Certification for mu-calculus with winning strategies

We define memory-efficient certificates for $\mu$-calculus model checking problems based on the well-known correspondence of the $\mu$-calculus model checking with winning certain parity games. Winning strategies can independently checked, in low polynomial time, by observing that there is no reachable strongly connected component in the graph of the parity game whose largest priority is odd. Winning strategies are computed by fixpoint iteration following the naive semantics of $\mu$-calculus. We instrument the usual fixpoint iteration of $\mu$-calculus model checking so that it produces evidence in the form of a winning strategy; these winning strategies can be computed in polynomial time in $|S|$ and in space $O(|S|^2 |{\phi}|^2)$, where $|S|$ is the size of the state space and $|\phi|$ the length of the formula $\phi$\@. The main technical contribution here is the notion and algebra of partial winning strategies. On the technical level our work can be seen as a new, simpler, and immediate constructive proof of the correspondence between $\mu$-calculus and parity games.