def hold(props, prop, duration, negation=False):
'''Creates a DFA which accepts a sequence of symbols all containing
proposition prop. The length of the sequence is duration+1 corresponding to
duration time intervals. If negation is True, then the symbols must not
contain prop instead.
'''
assert prop in props
guard = prop if not negation else '!' + prop
dfa = Fsa(props, directed=True, multi=False)
dfa.name = '(Hold {} {}{} )'.format(duration, 'not ' if negation else '', prop)
bitmaps = dfa.get_guard_bitmap(guard)
ngen = it.count()
nodes = [ngen.next() for _ in range(duration+2)]
attr_dict={'weight': 0, 'input': bitmaps, 'guard' : guard, 'label': guard}
dfa.g.add_path(nodes, **attr_dict)
u, v = nodes[0], nodes[-1]
dfa.init[u] = 1
dfa.final.add(v)
init_tree(dfa, operation=Op.hold)
logger.debug('[hold] Prop: {} Duration: {} Negation: {} Props: {}'.format(prop, duration, negation, props))
return dfa
def accept_prop(props, prop=None, boolean=None):
'''Creates a DFA which accepts:
1) all symbols which contain proposition prop, if prop is not None;
2) all symbols, if boolean is True;
3) no symbol, if boolean is False.
'''
if prop is not None:
assert prop in props
guard = prop
name = '(Prop ' + str(prop) + ')'
logger.debug('[accent_prop] Prop: {} Props: {}'.format(prop, props))
elif boolean is not None:
assert type(boolean) == bool
guard = '(1)' if boolean else '(0)'
name = '(Bool ' + str(boolean) + ')'
logger.debug('[accent_prop] Boolean: {} Props: {}'.format(boolean, props))
else:
raise AssertionError('Either prop or boolean must be given!')
dfa = Fsa(props, directed=True, multi=False)
dfa.name = name
bitmaps = dfa.get_guard_bitmap(guard)
ngen = it.count()
u, v = ngen.next(), ngen.next()
dfa.g.add_edge(u, v, attr_dict={'weight': 0, 'input': bitmaps, 'guard' : guard, 'label': guard})
dfa.init[u] = 1
dfa.final.add(v)
init_tree(dfa, operation=Op.accept)
return dfa
def concatenation(dfa1, dfa2):
'''Creates a DFA which accepts the language of concatenated word accepted by
dfa1 and dfa2. Is assumes that concatenation is non-ambiguous, i.e. every
word in the resulting language can be uniquely decomposed into a prefix word
from the language accepted by dfa1 and a suffix word from the language
accepted by dfa2.
Theorem: card(lang(concatenate(dfa1, dfa2))) ==
card(lang(dfa1)) x card(lang(dfa2))
'''
assert dfa1.directed == dfa2.directed and dfa1.multi == dfa2.multi
assert dfa1.props == dfa2.props
assert dfa1.alphabet == dfa2.alphabet
assert len(dfa1.init) == 1 and len(dfa2.init) == 1
assert len(dfa1.final) == 1 and len(dfa2.final) == 1
dfa = Fsa(dfa1.props, dfa1.directed, dfa1.multi)
dfa.name = '(Concat {} {} )'.format(dfa1.name, dfa2.name)
# relabel the two DFAs to avoid state name collisions and merge the final
# state of dfa1 with the initial state of dfa2
relabel_dfa(dfa1, start=0)
init2 = dfa2.init.keys()[0]
final1 = iter(dfa1.final).next()
relabel_dfa(dfa2, mapping={init2: final1}, start=dfa1.g.number_of_nodes())
assert len(set(dfa1.g.nodes()) & set(dfa2.g.nodes())) == 1
dfa.g.add_edges_from(dfa1.g.edges_iter(data=True))
dfa.g.add_edges_from(dfa2.g.edges_iter(data=True))
# define initial state and final state
dfa.init = dict(dfa1.init)
dfa.final = set(dfa2.final)
if getDFAType() == DFAType.Infinity:
mark_concatenation(dfa1, dfa2, dfa)
elif getDFAType() == DFAType.Normal:
# minimize the DFA
dfa = minimize_dfa(dfa)
else:
raise ValueError('DFA type must be either DFAType.Normal or ' +
'DFAType.Infinity! {} was given!'.format(getDFAType()))
logger.debug('[concatenation] DFA1: {} DFA2: {}'.format(dfa1.name, dfa2.name))
return dfa
def relabel_dfa(dfa, mapping='default', start=0, copy=False):
'''Relabels the DFA. The new labels are given by the mapping dictionary. By
default, it relabels the states with integers with the lowest one given by
start. The dictionary can be a partial mapping of the nodes. The states
which are not specified are labeled with integers starting from start.
If copy is True a new copy of the DFA is returned, otherwise it performs an
in-place relabeling.
'''
if mapping is 'default': # default mapping
mapping = dict()
keys = mapping.keys()
nodes = [u for u in dfa.g.nodes_iter() if u not in keys]
mapping.update(dict(zip(nodes, it.count(start))))
if copy: # create new dfa
ret = Fsa(dfa.props, dfa.directed, dfa.multi)
ret.name = str(dfa.name)
else: # in-place relabeling
ret = dfa
# relabel state, inital state and set of final states
ret.g = nx.relabel_nodes(dfa.g, mapping=mapping, copy=True)
ret.init = dict([(mapping[u], 1) for u in dfa.init.keys()])
ret.final = set([mapping[u] for u in dfa.final])
# copy tree
copy_tree(dfa, ret, mapping=mapping)
return ret
def gdtl2fsa(formula):
tl, ap = gdtl2ltl(formula)
ltl_formula = tl.formulaString(ltl=True)
assert tl.isSynCoSafe()
fsa = Fsa(multi=False)
fsa.from_formula(ltl_formula)
# optional add trap state
fsa.add_trap_state()
logging.info('Automaton size: %s', fsa.size())
logging.info('Automaton: %s', fsa)
return fsa, tl, ap
def construct_fsa():
ap = set(['a', 'b']) # set of atomic propositions
fsa = Fsa(props=ap, multi=False) # empty FSA with propsitions from `ap`
# add states
fsa.g.add_nodes_from(['s0', 's1', 's2', 's3'])
# add transitions
inputs = set(fsa.bitmap_of_props(value) for value in [set()])
fsa.g.add_edge('s0', 's0', attr_dict={'input': inputs})
inputs = set(fsa.bitmap_of_props(value) for value in [set(['a'])])
fsa.g.add_edge('s0', 's1', attr_dict={'input': inputs})
inputs = set(fsa.bitmap_of_props(value) for value in [set(['b'])])
fsa.g.add_edge('s0', 's2', attr_dict={'input': inputs})
inputs = set(fsa.bitmap_of_props(value) for value in [set(['a', 'b'])])
fsa.g.add_edge('s0', 's3', attr_dict={'input': inputs})
inputs = set(fsa.bitmap_of_props(value) for value in [set(), set(['a'])])
fsa.g.add_edge('s1', 's1', attr_dict={'input': inputs})
inputs = set(
fsa.bitmap_of_props(value)
for value in [set(['b']), set(['a', 'b'])])
fsa.g.add_edge('s1', 's3', attr_dict={'input': inputs})
inputs = set(fsa.bitmap_of_props(value) for value in [set(), set(['b'])])
fsa.g.add_edge('s2', 's2', attr_dict={'input': inputs})
inputs = set(
fsa.bitmap_of_props(value)
for value in [set(['a']), set(['a', 'b'])])
fsa.g.add_edge('s2', 's3', attr_dict={'input': inputs})
fsa.g.add_edge('s3', 's3', attr_dict={'input': fsa.alphabet})
# set the initial state
fsa.init['s0'] = 1
# add `s3` to set of final/accepting states
fsa.final.add('s3')
return fsa
def repeat(phi_dfa, low, high):
'''Creates a DFA which accepts the language associated with a within
operator which encloses the formula corresponding to phi_dfa.
'''
assert len(phi_dfa.init) == 1
assert len(phi_dfa.final) == 1
init_state = phi_dfa.init.keys()[0]
final_state = set(phi_dfa.final).pop()
# remove trap states if there are any
phi_dfa.remove_trap_states()
# initialize the resulting dfa
dfa = Fsa(phi_dfa.props, phi_dfa.directed, phi_dfa.multi)
dfa.name = '(Repeat {} {} {} )'.format(phi_dfa.name, low, high)
# compute the maximum number of restarts
b = nx.shortest_path_length(phi_dfa.g, source=init_state, target=final_state)
d = high - low - b + 2
# copy dfa to dfa_aux and initialize the list of restart states
inits = []
nstates = 0
for k in range(d):
# 1. relabel dfa_aux
mapping = dict(zip(phi_dfa.g.nodes_iter(),
range(nstates, nstates + phi_dfa.g.number_of_nodes())))
mapping[final_state] = -1 # mark final state as special
dfa_aux = relabel_dfa(phi_dfa, mapping, copy=True)
# 2. compute truncated dfa_aux
truncate_dfa(dfa_aux, cutoff=(high-low+1)-k)
# 3. add truncated dfa_aux to dfa
dfa.g.add_edges_from(dfa_aux.g.edges_iter(data=True))
inits.append(dfa_aux.init.keys()[0])
nstates += dfa_aux.g.number_of_nodes()
# set initial and final state
dfa.init = {inits[0] : 1}
dfa.final = set([-1])
# create restart transitions
current_states = set([inits[0]])
for rstate in inits[1:]: # current restart state
next_states = set([])
# connect current states to restart state for (else) symbols
for state in current_states:
bitmaps = set()
guard = '(else)'
for _, next_state, d in dfa.g.out_edges_iter(state, data=True):
bitmaps |= d['input']
next_states.add(next_state)
bitmaps = dfa.alphabet - bitmaps
if state not in dfa.final and bitmaps:
dfa.g.add_edge(state, rstate, attr_dict={'weight': 0, 'input': bitmaps, 'guard' : guard, 'label': guard})
# update current states
current_states = next_states | set([rstate])
# relabel states
relabel_dfa(dfa)
# add states to accept a prefix word of any symbol of length low
if low > 0:
guard = '(1)'
bitmaps = dfa.get_guard_bitmap(guard)
ngen = it.count(start=dfa.g.number_of_nodes())
nodes = [ngen.next() for _ in range(low)]
attr_dict={'weight': 0, 'input': bitmaps, 'guard' : guard, 'label': guard}
dfa.g.add_path(nodes, **attr_dict)
dfa.g.add_edge(nodes[-1], dfa.init.keys()[0], attr_dict)
dfa.init = {nodes[0] : 1}
logger.debug('[within] Low: {} High: {} DFA: {}'.format(low, high, phi_dfa.name))
return dfa
def union(dfa1, dfa2):
'''Creates a DFA which accepts the union of the languages corresponding to
the two DFAs. The disjunction operation of TWTL is mapped to union.
If an infinity DFA is generated, the corresponding meta-data is copied as
well.
'''
assert dfa1.directed == dfa2.directed and dfa1.multi == dfa2.multi
assert dfa1.props == dfa2.props
assert dfa1.alphabet == dfa2.alphabet
assert len(dfa1.init) == 1 and len(dfa2.init) == 1
assert len(dfa1.final) == 1 and len(dfa2.final) == 1
dfa = Fsa(dfa1.props, dfa1.directed, dfa1.multi)
dfa.name = '(Union {} {} )'.format(dfa1.name, dfa2.name)
# add self-loops on final states and trap states
attr_dict={'weight': 0, 'input': dfa.alphabet, 'guard' : '(1)', 'label': '(1)'}
dfa1.g.add_edges_from([(s, s, attr_dict) for s in dfa1.final])
dfa2.g.add_edges_from([(s, s, attr_dict) for s in dfa2.final])
dfa1.add_trap_state()
dfa2.add_trap_state()
dfa1.g.remove_edges_from([(s, s) for s in dfa1.final])
dfa2.g.remove_edges_from([(s, s) for s in dfa2.final])
init = list(it.product(dfa1.init.keys(), dfa2.init.keys()))
dfa.init = dict(zip(init, (1,)*len(init)))
assert len(dfa.init) == 1 # dfa1 and dfa2 are deterministic
stack = list(init)
while stack:
u1, u2 = stack.pop()
for _, v1, d1 in dfa1.g.edges_iter(u1, data=True):
for _, v2, d2 in dfa2.g.edges_iter(u2, data=True):
bitmaps = d1['input'] & d2['input']
if bitmaps:
if (v1, v2) not in dfa.g:
stack.append((v1, v2))
guard = '({}) & ({})'.format(d1['guard'], d2['guard'])
dfa.g.add_edge((u1, u2), (v1, v2), attr_dict={'weight': 0, 'input': bitmaps, 'guard' : guard, 'label': guard})
# compute set of final states
dfa.final = set([(u, v) for u, v in dfa.g.nodes_iter()
if u in dfa1.final or v in dfa2.final])
# remove trap states
dfa1.g.remove_nodes_from(['trap'])
dfa2.g.remove_nodes_from(['trap'])
dfa.g.remove_nodes_from([('trap', 'trap')])
# merge finals
if len(dfa.final) > 1:
final = (iter(dfa1.final).next(), iter(dfa2.final).next())
# satisfies both left and right sub-formulae
choices = dict([(u, Choice(both=d['input']))
for u, _, d in dfa.g.in_edges(final, data=True)])
for u, v, d in dfa.g.in_edges_iter(dfa.final - set([final]), data=True):
bitmaps = set(d['input'])
guard = d['guard']
if dfa.g.has_edge(u, final):
bitmaps |= dfa.g[u][final]['input']
guard = '({}) | ({})'.format(guard, dfa.g[u][final]['guard'])
dfa.g.add_edge(u, final, attr_dict={'weight': 0, 'input': bitmaps, 'guard' : guard, 'label': guard})
if v[0] in dfa1.final: # satisfies only the left sub-formula
assert v[1] not in dfa2.final
choices.setdefault(u, Choice()).left.update(d['input'])
if v[1] in dfa2.final: # satisfies only the right sub-formula
assert v[0] not in dfa1.final
choices.setdefault(u, Choice()).right.update(d['input'])
# remove all other final states
dfa.g.remove_nodes_from(dfa.final - set([final]))
dfa.final = set([final])
if getDFAType() == DFAType.Infinity:
mark_product(dfa1, dfa2, dfa, Op.union, choices)
elif getDFAType() == DFAType.Normal:
# minimize the DFA
dfa = minimize_dfa(dfa)
else:
raise ValueError('DFA type must be either DFAType.Normal or ' +
'DFAType.Infinity! {} was given!'.format(getDFAType()))
# relabel states
relabel_dfa(dfa)
logger.debug('[union] DFA1: {} DFA2: {}'.format(dfa1.name, dfa2.name))
return dfa
def intersection(dfa1, dfa2):
'''Creates a DFA which accepts the intersection of the languages
corresponding to the two DFAs. The conjunction operation of TWTL is mapped
to intersection.
If an infinity DFA is generated, the corresponding meta-data is copied as
well.
'''
assert dfa1.directed == dfa2.directed and dfa1.multi == dfa2.multi
assert dfa1.props == dfa2.props
assert dfa1.alphabet == dfa2.alphabet
assert len(dfa1.init) == 1 and len(dfa2.init) == 1
assert len(dfa1.final) == 1 and len(dfa2.final) == 1
dfa = Fsa(dfa1.props, dfa1.directed, dfa1.multi)
dfa.name = '(Intersection {} {} )'.format(dfa1.name, dfa2.name)
init = list(it.product(dfa1.init.keys(), dfa2.init.keys()))
dfa.init = dict(zip(init, (1,)*len(init)))
assert len(dfa.init) == 1
stack = list(init)
while stack:
u1, u2 = stack.pop()
for _, v1, d1 in dfa1.g.edges_iter(u1, data=True):
for _, v2, d2 in dfa2.g.edges_iter(u2, data=True):
bitmaps = d1['input'] & d2['input']
if bitmaps:
if (v1, v2) not in dfa.g:
stack.append((v1, v2))
guard = '({}) & ({})'.format(d1['guard'], d2['guard'])
dfa.g.add_edge((u1, u2), (v1, v2), attr_dict={'weight': 0, 'input': bitmaps, 'guard' : guard, 'label': guard})
if u1 in dfa1.final:
for _, v2, d2 in dfa2.g.edges_iter(u2, data=True):
if (u1, v2) not in dfa.g:
stack.append((u1, v2))
bitmaps = set(d2['input'])
guard = d2['guard']
dfa.g.add_edge((u1, u2), (u1, v2), attr_dict={'weight': 0, 'input': bitmaps, 'guard' : guard, 'label': guard})
if u2 in dfa2.final:
for _, v1, d1 in dfa1.g.edges_iter(u1, data=True):
if (v1, u2) not in dfa.g:
stack.append((v1, u2))
bitmaps = set(d1['input'])
guard = d1['guard']
dfa.g.add_edge((u1, u2), (v1, u2), attr_dict={'weight': 0, 'input': bitmaps, 'guard' : guard, 'label': guard})
# the set of final states is the product of final sets of dfa1 and dfa2
dfa.final = set(it.product(dfa1.final, dfa2.final))
assert len(dfa.final) == 1
if getDFAType() == DFAType.Infinity:
mark_product(dfa1, dfa2, dfa, Op.intersection)
elif getDFAType() == DFAType.Normal:
# minimize the DFA
dfa = minimize_dfa(dfa)
else:
raise ValueError('DFA type must be either DFAType.Normal or ' +
'DFAType.Infinity! {} was given!'.format(getDFAType()))
# relabel states
relabel_dfa(dfa)
logger.debug('[intersection] DFA1: {} DFA2: {}'.format(dfa1.name, dfa2.name))
return dfa