Spot uses BDD for multiple purposes.

The most common one is for labeling edges of automata: each edge stores a BDD representing its guard (i.e., a Boolean function over atomic propositions). Note that the automaton is still represented as a graph (with a vector of states and a vector of edges) and the BDD is only used for the guard. This differs from symbolic representations where the entire transition structure is represented as one large BDD.

There are other algorithms where BDDs are used from different tasks. For instance, our simulation-based reduction function computes a signature of each state as a BDD that is essentially the disjunction of all outgoing edges, represented by their guard, their acceptance sets, and their destination classes. Also, the translation of LTL formulas to transition-based generalized Büchi automata is using an intermediate representation of states that is similar to the aforementioned signatures, excepts that classes are replaced by subformulas.

From the point of view of the BDD library, BDDs are just DAGs with nodes labeled by BDD variables (numbered from 0). From the point of view of Spot's algorithm, these BDD variables have a meaning. For instance if we want to synchronize two automata that have guards over the atomic propositions $$a$$ and $$b$$, we need to make sure that both automata agree on the BDD variables used to represent $$a$$ and $$b$$.

The purpose of bdd_dict

The spot::bdd_dict object is in charge of allocating BDD variables, and ensuring that multiple users reuse the same variables for similar purpose. When a twa_graph automaton is constructed, it takes a bdd_dict as argument. Every time an atomic proposition is registered through the twa::register_ap() method, the bdd_dict is queried.

As an example, the following two automata share their bdd_dict, and although they do not declare their atomic propositions in the same order, they get compatible variable numbers.

#include <spot/twa/twagraph.hh>

int main()
{
spot::bdd_dict_ptr dict = spot::make_bdd_dict();
spot::twa_graph_ptr aut1 = spot::make_twa_graph(dict);
int ap1a = aut1->register_ap("a");
int ap1b = aut1->register_ap("b");
std::cout << "aut1: a=" << ap1a << " b=" << ap1b << '\n';

spot::twa_graph_ptr aut2 = spot::make_twa_graph(dict);
int ap2c = aut2->register_ap("c");
int ap2b = aut2->register_ap("b");
int ap2a = aut2->register_ap("a");
std::cout << "aut2: a=" << ap2a << " b=" << ap2b << " c=" << ap2c << '\n';
}

aut1: a=0 b=1
aut2: a=0 b=1 c=2


Contrast the above result with the following example, where the two automata use different bdd_dict:

#include <spot/twa/twagraph.hh>

int main()
{
spot::bdd_dict_ptr dict1 = spot::make_bdd_dict();
spot::twa_graph_ptr aut1 = spot::make_twa_graph(dict1);
int ap1a = aut1->register_ap("a");
int ap1b = aut1->register_ap("b");
std::cout << "aut1: a=" << ap1a << " b=" << ap1b << '\n';

spot::bdd_dict_ptr dict2 = spot::make_bdd_dict();
spot::twa_graph_ptr aut2 = spot::make_twa_graph(dict2);
int ap2c = aut2->register_ap("c");
int ap2b = aut2->register_ap("b");
int ap2a = aut2->register_ap("a");
std::cout << "aut2: a=" << ap2a << " b=" << ap2b << " c=" << ap2c << '\n';
}

aut1: a=0 b=1
aut2: a=2 b=1 c=0


For this reason, operations like spot::product(aut1, aut2) will require that aut1->get_dict() == aut2->get_dict().

In Python, many functions that would take an explicit bdd_dict in C++ will default to some global bdd_dict instead. So we can do:

import spot
aut1 = spot.make_twa_graph()
ap1a = aut1.register_ap("a")
ap1b = aut1.register_ap("b")
print(f"aut1: a={ap1a} b={ap1b}")
aut2 = spot.make_twa_graph()
ap2c = aut2.register_ap("c")
ap2b = aut2.register_ap("b")
ap2a = aut2.register_ap("a")
print(f"aut1: a={ap2a} b={ap2b} c={ap2c}")

aut1: a=0 b=1
aut1: a=0 b=1 c=2


In that case we did not mention any bdd_dict, but there is one that is implicitly used in both cases. Similarly, when we call spot.translate() the same global bdd_dict is used by default.

What really confuses people, is that the association between an atomic proposition (a, b, …) and a BDD variable (0, 1, …) will only be held by the bdd_dict for the lifetime of the objects (here the automata) that registered this association to the bdd_dict.

Here is a new C++ example where we use the bdd_dict::dump() method to display the contents of the bdd_dict (this method is only meant for debugging, please do not rely on its output).

#include <spot/twa/twagraph.hh>

int main()
{
spot::bdd_dict_ptr dict = spot::make_bdd_dict();

spot::twa_graph_ptr aut1 = spot::make_twa_graph(dict);
int ap1a = aut1->register_ap("a");
int ap1b = aut1->register_ap("b");
std::cout << "aut1@" << aut1 << ": a=" << ap1a << " b=" << ap1b << '\n';
dict->dump(std::cout) << "---\n";

spot::twa_graph_ptr aut2 = spot::make_twa_graph(dict);
int ap2c = aut2->register_ap("c");
int ap2b = aut2->register_ap("b");
std::cout << "aut2@" << aut2 << ": b=" << ap2b << " c=" << ap2c << '\n';
dict->dump(std::cout) << "---\n";

aut1 = nullptr;
std::cout << "aut1 destroyed\n";
dict->dump(std::cout) << "---\n";

aut2 = nullptr;
std::cout << "aut2 destroyed\n";
dict->dump(std::cout);
}

aut1@0x5560f2374400: a=0 b=1
Variable Map:
0 Var[a] x1 { 0x5560f2374400 }
1 Var[b] x1 { 0x5560f2374400 }
Anonymous lists:
[0]
Free list:

---
aut2@0x5560f2375990: b=1 c=2
Variable Map:
0 Var[a] x1 { 0x5560f2374400 }
1 Var[b] x2 { 0x5560f2374400 0x5560f2375990 }
2 Var[c] x1 { 0x5560f2375990 }
Anonymous lists:
[0]
Free list:

---
aut1 destroyed
Variable Map:
0 Free
1 Var[b] x1 { 0x5560f2375990 }
2 Var[c] x1 { 0x5560f2375990 }
Anonymous lists:
[0]
Free list:
(0, 1)
---
aut2 destroyed
Variable Map:
0 Free
1 Free
2 Free
Anonymous lists:
[0]
Free list:
(0, 3)


For each BDD variable registered to the bdd_dict, we have one line that gives: the variable number, its meaning (e.g. Var[b]), its registration count (x2), and a list of pointers to the objects that registered the association.

Every time twa::register_ap() is called, it calls a similar function in the bdd_dict to check for an existing association or register a new one. When aut1 is deleted, it unregisters all its variables, causing variable 0 to become free. The free list is actually a list of pairs representing ranges of free variables that can be reassigned by the BDD dict when needed. (The anonymous list serves when anonymous BDD variables are used.)

Such a low-level registration is usually handled by the following interface:

// return a BDD variable number for f
int bdd_dict::register_proposition(formula f, const void* for_me);
// release the BDD variable
void bdd_dict::unregister_variable(int var, const void* me);
// release all BDD variables registered by me
void bdd_dict::unregister_all_my_variables(const void* me);
// register the same variables as another object
void bdd_dict::register_all_variables_of(const void* from_other,
const void* for_me);


The last function may be a bit tricky to use, because we need to be sure that another object has registered some variables. You can rely on the fact that each twa automaton register its variables this way.

Now, in most cases, there is no need to worry about the bdd_dict. Automata will register and unregister variables as needed. Other objects like spot::twa_word will do the same.

There are at least two situations where one may need to deal with the bdd_dict:

1. One case is when creating a derived object that store some BDD representing a formula over atomic proposition (but without reference to their original automaton).
2. Another case is when more BDD variables (maybe unrelated to atomic propositions) are needed.

These two cases are discussed in the next sections.

Prolonging the association between a BDD variable and an atomic proposition

Let us implement an object representing a set of transitions of the form $$(src, guard, dest)$$. This can for instance be used to store all transition that belong to a certain acceptance set.

import spot

class trans_set:
def __init__(self, dict):
self.set = set()
self.dict = dict
def str_trans(self, src, guard, dst):
f = spot.bdd_format_formula(self.dict, guard)
return f"({src},{f},{dst})"
def __str__(self):
return '{' + ",".join([ self.str_trans(*t) for t in self.set ]) + '}'

def accepting_set(aut, num):
ts = trans_set(aut.get_dict())
for e in aut.edges():
if e.acc.has(num):
return ts


The above code has two definitions.

1. The trans_set class is a set of transitions that can be printed. It stores a bdd_dict so that it can print the guard of the transition.
2. The accepting_set function iterates over an automaton, and saves all transitions that belong to a given acceptance set number.

For instance, we can now translate an automaton, compute its acceptance set 0, and print it as follows:

aut = spot.translate('GF(a <-> XXa)')
ts = accepting_set(aut, 0)
print(ts)

{(2,!a,2),(0,a,3),(1,a,1),(3,!a,0)}


The code of trans_set is in fact bogus. The problem is that it assumes the association between the atomic propositions and the BDD variable is still available when the str_trans method is called. However, that might not be the case.

The following call sequence demonstrates the problem:

try:
ts = accepting_set(spot.translate('GF(a <-> XXa)'), 0)
print(ts)
except RuntimeError as e:
print("ERROR:", e)

ERROR: bdd_to_formula() was passed a bdd with a variable that is not in the dictionary


In this case, the temporary automaton constructed by spot.translate() and passed to the accepting_set() function is destroyed right after the ts object has been constructed. When the automaton is destroyed, it removes all its associations from the bdd_dict. This means that before the print(ts), the dictionary that was used by the automaton and that is still stored in the ts objects is now empty. Consequently, calling bdd_format_formula() raises an exception.

This can be fixed in a couple of ways. The easy way is to store the automaton inside the trans_set object, to ensure that it will live at least as long as the trans_set object. But maybe the automaton is too big, and we really want to get rid of it? In this case trans_set should tell the bdd_dict that it want to retain the associations. The easiest way in this case is to call the register_all_variables_of() method, because we know that each automaton registers its variables.

import spot

class trans_set:
def __init__(self, aut):
self.set = set()
self.dict = aut.get_dict()
self.dict.register_all_variables_of(aut, self)
def __del__(self):
self.dict.unregister_all_my_variables(self)
def str_trans(self, src, guard, dest):
f = spot.bdd_format_formula(self.dict, guard)
return f"({src},{f},{dest})"
def __str__(self):
return '{' + ",".join([ self.str_trans(*t) for t in self.set ]) + '}'

def accepting_set(aut, num):
ts = trans_set(aut)
for e in aut.edges():
if e.acc.has(num):
return ts

try:
ts = accepting_set(spot.translate('GF(a <-> XXa)'), 0)
print(ts)
except RuntimeError as e:
print("ERROR:", e)

{(2,!a,2),(0,a,3),(1,a,1),(3,!a,0)}


Notice that we have also added a destructor to trans_set to unregister all the variables.

Anonymous BDD variables

Another scenario where working with a bdd_dict is needed is when one needs to allocate anonymous BDD variables. These are variables that are not attached to any atomic proposition, and that can be used by one algorithm privately. If multiple algorithms (or objects) register anonymous variables, the bdd_dict will reuse anonymous variables allocated to other algorithms. One can allocate multiple anonymous variables with the following bdd_dict method:

int bdd_dict::register_anonymous_variables(int n, const void* for_me);


A range of n variables will be allocated starting at the returned index.

For instance, let's say that our trans_set should now store a symbolic representation of a transition relation. For simplicity, we assume we just want to store set of pairs (src,dst): each pair will be a conjunction $$v_{src}\land v'_{dst}$$ between two BDD variables taken from two ranges ($$v_i$$ representing a source state $$i$$ and $$v'i$$ representing a destination state $$i$$), and the entire set will be a disjunction of all these pairs. If the automaton has $$n$$ states, we want to allocate $$2n$$ BDD variables for this purpose. We call these variables anonymous because their meaning is unknown to the bdd_dict.

import spot
from buddy import *

class trans_set:
def __init__(self, aut):
self.dict = d = aut.get_dict()
self.num_states = n = aut.num_states()
self.anonbase = b = d.register_anonymous_variables(2*n, self)
s = bddfalse
for e in aut.edges():
s |= self.src(e.src) & self.dst(e.dst)
self.rel = s
def src(self, n):
return bdd_ithvar(self.anonbase + n)
def dst(self, n):
return bdd_ithvar(self.anonbase + n + self.num_states)
def __del__(self):
self.dict.unregister_all_my_variables(self)
def __str__(self):
isop = spot.minato_isop(self.rel)
i = isop.next()
res = []
while i != bddfalse:
s = bdd_var(i) - self.anonbase
d = bdd_var(bdd_high(i)) - self.anonbase - self.num_states
res.append((s, d))
i = isop.next()
return str(res)

ts = trans_set(spot.translate('GF(a <-> XXa)'))
print(ts)

[(0, 2), (0, 3), (1, 0), (1, 1), (2, 2), (2, 3), (3, 0), (3, 1)]