This notebook discusses how explicit automata (the spot::twa_graph_ptr objects in C++) are stored by Spot. The Python bindings do not expose all of the internals available in C++, however they have some graphical representation that are convenient to present those inner workings.

In [1]:
import buddy
import spot
spot.setup(show_default='.n')
from IPython.display import display

The two-vector representation

Let's consider a small automaton, generated from an LTL formula.

In [2]:
aut = spot.translate('GF(a <-> Xa) & FGb', 'det', 'gen')
In [3]:
aut
Out[3]:
Fin( ) & Inf( ) [Rabin 1] 0 0 I->0 0->0 a & b 0->0 a & !b 1 1 0->1 !a & b 0->1 !a & !b 1->0 a & b 1->0 a & !b 1->1 !a & b 1->1 !a & !b

The graphical representation above is just a convenient representation of that automaton and hides some details. Internally, this automaton is stored as two vectors plus some additional data. All of those can be displayed using the show_storage() method. The two vectors are the states and edges vectors. The additional data gives the initial state, number of acceptance sets, acceptance condition, list of atomic propositions, as well as a bunch of property flags on the automaton. All those properties default to maybe, but some algorithms will turn them to yes or no whenever that property can be decided at very low cost (usually a side effect of the algorithm). In this example we asked for a deterministic automaton, so the output of the construction is necessarily universal (this means no existantial branching, hence deterministic for our purpose), and this property implies unambiguous and semi_deterministic.

In [4]:
aut.show_storage()
Out[4]:
g states states 0 1 succ 1 5 succ_tail 4 8 edges edges 1 2 3 4 5 6 7 8 cond a & b a & !b !a & b !a & !b a & b a & !b !a & b !a & !b acc {1} {0,1} {} {0} {} {0} {1} {0,1} dst 0 0 1 1 0 0 1 1 next_succ 2 3 4 0 6 7 8 0 src 0 0 0 0 1 1 1 1 meta init_state: 0 num_sets: 2 acceptance: Fin(0) & Inf(1) ap_vars: b a props prop_state_acc: maybe prop_inherently_weak: maybe prop_terminal: no prop_weak: maybe prop_very_weak: maybe prop_complete: maybe prop_universal: yes prop_unambiguous: yes prop_semi_deterministic: yes prop_stutter_invariant: maybe

Each state is represented by an integer that is a 0-based index into the states array. Each edge is also represented by an integer that is a 1-based index into the edges array. In the above picture, yellow and cyan denote state and edge indices respectively.

Adding a new edge, for instance, will augment the size of the edges array and return the index of the newly added edge:

In [5]:
s = aut.new_state()
aut.set_init_state(s)
aut.new_edge(s, 0, buddy.bddtrue)
Out[5]:
9
In [6]:
display(aut, aut.show_storage("v"))  # "v" displays only the states and edges Vectors
Fin( ) & Inf( ) [Rabin 1] 2 2 I->2 0 0 2->0 1 0->0 a & b 0->0 a & !b 1 1 0->1 !a & b 0->1 !a & !b 1->0 a & b 1->0 a & !b 1->1 !a & b 1->1 !a & !b
g states states 0 1 2 succ 1 5 9 succ_tail 4 8 9 edges edges 1 2 3 4 5 6 7 8 9 cond a & b a & !b !a & b !a & !b a & b a & !b !a & b !a & !b 1 acc {1} {0,1} {} {0} {} {0} {1} {0,1} {} dst 0 0 1 1 0 0 1 1 0 next_succ 2 3 4 0 6 7 8 0 0 src 0 0 0 0 1 1 1 1 2

For each state, the states vector stores two edge indices: succ is the index of the first outgoing edge, and succ_tail is the index of the last outgoing edge. Since there is no edge at index 0, that value is used to indicate that there is no outgoing edge.

In the edges vector, the field next_succ is used to organize the outgoing edges of a state as a linked list. For instance to iterate over all successors of state 0, we would start at the edge e = states[0].succ (if it's not 0), then move to the next successor with e = edges[e].next_succ, and repeat until e becomes 0. This code cannot be executed in Python because the automaton class won't let us access the states vector. However this iteration mechanism is what is used into the out() method: out() simply provides an iterator over some columns of the edges vector, following the next_succ links. When we have a reference to a column of edges as returned by out(), we can convert it into an edge index with the edge_number() method.

In [7]:
for ed in aut.out(0):
    print("edges[{e}].src={src}, edges[{e}].dst={dst}".format(e=aut.edge_number(ed), src=ed.src, dst=ed.dst))
edges[1].src=0, edges[1].dst=0
edges[2].src=0, edges[2].dst=0
edges[3].src=0, edges[3].dst=1
edges[4].src=0, edges[4].dst=1

The other fields of the edges vector probably speak for themselves. cond is a BDD representing the boolean combination of atomic propositions expected by the edge, acc is an instance of spot::acc_cond::mark_t, i.e., a bit set representing the set of acceptance sets the edge belongs to, src and dst are the source and destination of the transition. Of course when iterating over the successors of a state with aut.out(src), the source is well known, but there are other situations where it is convenient to retrieve the source from the edge (e.g., when iterating over all edges of an automaton, or when storing edges indices for later processing).

You can access one column of the edges vector using the edge_storage() method. For instance let's modify edge 3:

In [8]:
aut.edge_storage(3).acc.set(1)
aut.edge_storage(3).dst = 0
display(aut)
aut.show_storage("v")
Fin( ) & Inf( ) [Rabin 1] 2 2 I->2 0 0 2->0 1 0->0 a & b 0->0 a & !b 0->0 !a & b 1 1 0->1 !a & !b 1->0 a & b 1->0 a & !b 1->1 !a & b 1->1 !a & !b
Out[8]:
g states states 0 1 2 succ 1 5 9 succ_tail 4 8 9 edges edges 1 2 3 4 5 6 7 8 9 cond a & b a & !b !a & b !a & !b a & b a & !b !a & b !a & !b 1 acc {1} {0,1} {1} {0} {} {0} {1} {0,1} {} dst 0 0 0 1 0 0 1 1 0 next_succ 2 3 4 0 6 7 8 0 0 src 0 0 0 0 1 1 1 1 2

Having the source into the edges vector also allows us to easily sort that vector to put all sibling transitions (i.e., transition leaving the same state) together to improve data locality. The merge_edges() method will do that and a bit more: edges are first sorted by (src, dst, acc) making possible to merge the cond field of edges with identical (src, dst, acc). On Fin-less automata (not our example), the a second pass is perform to sort the edge (src, dst, cond) and then merge the acc fields of edges that share the other fields.

In our example, merge_edges() will merge edges 1 and 3.

In [9]:
aut.merge_edges()
display(aut, aut.show_storage("v"))
Fin( ) & Inf( ) [Rabin 1] 2 2 I->2 0 0 2->0 1 0->0 b 0->0 a & !b 1 1 0->1 !a & !b 1->0 a & b 1->0 a & !b 1->1 !a & b 1->1 !a & !b
g states states 0 1 2 succ 1 4 8 succ_tail 3 7 8 edges edges 1 2 3 4 5 6 7 8 cond b a & !b !a & !b a & b a & !b !a & b !a & !b 1 acc {1} {0,1} {0} {} {0} {1} {0,1} {} dst 0 0 1 0 0 1 1 0 next_succ 2 3 0 5 6 7 0 0 src 0 0 0 1 1 1 1 2

Note that the succ_tail field of the states vector is seldom used when reading automata as the linked list of edges ends when next_succ (or succ) equals 0. Its main use is during calls to new_edge(): new edges are always created at the end of the list (otherwise it would be hard to preserve the order of edges when parsing and automaton and printing it).

The property-update issue

Properties like prop_complete(), prop_universal(), and the like are normally updated by algorithms that modify the automaton in place. They are not updated when we modify the automaton using low-level methods like new_state() or new_edges() as we have done so far.

For instance we could add a new edge to the automaton to introduce some non-determinism, and the automaton would still pretend it is universal.

In [10]:
aut.new_edge(1, 1, buddy.bddtrue, [1, 0])
display(aut, aut.show_storage())
Fin( ) & Inf( ) [Rabin 1] 2 2 I->2 0 0 2->0 1 0->0 b 0->0 a & !b 1 1 0->1 !a & !b 1->0 a & b 1->0 a & !b 1->1 !a & b 1->1 !a & !b 1->1 1
g states states 0 1 2 succ 1 4 8 succ_tail 3 9 8 edges edges 1 2 3 4 5 6 7 8 9 cond b a & !b !a & !b a & b a & !b !a & b !a & !b 1 1 acc {1} {0,1} {0} {} {0} {1} {0,1} {} {0,1} dst 0 0 1 0 0 1 1 0 1 next_succ 2 3 0 5 6 7 9 0 0 src 0 0 0 1 1 1 1 2 1 meta init_state: 2 num_sets: 2 acceptance: Fin(0) & Inf(1) ap_vars: b a props prop_state_acc: maybe prop_inherently_weak: maybe prop_terminal: no prop_weak: maybe prop_very_weak: maybe prop_complete: maybe prop_universal: yes prop_unambiguous: yes prop_semi_deterministic: yes prop_stutter_invariant: maybe

Such an inconsistency will cause many issues when the automaton is passed to algorithm with specialized handling of universal automata. When writing an algorithm that modify the automaton, it is your responsibility to update the property bits as well. In this case it could be fixed by calling aut.prop_universal(False); aut.prop_unambiguous(spot.trival_maybe()); ... for each property, or by reseting all properties to maybe with prop_reset():

In [11]:
aut.prop_reset()
aut.show_storage("p")  # "p" displays only the properties
Out[11]:
g props prop_state_acc: maybe prop_inherently_weak: maybe prop_terminal: maybe prop_weak: maybe prop_very_weak: maybe prop_complete: maybe prop_universal: maybe prop_unambiguous: maybe prop_semi_deterministic: maybe prop_stutter_invariant: maybe

Erasing edges

Erasing a single edge, denoted by its edge index i, is not convenient because of the linked structure of the edges: the next_succ of the previous (but unknown given i) edge would have to be updated (or maybe the succ or succ_tail fields of states vector have to be updated).

The out_iteraser(s) method provides a way to iterate over the outgoing edges of state s that allows erasing edges. The iteration does not follow the usual Python pattern because once you have looked at the current edge using current(), you have two choices: advance() to the next one, or erase() the current one (and advance to the next). Note that it.current() and it.advance() are written *it and ++it in C++.

The following example erases all the outgoing transitions of state 0 that belong to acceptance set 1.

In [12]:
it = aut.out_iteraser(0)
while it:
    e = it.current()
    toerase = e.acc.has(1)
    print("pos={}, acc={}, toerase={}".format(aut.edge_number(e), e.acc, toerase))
    if toerase:
        it.erase()
    else:
        it.advance()
pos=1, acc={1}, toerase=True
pos=2, acc={0,1}, toerase=True
pos=3, acc={0}, toerase=False
In [13]:
aut.show_storage("v")
Out[13]:
g states states 0 1 2 succ 3 4 8 succ_tail 3 9 8 edges edges 1 2 3 4 5 6 7 8 9 cond b a & !b !a & !b a & b a & !b !a & b !a & !b 1 1 acc {1} {0,1} {0} {} {0} {1} {0,1} {} {0,1} dst 0 0 1 0 0 1 1 0 1 next_succ 1 2 0 5 6 7 9 0 0 src 0 0 0 1 1 1 1 2 1

Notice that the edges vector hasn't been resized, as doing so would renumber edges. Instead, erased edges have removed from the linked list of outgoing edges of 0, and their next_succ field has been changed to point to themselves.

You can test whether an edges has been erased with is_dead_edge():

In [14]:
aut.is_dead_edge(2)
Out[14]:
True

However you usually do not have to care, because iterator methods will skip such dead edges. For instance:

In [15]:
for e in aut.edges():  # iterate over all non-erased edges
    print("edges[{e}].src={src}, edges[{e}].dst={dst}".format(e=aut.edge_number(e), src=e.src, dst=e.dst))
edges[3].src=0, edges[3].dst=1
edges[4].src=1, edges[4].dst=0
edges[5].src=1, edges[5].dst=0
edges[6].src=1, edges[6].dst=1
edges[7].src=1, edges[7].dst=1
edges[8].src=2, edges[8].dst=0
edges[9].src=1, edges[9].dst=1

Similarly, num_edges() returns the count of non-erased edges.

In [16]:
aut.num_edges()
Out[16]:
7

Erased edges are actually removed by merge_edges():

In [17]:
aut.merge_edges()
aut.show_storage("v")
Out[17]:
g states states 0 1 2 succ 1 2 6 succ_tail 1 5 6 edges edges 1 2 3 4 5 6 cond !a & !b a & b a & !b !a & b 1 1 acc {0} {} {0} {1} {0,1} {} dst 1 0 0 1 1 0 next_succ 0 3 4 5 0 0 src 0 1 1 1 1 2

Another way to erase an edge, is to set its cond field to bddfalse. Strictly speaking, this does not really erase the edge, and it will still be iterated upon. However a subsequent call to merge_edges() will perform the removal of that edge.

In [18]:
aut.edge_storage(3).cond = buddy.bddfalse
aut.show_storage("v")
Out[18]:
g states states 0 1 2 succ 1 2 6 succ_tail 1 5 6 edges edges 1 2 3 4 5 6 cond !a & !b a & b 0 !a & b 1 1 acc {0} {} {0} {1} {0,1} {} dst 1 0 0 1 1 0 next_succ 0 3 4 5 0 0 src 0 1 1 1 1 2
In [19]:
for e in aut.edges():  # iterate over all non-erased edges
    print("edges[{e}].src={src}, edges[{e}].dst={dst}".format(e=aut.edge_number(e), src=e.src, dst=e.dst))
edges[1].src=0, edges[1].dst=1
edges[2].src=1, edges[2].dst=0
edges[3].src=1, edges[3].dst=0
edges[4].src=1, edges[4].dst=1
edges[5].src=1, edges[5].dst=1
edges[6].src=2, edges[6].dst=0
In [20]:
aut.is_dead_edge(3)
Out[20]:
False
In [21]:
aut.merge_edges()
aut.show_storage("v")
Out[21]:
g states states 0 1 2 succ 1 2 5 succ_tail 1 4 5 edges edges 1 2 3 4 5 cond !a & !b a & b !a & b 1 1 acc {0} {} {1} {0,1} {} dst 1 0 1 1 0 next_succ 0 3 4 0 0 src 0 1 1 1 2

Alternation

The data structures seen so far only support a single destination per edge. Support for universal branching therefore calls for something new.

Let's add some universal branching in our example automaton.

In [22]:
a = buddy.bdd_ithvar(aut.register_ap('a'))
s = aut.new_state()
aut.new_univ_edge(0, [0, s], a, [1])
aut.new_univ_edge(s, [0, 1], -a, [0])
aut
Out[22]:
Fin( ) & Inf( ) [Rabin 1] 2 2 I->2 0 0 2->0 1 1 1 0->1 !a & !b -1 0->-1 a 1->0 a & b 1->1 !a & b 1->1 1 -1->0 3 3 -1->3 -4 3->-4 !a -4->0 -4->1
In [23]:
aut.show_storage()
Out[23]:
g states states 0 1 2 3 succ 1 2 5 7 succ_tail 6 4 5 7 edges edges 1 2 3 4 5 6 7 cond !a & !b a & b !a & b 1 1 a !a acc {0} {} {1} {0,1} {} {1} {0} dst 1 0 1 1 0 ~0 ~3 next_succ 6 3 4 0 0 0 0 src 0 1 1 1 2 0 3 dests dests ~0 ~3 #cnt/dst #2 0 3 #2 0 1 meta init_state: 2 num_sets: 2 acceptance: Fin(0) & Inf(1) ap_vars: b a props prop_state_acc: maybe prop_inherently_weak: maybe prop_terminal: maybe prop_weak: maybe prop_very_weak: maybe prop_complete: maybe prop_universal: maybe prop_unambiguous: maybe prop_semi_deterministic: maybe prop_stutter_invariant: maybe

Here we have created two universal transitions: 0->[0,2] and 2->[0,1]. The destination groups [0,2] and [0,1] are stored in a integer vector called dests. Each group is encoded by its size immediately followed by the state numbers of the destinations. So group [0,2] get encoded as 2,0,2 at position 0 of dests, and group [0,1] is encoded as 2,0,1 at position 3. Each group is denoted by the index of its size in the dests vector. When an edge targets a destination group, the complement of that destination index is written in the dst field of the edges entry, hence that ~0 and ~3 that appear here. Using a complement like this allows us to quickly detect universal edges by looking at the sign bit if their dst entry.

To work on alternating automata, one can no longuer just blindingly use the dst field of outgoing iterations:

In [24]:
for e in aut.edges():
    print(e.dst)
1
0
1
1
0
4294967295
4294967292

Using such a large e.dst value as an index in states would likely crash the program. Instead we should iterate over all the successor of an edge using the univ_dests() method. Note that univ_dests() can be applied to regular edges as well.

In [25]:
for e in aut.edges():
    print([d for d in aut.univ_dests(e.dst)])
[1]
[0]
[1]
[1]
[0]
[0, 3]
[0, 1]

Note that univ_dests() can be applied to e.dst or e.

In [26]:
for e in aut.edges():
    print([d for d in aut.univ_dests(e)])
[1]
[0]
[1]
[1]
[0]
[0, 3]
[0, 1]

Note that the initial state get also use universal branching:

In [27]:
aut.set_univ_init_state([0, 1, 2])
In [28]:
aut
Out[28]:
Fin( ) & Inf( ) [Rabin 1] -7 I->-7 0 0 -7->0 1 1 -7->1 2 2 -7->2 0->1 !a & !b -1 0->-1 a 1->0 a & b 1->1 !a & b 1->1 1 2->0 1 -1->0 3 3 -1->3 -4 3->-4 !a -4->0 -4->1
In [29]:
print([d for d in aut.univ_dests(aut.get_init_state_number())])
[0, 1, 2]
In [30]:
aut.show_storage()
Out[30]:
g states states 0 1 2 3 succ 1 2 5 7 succ_tail 6 4 5 7 edges edges 1 2 3 4 5 6 7 cond !a & !b a & b !a & b 1 1 a !a acc {0} {} {1} {0,1} {} {1} {0} dst 1 0 1 1 0 ~0 ~3 next_succ 6 3 4 0 0 0 0 src 0 1 1 1 2 0 3 dests dests ~0 ~3 ~6 #cnt/dst #2 0 3 #2 0 1 #3 0 1 2 meta init_state: ~6 num_sets: 2 acceptance: Fin(0) & Inf(1) ap_vars: b a props prop_state_acc: maybe prop_inherently_weak: maybe prop_terminal: maybe prop_weak: maybe prop_very_weak: maybe prop_complete: maybe prop_universal: maybe prop_unambiguous: maybe prop_semi_deterministic: maybe prop_stutter_invariant: maybe

Adding several transitions with the same destination groups may result in duplicates in the dests vector. Those groups get merged during merge_edges().

In [31]:
aut.new_univ_edge(3, [0,3], buddy.bddtrue, [0])
display(aut, aut.show_storage("vd"))
aut.merge_edges()
display(aut, aut.show_storage("vd"))
Fin( ) & Inf( ) [Rabin 1] -7 I->-7 0 0 -7->0 1 1 -7->1 2 2 -7->2 0->1 !a & !b -1 0->-1 a 1->0 a & b 1->1 !a & b 1->1 1 2->0 1 -1->0 3 3 -1->3 -4 3->-4 !a -11 3->-11 1 -4->0 -4->1 -11->0 -11->3
g states states 0 1 2 3 succ 1 2 5 7 succ_tail 6 4 5 8 edges edges 1 2 3 4 5 6 7 8 cond !a & !b a & b !a & b 1 1 a !a 1 acc {0} {} {1} {0,1} {} {1} {0} {0} dst 1 0 1 1 0 ~0 ~3 ~10 next_succ 6 3 4 0 0 0 8 0 src 0 1 1 1 2 0 3 3 dests dests ~0 ~3 ~6 ~10 #cnt/dst #2 0 3 #2 0 1 #3 0 1 2 #2 0 3 meta init_state: ~6 num_sets: 2 acceptance: Fin(0) & Inf(1) ap_vars: b a
Fin( ) & Inf( ) [Rabin 1] -7 I->-7 0 0 -7->0 1 1 -7->1 2 2 -7->2 0->1 !a & !b -1 0->-1 a 1->0 a & b 1->1 !a & b 1->1 1 2->0 1 -1->0 3 3 -1->3 3->-1 1 -4 3->-4 !a -4->0 -4->1
g states states 0 1 2 3 succ 1 3 6 7 succ_tail 2 5 6 8 edges edges 1 2 3 4 5 6 7 8 cond !a & !b a a & b !a & b 1 1 !a 1 acc {0} {1} {} {1} {0,1} {} {0} {0} dst 1 ~0 0 1 1 0 ~3 ~0 next_succ 2 0 4 5 0 0 8 0 src 0 0 1 1 1 2 3 3 dests dests ~0 ~3 ~6 #cnt/dst #2 0 3 #2 0 1 #3 0 1 2 meta init_state: ~6 num_sets: 2 acceptance: Fin(0) & Inf(1) ap_vars: b a

Above group ~0 and ~10 have been merged.

Named properties

Finally automata can also been attached arbitrarily named properties. The show_storage() method will only display the name of these properties, not their contents. Properties like automaton-name are used to store a name for the automaton, product-states is filled by product() and holds a vector of pairs representing the source states in the product's operands, etc.

In [32]:
aub = spot.translate('a U b')
gfa = spot.translate('GFa')
prod = spot.product(aub, gfa)
prod.set_name("aub * gfa")
display(prod, prod.show_storage())
aub * gfa aub * gfa Inf( ) [Büchi] 0 1,0 I->0 0->0 a & !b 1 0,0 0->1 !a & b 0->1 a & b 1->1 !a 1->1 a
g states states 0 1 succ 1 4 succ_tail 3 5 edges edges 1 2 3 4 5 cond !a & b a & b a & !b !a a acc {} {} {} {} {0} dst 1 1 0 1 1 next_succ 2 3 0 5 0 src 0 0 0 1 1 meta init_state: 0 num_sets: 1 acceptance: Inf(0) ap_vars: b a props prop_state_acc: maybe prop_inherently_weak: maybe prop_terminal: maybe prop_weak: maybe prop_very_weak: maybe prop_complete: maybe prop_universal: yes prop_unambiguous: yes prop_semi_deterministic: yes prop_stutter_invariant: yes namedprops named properties: automaton-name product-states

These properties also need to be updated by algorithms. They can be reset with:

In [33]:
prod.release_named_properties()
display(prod, prod.show_storage())
Inf( ) [Büchi] 0 0 I->0 0->0 a & !b 1 1 0->1 !a & b 0->1 a & b 1->1 !a 1->1 a
g states states 0 1 succ 1 4 succ_tail 3 5 edges edges 1 2 3 4 5 cond !a & b a & b a & !b !a a acc {} {} {} {} {0} dst 1 1 0 1 1 next_succ 2 3 0 5 0 src 0 0 0 1 1 meta init_state: 0 num_sets: 1 acceptance: Inf(0) ap_vars: b a props prop_state_acc: maybe prop_inherently_weak: maybe prop_terminal: maybe prop_weak: maybe prop_very_weak: maybe prop_complete: maybe prop_universal: yes prop_unambiguous: yes prop_semi_deterministic: yes prop_stutter_invariant: yes