Abstract. We propose and evaluate a new algorithm for checking the universality of nondeterministic finite automata. In contrast to the standard algorithm, which uses the subset construction to explicitly determinize the automaton, we keep the determinization step implicit. Our algorithm computes the least fixed point of a monotone function on the lattice of antichains of state sets. We evaluate the performance of our algorithm experimentally using the random automaton model recently proposed by Tabakov and Vardi. We show that on the difficult instances of this probabilistic model, the antichain algorithm outperforms the standard one by several orders of magnitude. We also show how variations of the antichain method can be used for solving the language-inclusion problem for nondeterministic finite automata, and the emptiness problem for alternating finite automata. 1

We propose and evaluate new algorithms to solve the universality and language inclusion problems for nondeterministic Büchi automata. To obtain those new algorithms, we establish the existence of pre-orders that can be exploited to efficiently evaluate fixed points on the automata defined during the complementation step (that we keep implicit in our approach). We evaluate the performance of the new algorithm to check the universality of Büchi automata using the random automaton model recently proposed by Tabakov and Vardi. We show that on the difficult instances of this probabilistic model, our algorithm outperforms the standard ones by several orders of magnitude.

Abstract. We compare tools for complementing nondeterministic Büchi automata with a recent termination-analysis algorithm. Complementation of Büchi automata is a key step in program verification. Early constructions using a Ramsey-based argument have been supplanted by rank-based constructions with exponentially better bounds. In 2001 Lee et al. presented the size-change termination (SCT) problem, along with both a reduction to Büchi automata and a Ramsey-based algorithm. This algorithm strongly resembles the initial complementation constructions for Büchi automata. We prove that the SCT algorithm is a specialized realization of the Ramseybased complementation construction. Surprisingly, empirical analysis suggests Ramsey-based approaches are superior over the domain of SCT problems. Upon further analysis we discover an interesting property of the problem space that both explains this result and provides a chance to improve rank-based tools. With these improvements, we show that theoretical gains in efficiency are mirrored in empirical performance. 1

Abstract—We introduce bisimulation up to congruence as a technique for proving language equivalence of non-deterministic finite automata. Exploiting this technique, we devise an optimisation of the classical algorithm by Hopcroft and Karp [12] that, instead of computing the whole determinised automata, explores only a small portion of it. Although the optimised algorithm remains exponential in worst case (the problem is PSPACEcomplete), experimental results show improvements of several orders of magnitude over the standard algorithm. I.

This paper is dedicated to candidate abstractions to capture relevant aspects of the integer weighted automata. The expected effect of applying these abstractions is studied to build the deterministic reachability graphs allowing us to semi-decide the positivity problem on these automata. Moreover, the papers reports on the implementations and experimental results, and discusses other encodings.

Abstract. We present a model checking algorithm for HFL1, the firstorder fragment of Higher-Order Fixpoint Logic. This logic is capable of expressing many interesting properties which are not regular and, hence, not expressible in the modal µ-calculus. The algorithm avoids best-case exponential behaviour by localising the computation of functions and can be implemented symbolically using BDDs. We show how insight into the behaviour of this procedure, when run on a fixed formula, can be used to obtain specialised algorithms for particular problems. This yields, for example, the competitive antichain algorithm for NFA universality but also a new algorithm for a string matching problem. 1

Abstract. We present a general theory that exploits simulation relations on transition systems to obtain antichain algorithms for solving the reachability and repeated reachability problems. Antichains are more succinct than the sets of states manipulated by the traditional fixpoint algorithms. The theory justifies the correctness of the antichain algorithms, and applications such as the universality problem for finite automata illustrate efficiency improvements. Finally, we show that new and provably better antichain algorithms can be obtained for the emptiness problem of alternating automata over finite and infinite words. 1

Minimal deterministic finite automata (dfas) can be reduced further at the expense of a finite number of errors. Recently, such minimization algorithms have been improved to run in time O(n log n), where n is the number of states of the input dfa, by [Gawrychowski and Je»: Hyper-minimisation made e cient. Proc. Mfcs, Lncs 5734, 2009] and [Holzer and Maletti: An n log n algorithm for hyper-minimizing a (minimized) deterministic automaton. Theor. Comput. Sci. 411, 2010]. Both algorithms return a dfa that is as small as possible, while only committing a nite number of errors. These algorithms are further improved to return a dfa that commits the least number of errors at the expense of an increased (quadratic) run-time. This solves an open problem of [Badr, Geffert, and Shipman: Hyper-minimizing minimized deterministic finite state automata. Rairo Theor. Inf. Appl. 43, 2009]. In addition, an experimental study on random automata is performed and the effects of the existing algorithms and the new algorithm are reported.

Most temporal logics which have been introduced and studied in the past decades can be embedded into the modal L µ. This is the case for e.g. PDL, CTL, CTL ∗ , ECTL, LTL, etc. and entails that these logics cannot express non-regular program properties. In recent years, some novel approaches towards an increase in expressive power have been made: Fixpoint Logic with Chop enriches L µ with a sequential composition operator and thereby allows to characterise context-free processes. The Modal Iteration Calculus uses inflationary fixpoints to exceed the expressive power of L µ. Higher-Order Fixpoint Logic (HFL) incorporates a simply typed λ-calculus into a setting with extremal fixpoint operators and even exceeds the expressive power of Fixpoint Logic with Chop. But also PDL has been equipped with context-free programs instead of regular ones. In terms of expressivity there is a natural demand for richer frameworks since program property specifications are simply not limited to the regular sphere. Expressivity however usually comes at the price of an increased computational complexity of logic-related decision problems. For instance are the satisfiability problems for the above mentioned logics