig).short_error(a),
use_trans=argv.use_trans),
merge_some_signals(cube, C, aig, argv))
elif argv.decomp == 2:
raise NotImplementedError
def bdd2aig(self, a_bdd):
"""
Walk given BDD node (recursively). If given input BDD requires
intermediate AND gates for its representation, the function adds them.
Literal representing given input BDD is `not` added to the spec.
"""
if a_bdd in self.bdd_gate_cache:
return self.bdd_gate_cache[a_bdd]
if a_bdd.is_constant():
res = int(a_bdd == BDD.true()) # in aiger 0/1 = False/True
return res
# get an index of variable,
# all variables used in bdds also introduced in aiger,
# except fake error latch literal,
# but fake error latch will not be used in output functions (at least
# we don't need this..)
a_lit = a_bdd.get_index()
assert (a_lit != self.error_fake_latch.lit),\
("using error latch in the " +
"definition of output " +
"function is not allowed")
t_bdd = a_bdd.then_child()
e_bdd = a_bdd.else_child()
t_lit = self.bdd2aig(t_bdd)
e_lit = self.bdd2aig(e_bdd)
# ite(a_bdd, then_bdd, else_bdd)
# = a*then + !a*else
# = !(!(a*then) * !(!a*else))
# -> in general case we need 3 more ANDs
a_t_lit = self.get_optimized_and_lit(a_lit, t_lit)
na_e_lit = self.get_optimized_and_lit(negate_lit(a_lit), e_lit)
n_a_t_lit = negate_lit(a_t_lit)
n_na_e_lit = negate_lit(na_e_lit)
ite_lit = self.get_optimized_and_lit(n_a_t_lit, n_na_e_lit)
res = negate_lit(ite_lit)
if a_bdd.is_complement():
res = negate_lit(res)
# cache result
self.bdd_gate_cache[a_bdd] = res
return res
def decompose(aig, argv):
if argv.decomp == 1:
if lit_is_negated(aig.error_fake_latch.next):
log.DBG_MSG("Decomposition opt possible (BIG OR case)")
(A, B) = aig.get_1l_land(strip_lit(aig.error_fake_latch.next))
return imap(lambda a: ConcGame(
BDDAIG(aig).short_error(a),
use_trans=argv.use_trans),
merge_some_signals(BDD.true(), A, aig, argv))
else:
(A, B) = aig.get_1l_land(aig.error_fake_latch.next)
if not B:
log.DBG_MSG("No decomposition opt possible")
return None
else:
log.DBG_MSG("Decomposition opt possible (A ^ [C v D] case)")
log.DBG_MSG(str(len(A)) + " AND leaves: " + str(A))
# critical heuristic: which OR leaf do we distribute?
# here I propose to choose the one with the most children
b = B.pop()
(C, D) = aig.get_1l_land(b)
for bp in B:
(Cp, Dp) = aig.get_1l_land(bp)
if len(Cp) > len(C):
b = bp
C = Cp
log.DBG_MSG("Chosen OR: " + str(b))
rem_AND_leaves = filter(lambda x: strip_lit(x) != b, A)
rdeps = set()
for r in rem_AND_leaves:
rdeps |= aig.get_lit_deps(strip_lit(r))
log.DBG_MSG("Rem. AND leaves' deps: " + str(rdeps))
cube = BDD.make_cube(map(aig.lit2bdd, rem_AND_leaves))
log.DBG_MSG(str(len(C)) + " OR leaves: " +
str(map(aig.get_lit_name, C)))
return imap(lambda a: ConcGame(
BDDAIG(aig).short_error(a),
use_trans=argv.use_trans), merge_some_signals(cube, C, aig,
argv))
elif argv.decomp == 2:
raise NotImplementedError
def extract_output_funs(self, strategy, care_set=None):
"""
Calculate BDDs for output functions given non-deterministic winning
strategy.
"""
if care_set is None:
care_set = BDD.true()
output_models = dict()
all_outputs = [BDD(x.lit) for x in self.iterate_controllable_inputs()]
for c_symb in self.iterate_controllable_inputs():
c = BDD(c_symb.lit)
others = set(set(all_outputs) - set([c]))
if others:
others_cube = BDD.make_cube(others)
c_arena = strategy.exist_abstract(others_cube)
else:
c_arena = strategy
# pairs (x,u) in which c can be true
can_be_true = c_arena.cofactor(c)
# pairs (x,u) in which c can be false
can_be_false = c_arena.cofactor(~c)
must_be_true = (~can_be_false) & can_be_true
must_be_false = (~can_be_true) & can_be_false
local_care_set = care_set & (must_be_true | must_be_false)
# Restrict operation:
# on care_set: must_be_true.restrict(care_set) <-> must_be_true
c_model = min([
must_be_true.safe_restrict(local_care_set),
(~must_be_false).safe_restrict(local_care_set)
],
key=lambda x: x.dag_size())
output_models[c_symb.lit] = c_model
log.DBG_MSG("Size of function for " + str(c.get_index()) + " = " +
str(c_model.dag_size()))
strategy &= BDD.make_eq(c, c_model)
return output_models
def init_state_bdd(self):
b = BDD.true()
for x in self.iterate_latches():
b &= ~BDD(x.lit)
return b
def comp_synth4(games, gen_game):
s = None
cum_s = None
cum_w = None
cnt = 0
triple_list = []
for game in games:
assert isinstance(game, BackwardGame)
w = backward_safety_synth(game)
cnt += 1
# short-circuit a negative response
if w is None:
log.DBG_MSG("Short-circuit exit 1 after sub-game #" + str(cnt))
return None
s = game.cpre(w, get_strat=True)
if cum_s is None:
cum_s = s
cum_w = w
else:
cum_s &= s
cum_w &= w
# another short-circuit exit
if (not cum_s or not game.init() & cum_s):
log.DBG_MSG("Short-circuit exit 2 after sub-game #" + str(cnt))
return None
triple_list.append((game, s, w))
log.DBG_MSG("Solved " + str(cnt) + " sub games.")
# lets simplify transition functions
gen_game.aig.restrict_latch_next_funs(cum_s)
# what comes next is a fixpoint computation using a UPRE
# step at a time in the global game and using it to get more
# information from the local sub-games
lose = BDD.true()
lose_next = ~cum_w | gen_game.error()
while lose_next != lose:
lose = lose_next
log.DBG_MSG("Doing global UPRE")
lose_next = lose | gen_game.upre(lose)
for i in range(len(triple_list)):
wt = triple_list[i][2]
gamet = triple_list[i][0]
local_deps = set([x.lit for x in gamet.aig.iterate_latches()])
rem_lats = gen_game.aig.get_bdd_latch_deps(lose_next) - local_deps
pt = lose_next
if rem_lats:
pt = lose_next.univ_abstract(BDD.make_cube(map(BDD, rem_lats)))
# log.BDD_DMP(lose_next, "global losing area iterate")
# log.BDD_DMP(pt, "new losing area")
assert BDD.make_impl(~wt, pt) == BDD.true()
if BDD.make_impl(pt, ~wt) != BDD.true():
gamet.short_error = pt
wt = backward_safety_synth(gamet)
if (wt is None or not gamet.init() & wt):
log.DBG_MSG("Short-circuit exit 3")
return None
st = gamet.cpre(wt, get_strat=True)
gen_game.aig.restrict_latch_next_funs(wt)
triple_list[i] = (gamet, st, wt)
for t in triple_list:
lose_next |= ~t[2]
# after the fixpoint has been reached we can compute the error
win = ~lose
if (not win or not gen_game.init() & win):
return None
else:
return win
def init_state_bdd(self):
b = BDD.true()
for x in self.iterate_latches():
b &= ~BDD(x.lit)
return b
def comp_synth4(games, gen_game):
s = None
cum_s = None
cum_w = None
cnt = 0
triple_list = []
for game in games:
assert isinstance(game, BackwardGame)
w = backward_safety_synth(game)
cnt += 1
# short-circuit a negative response
if w is None:
log.DBG_MSG("Short-circuit exit 1 after sub-game #" + str(cnt))
return None
s = game.cpre(w, get_strat=True)
if cum_s is None:
cum_s = s
cum_w = w
else:
cum_s &= s
cum_w &= w
# another short-circuit exit
if (not cum_s or not game.init() & cum_s):
log.DBG_MSG("Short-circuit exit 2 after sub-game #" + str(cnt))
return None
triple_list.append((game, s, w))
log.DBG_MSG("Solved " + str(cnt) + " sub games.")
# lets simplify transition functions
gen_game.aig.restrict_latch_next_funs(cum_s)
# what comes next is a fixpoint computation using a UPRE
# step at a time in the global game and using it to get more
# information from the local sub-games
lose = BDD.true()
lose_next = ~cum_w | gen_game.error()
while lose_next != lose:
lose = lose_next
log.DBG_MSG("Doing global UPRE")
lose_next = lose | gen_game.upre(lose)
for i in range(len(triple_list)):
wt = triple_list[i][2]
gamet = triple_list[i][0]
local_deps = set([x.lit for x in gamet.aig.iterate_latches()])
rem_lats = gen_game.aig.get_bdd_latch_deps(lose_next) - local_deps
pt = lose_next
if rem_lats:
pt = lose_next.univ_abstract(
BDD.make_cube(map(BDD, rem_lats)))
# log.BDD_DMP(lose_next, "global losing area iterate")
# log.BDD_DMP(pt, "new losing area")
assert BDD.make_impl(~wt, pt) == BDD.true()
if BDD.make_impl(pt, ~wt) != BDD.true():
gamet.short_error = pt
wt = backward_safety_synth(gamet)
if (wt is None or not gamet.init() & wt):
log.DBG_MSG("Short-circuit exit 3")
return None
st = gamet.cpre(wt, get_strat=True)
gen_game.aig.restrict_latch_next_funs(wt)
triple_list[i] = (gamet, st, wt)
for t in triple_list:
lose_next |= ~t[2]
# after the fixpoint has been reached we can compute the error
win = ~lose
if (not win or not gen_game.init() & win):
return None
else:
return win
