We study some existing techniques for solving mean-payoff games (MPGs), improve them, and design a randomized algorithm for solving MPGs with currently the best expected complexity.
We study observation-based strategies for two-player turn-based games on graphs with omega-regular objectives. An observation-based strategy relies on incomplete information about the history of a play, namely, on the past sequence of observations. Such games occur in the synthesis of a controller that does not see the private state of the plant. Our main results are twofold. First, we give a fixpoint algorithm for computing the set of states from which a player can win with a deterministic observation-based strategy for any omega-regular objective. The fixpoint is computed in the lattice of antichains of state sets. This algorithm has the advantages of being directed by the objective and of avoiding an explicit subset construction on the game graph. Second, we give an algorithm for computing the set of states from which a player can win with probability 1 with a randomized observationbased strategy for a Büchi objective. This set is of interest because in the absence of complete information, randomized strategies are more powerful than deterministic ones. We show that our algorithms are optimal by proving matching lower bounds. *
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.
Abstract. We present Acacia+, a tool for solving the LTL realizability and synthesis problems. We use recent approaches that reduce these problems to safety games, that can be solved efficiently by symbolic incremental algorithms based on antichains. The reduction to safety games offers very interesting properties in practice: the construction of compact solutions (when they exist) and a compositional approach for large conjunctions of LTL formulas. Our tool does not use BDDs but rather efficiently treat the underlying antichains. LTL realizability and synthesis problemsThe realizability problem is central when reasoning about specifications for reactive systems: the uncontrollable input signals are generated by the environment whereas the controllable output signals are generated by the system which tries to satisfy the specification against any behavior of the environment. Formally, the LTL realizability problem is stated as a two-player game as follows. Let φ be an LTL formula over a set P partitioned into O (output signals controlled by Player O, the system) and I (input signals controlled by Player I, the environment). In the first round of the play, Player O starts 1 by giving a subset o 1 ⊆ O and Player I reponds by giving a subset i 1 ⊆ I. Then the second round starts, Player O gives o 2 ⊆ O and Player I reponds by i 2 ⊆ I, and so on for an infinite number of rounds. The outcome of this interaction is the infinite word w = (Player O wins the play if w satisfies φ, otherwise Player I wins. The realizability problem asks to decide whether Player O has a winning strategy to satisfy φ against any strategy of Player I. The LTL synthesis problem asks to produce such a winning strategy when φ is realizable. Both problems have been first studied in the seminal works by Pnueli and Rosner [18], and Abadi, Lamport and Wolper [7]. The proposed solution is based on the costly Safra's procedure for the determinization of Rabin automata [20]. The LTL realizability problem is 2ExpTime-Complete and it is known that finite-memory strategies suffice to win the realizability game [18,19]. In [16], Kupferman and Vardi proposed a so called Safraless procedure that avoids the determinization step by reducing the LTL realizability problem to Büchi games. It has been implemented in the tool Lily [3,15]. Another Safraless approach has been recently proposed in [21] for the distributed LTL synthesis problem. It is based on a novel emptiness-preserving translation from LTL to safety tree automata. In [10,5], Elhers proposed a procedure for LTL synthesis problem, implemented in the tool Unbeast, based on the approach of [21] and symbolic game solving with BDDs.
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