  Parity games play an important role for LTL synthesis as evidenced by recent
breakthroughs on LTL synthesis, which rely in part on parity game solving. Yet
state space explosion remains a major issue if we want to scale to larger
systems or specifications. In order to combat this problem, we need to
investigate symbolic methods such as BDDs, which have been successful in the
past to tackle exponentially large systems. It is therefore essential to have
symbolic parity game solving algorithms, operating using BDDs, that are fast
and that can produce the winning strategies used to synthesize the controller
in LTL synthesis.
  Current symbolic parity game solving algorithms do not yield winning
strategies. We now propose two symbolic algorithms that yield winning
strategies, based on two recently proposed fixpoint algorithms. We implement
the algorithms and empirically evaluate them using benchmarks obtained from
SYNTCOMP 2020. Our conclusion is that the algorithms are competitive with or
faster than an earlier symbolic implementation of Zielonka's recursive
algorithm, while also providing the winning strategies.
