def l1_1(dfa):
Z, witness_z = dfa.inEquality()
ZK = tarjan2dict(dfa.tarjan())
R, witness_wv = _calcW(dfa)
for (p, q) in Z:
if q in ZK[p]:
w = witness_wv[p][q]
v = witness_wv[q][p]
z = witness_z[(p, q)]
return {'w': w, 'z': z, 'v': v, 'p': p, 'q': q}
return {}
def b1(dfa):
"""
:param dfa:
:return:
"""
#A2 = dfa ** 2
A3 = dfa ** 3
ZK = tarjan2dict(dfa.tarjan())
Z, witness_z = dfa.inEquality()
T = A3.trivialComponents()
for q1, q5, q4 in A3.Q:
for (q1_, q5_, q4_), u in A3.search((q1, q5, q4), True):
if q1_ == q5_:
for q2 in ZK[q1]:
for (q2_, q1__, q4__), v in A3.search((q2, q1_, q4_), True):
if q2_ == q4 and q5 == q1__ and q4 == q4__:
if (q1, q5, q4) not in T\
and (q2, q5_, q4_) not in T\
and ((q5_, q4_) in Z or (q4_, q5_) in Z):
witness = A3.reachable((q1, q5, q4), (q1, q5, q4))
w_ = A3.reachable((q2, q5_, q4_), (q2, q5_, q4_))
z = witness_z[(q5_, q4_)] if ((q5_, q4_) in Z)\
else witness_z[(q4_, q5_)]
return {
"w": witness, "z": z, "v": v, "u": u,
"w'": w_,
"y": dfa.reachable(q1, q2),
"y'": dfa.reachable(q2, q1),
"q1": q1,
"q2": q2,
"q3": q5_,
"q4": q4,
"q5": q5,
"q6": q4_,
"d": dfa(q5_, z),
"f": dfa(q4_, z)
}
return {}
def l32(dfa):
"""
"""
def _calcAlphaComp(A, K):
alpha = set()
for (s, a), e in A.d.items():
if s in K and e in K:
alpha.add(a)
return alpha
def _witness_v(K, q):
def _removeEdge(state, word):
prev = state
nxt = state
for symbol in word:
nxt = dfa(prev, symbol)
visited.add(( prev, nxt, symbol ))
prev = nxt
E = set()
for (s, a), e in A2.d.items():
if s in K and e in K:
E.add((s, e, a))
R, paths = _calcW(A2, K)
witness = ""
visited = set()
for (start, end, a) in E:
if (start, end, a) not in visited:
w = paths[q][start]
v = paths[end][q]
visited.add((start, end, a))
_removeEdge(start, witness)
_removeEdge(end, v)
witness += w + a + v
#loop invariants
#assert visited <= E
assert A2(q, witness) == q
#our witness is satisfied with all symbols
if alpha_v[q] == set(witness): break
return witness
A2 = dfa ** 2
ZK = A2.tarjan()
Z, witness_z = dfa.inEquality()
alpha_v = {}
for K in ZK:
alpha = _calcAlphaComp(A2, K)
for q in K:
alpha_v[q] = alpha
_witness_z = lambda a, b: witness_z[a, b]\
if (a, b) in witness_z\
else witness_z[b, a]
ZK = tarjan2dict(ZK)
for p, q in A2.Q:
if (p, q) in Z or (q, p) in Z:
witness_w = dfa.reachable(p, q, False, alpha_v[(p, q)])
if witness_w:
z = _witness_z(p, q)
v = _witness_v(ZK[(p, q)], (p, q))
return {"z": z, "w": witness_w, "v": v, "p": p, "q": q}
return {}
