The complementation problem for nondeterministic word automata has numerous applications in formal verification. In particular, the language-containment problem, to which many verification problems are reduced, involves complementation. For automata on finite words, which correspond to safety properties, complementation involves determinization. The 2n blow-up that is caused by the subset construction is justified by a tight lower bound. For Büchi automata on infinite words, which are required for the modeling of liveness properties, optimal complementation constructions are quite complicated, as the subset construction is not sufficient. We review here progress on this problem, which dates back to its introduction in Büchi's seminal 1962 paper.

Automata on infinite objects are extensively used in system specification, verification, and synthesis. While some applications of the automata-theoretic approach have been well accepted by the industry, some have not yet been reduced to practice. Applications that involve determinization of automata on infinite words have been doomed to belong to the second category. This has to do with the intricacy of Safra’s optimal determinization construction, the fact that the state space that results from determinization is awfully complex and is not amenable to optimizations and a symbolic implementation, and the fact that determinization requires the introduction of acceptance conditions that are more complex than the Büchi acceptance condition. Examples of applications that involve determinization and belong to the unfortunate second category include model checking of ω-regular properties, decidability of branching temporal logics, and synthesis and control of open systems. We offer an alternative to the standard automatatheoretic approach. The crux of our approach is avoiding determinization. Our approach goes instead via universal co-Büchi automata. Like nondeterministic automata, universal automata may have several runs on every input. Here, however, an input is accepted if all of the runs are accepting. We show how the use of universal automata simplifies significantly known complementation constructions for automata on infinite words, known decision procedures for branching temporal logics, known synthesis algorithms, and other applications that are now based on determinization. Our algorithms are less difficult to implement and have practical advantages like being amenable to optimizations and a symbolic implementation.

Abstract. In this paper, we first introduce a new lower bound technique for the state complexity of transformations of automata. Namely we suggest considering the class of full automata in lower bound analysis. Then we apply such technique to the complementation of nondeterministic ωautomata and obtain several lower bound results. Particularly, we prove an Ω((0.76n) n) lower bound for Büchi complementation, which also holds for almost every complementation and determinization transformation of nondeterministic ω-automata, and prove an optimal (Ω(nk)) n lower bound for the complementation of generalized Büchi automata, which holds for Streett automata as well. 1

This article studies the expressive power of finite-state automata recognizing sets of real numbers encoded positionally. It is known that the sets that are definable in the first-order additive theory of real and integer variables 〈R, Z, +, < 〉 can all be recognized by weak deterministic Büchi automata, regardless of the encoding base r> 1. In this article, we prove the reciprocal property, i.e., that a subset of R that is recognizable by weak deterministic automata in every base r> 1 is necessarily definable in 〈R, Z, +, <〉. This result generalizes to real numbers the well-known Cobham’s theorem on the finite-state recognizability of sets of integers. Our proof gives interesting insight into the internal structure of automata recognizing sets of real numbers, which may lead to efficient data structures for handling these sets.

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. 1