On the complexity of ω-pushdown automata

Abstract

Finite automata over infinite words (called $\omega$-automata) play an important role in the automata-theoretic approach to system verification.Different types of $\omega$-automata differ in their succinctness and complexity of their emptiness problems, as a result, theory of $\omega$-automata hasreceived considerable research attention. Pushdown automata over infinite words (called $\omega$-PDAs), a generalization of $\omega$-automata, are a naturalmodel of recursive programs. Our goal in this paper is to conduct a relatively complete investigation on the complexity of the emptiness problems for variantsof $\omega$-PDAs. For this purpose, we consider $\omega$-PDAs of five standard acceptance types: Buchi, Parity, Rabin, Streett and Muller acceptances.Based on the transformation for $\omega$-automata and the efficient algorithm proposed by Esparza et al. in CAV00 for verifying the emptiness problem of $\omega$-PDAswith Buchi acceptance, it is trivial to check the emptiness problem of other $\omega$-PDAs. However, this naive approach is not optimal. In this paper,we propose novel algorithms for the emptiness problem of $\omega$-PDAs based on the observations of the structure of accepting runs. Our algorithms outperform algorithmsthat go through Buchi PDAs. In particular, the space complexity of the algorithm for Streett acceptance that goes through Buchi acceptance is exponential,while ours is polynomial. The algorithm for Parity acceptance that goes through Buchi acceptance is in ${\boldsymbol O}(k^3n^2m)$ time and ${\boldsymbol O}(k^2nm)$ space,while ours is in ${\boldsymbol O}(kn^2m)$ time and ${\boldsymbol O}(nm)$ space, where $n$ (resp. $m$ and $k$) is the number of control states (resp. transitions and index).Finally, we show that our algorithms yield a better solution for the pushdown model checking problem against linear temporal logic with fairness.