Buchi Acceptance Research Articles

On the complexity of ω-pushdown automata

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.

Variations on parallel explicit emptiness checks for generalized Büchi automata

We present new parallel explicit emptiness checks for LTL model checking. Unlike existing parallel emptiness checks, these are based on a Strongly Connected Component (SCC) enumeration, support generalized Buchi acceptance, and require no synchronization points nor recomputing procedures. A salient feature of our algorithms is the use of a global union-find data structure in which multiple threads share structural information about the automaton checked. Besides these basic algorithms, we present one architectural variant isolating threads that write to the union-find, and one extension that decomposes the automaton based on the strength of its SCCs to use more optimized emptiness checks. The results from an extensive experimentation of our algorithms and their variations show encouraging performances, especially when the decomposition technique is used.

Wadge Degrees of Infinitary Rational Relations

We show that, from the topological point of view, 2-tape Buchi automata have the same accepting power as Turing machines equipped with a Buchi acceptance condition. The Borel and the Wadge hierarchies of the class RAT_omega of infinitary rational relations accepted by 2-tape Buchi automata are equal to the Borel and the Wadge hierarchies of omega-languages accepted by real-time Buchi 1-counter automata or by Buchi Turing machines. In particular, for every non-null recursive ordinal $\alpha$, there exist some $\Sigma^0_\alpha$-complete and some $\Pi^0_\alpha$-complete infinitary rational relations. And the supremum of the set of Borel ranks of infinitary rational relations is an ordinal $\gamma^1_2$ which is strictly greater than the first non-recursive ordinal $\omega_1^{CK}$. This very surprising result gives answers to questions of Simonnet (1992) and of Lescow and Thomas (1988,1994).

Borel ranks and Wadge degrees of context free $\omega$-languages

We show that the Borel hierarchy of the class of context free $\omega$-languages, or even of the class of $\omega$-languages accepted by Buchi 1-counter automata, is the same as the Borel hierarchy of the class of $\omega$-languages accepted by Turing machines with a Buchi acceptance condition. In particular, for each recursive non-null ordinal $\alpha$, there exist some ${\bf \Sigma}^0_\alpha$-complete and some ${\bf \Pi}^0_\alpha$-complete $\omega$-languages accepted by Buchi 1-counter automata. And the supremum of the set of Borel ranks of context free $\omega$-languages is an ordinal $\gamma_2^1$ that is strictly greater than the first non-recursive ordinal $\omega_1^{\mathrm{CK}}$. We then extend this result, proving that the Wadge hierarchy of context free $\omega$-languages, or even of $\omega$-languages accepted by Buchi 1-counter automata, is the same as the Wadge hierarchy of $\omega$-languages accepted by Turing machines with a Buchi or a Muller acceptance condition.