The long search for an optimal complementation construction for Büchi automata climaxed with the work of Schewe, who proposed a worst-case optimal rank-based procedure that generates complements of a size matching the theoretical lower bound of (0.76n) n , modulo a polynomial factor of O(n 2 ). Although worst-case optimal, the procedure in many cases produces automata that are unnecessarily large. In this paper, we propose several ways of how to use the direct and delayed simulation relations to reduce the size of the automaton obtained in the rank-based complementation procedure. Our techniques are based on either (i) ignoring macrostates that cannot be used for accepting a word in the complement or (ii) saturating macrostates with simulation-smaller states, in order to decrease their total number. We experimentally showed that our techniques can indeed considerably decrease the size of the output of the complementation. ACM Subject Classification Theory of computation → Automata over infinite objectsBüchi automata (BA) complementation is a fundamental problem in formal verification, from both theoretical and practical angles. It is, for instance, a critical step in language inclusion testing, which is used in automata-based program termination analysis [15,8], or a component of decision procedures of some logics, such S1S capturing a decidable fragment of second-order arithmetic [5] or the temporal logics ETL and QPTL [32].The study of the BA complementation problem can be traced back to 1962, when Büchi introduced his automaton model in the seminal paper [5] in the context of a decision procedure for the S1S fragment of second-order arithmetic. In the paper, a doubly exponential complementation algorithm based on the infinite Ramsey theorem is proposed. In 1988, Safra [29] introduced a complementation procedure with an n O(n) upper bound and, in the same year, Michel [25] established an n! lower bound. From the traditional theoretical point of view, the problem was already solved, since exponents in the two bounds matched under the O notation (recall that n! is approximately (n/e) n ). From a more practical point of view, a linear factor in an exponent has a significant impact on real-world applications. It was established that the upper bound of Safra's construction is 2 2n , so the hunt for an

We present the tool Ranker for complementing Büchi automata (BAs). Ranker builds on our previous optimizations of rank-based BA complementation and pushes them even further using numerous heuristics to produce even smaller automata. Moreover, it contains novel optimizations of specialized constructions for complementing (i) inherently weak automata and (ii) semi-deterministic automata, all delivered in a robust tool. The optimizations significantly improve the usability of Ranker, as shown in an extensive experimental evaluation with real-world benchmarks, where Ranker produced in the majority of cases a strictly smaller complement than other state-of-the-art tools.

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We consider the problem of approximate reduction of non-deterministic automata that appear in hardware-accelerated network intrusion detection systems (NIDSes). We define an error distance of a reduced automaton from the original one as the probability of packets being incorrectly classified by the reduced automaton (wrt the probabilistic distribution of packets in the network traffic). We use this notion to design an approximate reduction procedure that achieves a great size reduction (much beyond the state-of-the-art language preserving techniques) with a controlled and small error. We have implemented our approach and evaluated it on use cases from Snort, a popular NIDS. Our results provide experimental evidence that the method can be highly efficient in practice, allowing NIDSes to follow the rapid growth in the speed of networks.

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