def trap_predicate(self, number: int) -> mona.Formula:
guard = self.guard_as_mona()
variables = sorted([cast(mona.Variable, v.as_mona())
for v in self.free_variables], key=str)
ports = self.ports.predicates
free_pre = mona.Disjunction([p.hit_pre() for p in ports])
free_post = mona.Disjunction([p.hit_post() for p in ports])
safe_post: List[mona.Formula] = [free_post]
broadcast_local: List[mona.Formula] = []
if self.broadcasts:
broadcasts_one_post = mona.Disjunction([b.one_post()
for b in self.broadcasts])
safe_post.append(broadcasts_one_post)
broadcasts_vertical = mona.Conjunction(
[cast(mona.Formula, mona.Negation(free_pre))] +
[b.vertical_hit() for b in self.broadcasts])
broadcast_local.append(broadcasts_vertical)
inner = mona.Disjunction([
mona.Disjunction(safe_post),
mona.Conjunction([mona.Negation(free_pre),
mona.Conjunction(broadcast_local)])
])
formula = mona.UniversalFirstOrder(variables,
mona.Implication(guard, inner))
return mona.PredicateDefinition(f"trap_transition_{number}",
self.system.state_variables,
[], formula).simplify()
def unique_intersection_predicate(self) -> mona.Formula:
x = mona.Variable("x")
y = mona.Variable("y")
one_states = [mona.Variable(f"one{s}") for s in self.system.states]
two_states = [mona.Variable(f"two{s}") for s in self.system.states]
pairs = list(zip(one_states, two_states))
statements: List[mona.Formula] = []
# fix x in intersection of one state:
for pair in pairs:
pos = [cast(mona.Formula, mona.ElementIn(x, state))
for state in pair]
different_pairs = [p for p in pairs if p != pair]
neg = [ # negated statement from above
cast(mona.Formula, mona.Negation(
mona.Conjunction(
[mona.ElementIn(x, state) for state in pair])))
# for all other pairs
for pair in different_pairs]
statements.append(mona.Conjunction(pos + neg))
fix_x = mona.Disjunction(statements)
# make sure x is unique:
# y is in intersection
y_in_intersection = mona.Disjunction([
mona.Conjunction([mona.ElementIn(y, state) for state in pair])
for pair in pairs])
# y is x if y is in intersection
unique_x = mona.UniversalFirstOrder(
[y],
mona.Implication(y_in_intersection, mona.Equal(x, y)))
formula = mona.ExistentialFirstOrder([x], mona.Conjunction([fix_x,
unique_x]))
return mona.PredicateDefinition(
"unique_intersection",
one_states + two_states, [], formula).simplify()
def marking_predicate(self) -> mona.Formula:
m = mona.Variable("m")
unique_in_component = mona.UniversalFirstOrder(
[m],
mona.Conjunction([
mona.Disjunction([
mona.Conjunction(
[cast(mona.Formula, mona.ElementIn(m, pos))]
+ [cast(mona.Formula, mona.ElementNotIn(m, neg))
for neg in c.state_variables
if neg != pos])
for pos in c.state_variables])
for c in self.system.components]))
flow_invariant = mona.PredicateCall("flow_invariant",
self.system.state_variables)
trap_invariant = mona.PredicateCall("trap_invariant",
self.system.state_variables)
return mona.PredicateDefinition(
"marking",
self.system.state_variables,
[],
mona.Conjunction([
unique_in_component,
flow_invariant,
trap_invariant,
])).simplify()
def intersects_initial_predicate(self) -> mona.Formula:
x = mona.Variable("x")
initial_states = [mona.Variable(c.initial_state)
for c in self.system.components]
x_initial = mona.Disjunction(
[mona.ElementIn(x, init) for init in initial_states])
formula = mona.ExistentialFirstOrder([x], x_initial)
return mona.PredicateDefinition("intersects_initial",
self.system.state_variables,
[], formula).simplify()
def deadlock_predicate(self) -> mona.Formula:
inner = mona.Conjunction(
[mona.PredicateCall(f"dead_transition_{number}",
self.system.state_variables)
for number in range(1, len(self.clauses) + 1)])
return mona.PredicateDefinition(
"deadlock",
self.system.state_variables,
[],
inner).simplify()
def initially_marked_trap_predicate(self) -> mona.Formula:
inner = mona.Conjunction([
mona.PredicateCall("trap",
self.system.state_variables),
mona.PredicateCall("intersects_initial",
self.system.state_variables)])
return mona.PredicateDefinition(
"initially_marked_trap",
self.system.state_variables,
[],
inner).simplify()
def intersection_predicate(self) -> mona.Formula:
x = mona.Variable("x")
one_states = [mona.Variable(f"one{s}") for s in self.system.states]
two_states = [mona.Variable(f"two{s}") for s in self.system.states]
in_both_states = cast(List[mona.Formula],
[mona.Conjunction([mona.ElementIn(x, o),
mona.ElementIn(x, t)])
for o, t in zip(one_states, two_states)])
quantified_formula = mona.ExistentialFirstOrder(
[x], mona.Disjunction(in_both_states))
return mona.PredicateDefinition("intersection",
one_states + two_states,
[], quantified_formula
).simplify()
def trap_invariant_predicate(self) -> mona.Formula:
trap_states = [mona.Variable(f"T{s}") for s in self.system.states]
precondition = mona.PredicateCall("initially_marked_trap", trap_states)
postcondition = mona.PredicateCall(
"intersection",
trap_states + self.system.state_variables)
return mona.PredicateDefinition(
"trap_invariant",
self.system.state_variables,
[],
mona.UniversalSecondOrder(trap_states,
mona.Implication(precondition,
postcondition))
).simplify()
def flow_invariant_predicate(self) -> mona.Formula:
flow_states = [mona.Variable(f"F{s}") for s in self.system.states]
precondition = mona.PredicateCall(
"initially_uniquely_marked_flow",
flow_states)
postcondition = mona.PredicateCall(
"unique_intersection",
flow_states + self.system.state_variables)
return mona.PredicateDefinition(
"flow_invariant",
self.system.state_variables,
[], mona.UniversalSecondOrder(
flow_states, mona.Implication(precondition, postcondition))
).simplify()
def is_dead_predicate(self, number: int) -> mona.Formula:
dead_free = mona.Disjunction(
[p.miss_pre() for p in self.ports.predicates])
dead_broadcasts = mona.Disjunction(
[b.is_dead() for b in self.broadcasts])
guard = self.guard_as_mona()
inner = mona.Implication(
guard,
mona.Disjunction([dead_free, dead_broadcasts]))
formula = mona.UniversalFirstOrder([cast(mona.Variable, v.as_mona())
for v in self.free_variables],
inner)
return mona.PredicateDefinition(f"dead_transition_{number}",
self.system.state_variables,
[], formula).simplify()
def invariant_predicate(self, number: int) -> mona.Formula:
inner = mona.Disjunction([
# disjoint pre and post
mona.Conjunction([self.disjoint_all_pre(),
self.disjoint_all_post()]),
# unique pre and post
mona.Conjunction([self.one_in_pre(), self.one_in_post()]),
# more than one in pre
mona.Conjunction([mona.Negation(self.disjoint_all_pre()),
mona.Negation(self.one_in_pre())]),
])
variables = sorted([cast(mona.Variable, v.as_mona())
for v in self.free_variables], key=str)
guard = self.guard_as_mona()
quantification = mona.UniversalFirstOrder(
variables,
mona.Implication(guard, inner))
return mona.PredicateDefinition(
f"invariant_transition_{number}",
self.system.state_variables,
[],
quantification).simplify()
def uniquely_intersects_initial_predicate(self) -> mona.Formula:
x = mona.Variable("x")
y = mona.Variable("y")
initial_states = [mona.Variable(c.initial_state)
for c in self.system.components]
statements: List[mona.Formula] = []
for init in initial_states:
other_initial_states = [i for i in initial_states if i != init]
x_only_in_init = mona.Conjunction(
[cast(mona.Formula, mona.ElementIn(x, init))]
+ [cast(mona.Formula, mona.ElementNotIn(x, i))
for i in other_initial_states])
statements.append(x_only_in_init)
x_in_only_one_initial = mona.Disjunction(statements)
y_in_initial = mona.Disjunction(
[mona.ElementIn(y, i) for i in initial_states])
x_unique = mona.UniversalFirstOrder([y], mona.Implication(
y_in_initial, mona.Equal(x, y)))
formula = mona.ExistentialFirstOrder(
[x],
mona.Conjunction([x_in_only_one_initial, x_unique]))
return mona.PredicateDefinition(
"uniquely_intersects_initial",
self.system.state_variables, [], formula).simplify()
def custom_property(self, name: str, formula: str) -> mona.Formula:
return mona.PredicateDefinition(
name,
self.system.state_variables,
[],
mona.RawFormula(formula)).simplify()
