Alternation Removal in Büchi Automata

Abstract

Abstract. Alternating automata play a key role in the automata-theoretic approach to specification, verification, and synthesis of reactive systems. Many algorithms on alternating automata, and in particular, their nonemptiness test, involve removal of alternation: a translation of the alternating automaton to an equivalent nondeterministic one. For alternating Büchi automata, the best known translation uses the "breakpoint construction" and involves an O(3 n ) state blowup. The translation was described by Miyano and Hayashi in 1984, and is widely used since, in both theory and practice. Yet, the best known lower bound is only 2 n . In this paper we develop and present a complete picture of the problem of alternation removal in alternating Büchi automata. In the lower bound front, we show that the breakpoint construction captures the accurate essence of alternation removal, and provide a matching Ω(3 n ) lower bound. Our lower bound holds already for universal (rather than alternating) automata with an alphabet of a constant size. In the upper-bound front, we point to a class of alternating Büchi automata for which the breakpoint construction can be replaced by a simpler n2 n construction. Our class, of ordered alternating Büchi automata, strictly contains the class of very-weak alternating automata, for which an n2 n construction is known.

The Blow-Up in Translating LTL to Deterministic Automata

“…In [BKR10], the authors define the class of ordered alternating automata. In ordered automata, the non-accepting states of the automaton are ordered, and transitions between non-accepting states must respect the order.…”

“…Unlike general alternating automata, for which removal of alternation involves that break-point construction and a 3 n blow up [MH84], alternation of ordered automata (as well as very weak alternating automata, which are a special case of ordered automata [GO01]) can be removed with only an n2 n blow-up. Moreover, it is shown in [BKR10] that for ordered automata with m letters, alternation can be removed with a 2 m+n blow-up, in a construction that makes use of the fact that the break-point construction can be based on subsets of letters rather than subsets of states. Our results here motivate further study of constructions that explicitly refer to the set of letters.…”

Co-Büching Them All

“…As in the case of translating an NBW, we can further determinize the resulting augmented subset construction, getting a 3 n upper bound for the translation of NSW to DCW. Both bounds are asymptotically tight, having matching lower bounds by the special cases of translating NBW to NCW [2] and NCW to DCW [3]. The above good news apply also to the parity and the generalized-Büchi acceptance conditions, as they are special cases of the Streett condition.…”

“…By [3], one cannot avoid the 3 n state blow-up for translating an NCW to a DCW. Since this lower bound clearly holds also for the stronger conditions, we can conclude with the following.…”

From LTL Formulae to Büchi Automata: A Direct Translation Using On-the-Fly De-Generalization

“…Finally, the TGBA is degeneralized into a BA. The time complexity of translating an LTL formula into a VWAA is O(n2 n ) (n is the number of states in VWAA) [6], which is the same magnitude as LTL-model checking algorithm. A. Duret-Lutz [7] introduced many improvements to improve performance of the algorithm proposed by J. M. Couvreur [8].…”