Given two deterministic parity automata $A_1=(Q_1,\Sigma,\delta,q_{01},c_1)$ and $A_2=(Q_2,\Sigma,\delta,q_{02},c_2)$ with the finite set of states $Q_i$, the finite alphabet $\Sigma_i$, the transition function $\delta_i : Q_i \times \Sigma \rightarrow Q_i$, the initial state $q_{0i}$ in $Q_i$, and the coloring function $c_i : Q_i \rightarrow \mathbb{N}$, with $i \in\{0,1\}$.

How is the intersection of two deterministic parity automata formally defined? In particular, how are the coloring functions combined?

  • $\begingroup$ Hint: you can start by proving that there is no way to define a correct coloring function for the intersection on the states $Q_1\times Q_2$ of the product automaton. $\endgroup$
    – Denis
    Commented Jun 20, 2019 at 15:21

1 Answer 1


The following paper: http://www.faculty.idc.ac.il/udiboker/files/AutomataTypes.pdf shows that the intersection (and union) of deterministic parity automata may involve an exponential blowup.

Thus, you cannot do much better than converting to nondeterministic Buchi automata, and taking the intersection there (and determinizing back to parity).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.