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.

Abstract. In this paper, we study the structure of underlying automata based constructions for solving the LTL realizability and synthesis problem. We show how to reduce the LTL realizability problem to a game with an observer that checks that the game visits a bounded number of times accepting states of a universal co-Büchi word automaton. We show that such an observer can be made deterministic and that this deterministic observer has a nice structure which can be exploited by an incremental algorithm that manipulates antichains of game positions. We have implemented this new algorithm and our first results are very encouraging.

In this paper, we present new monolithic and compositional algorithms to solve the LTL realizability problem. Those new algorithms are based on a reduction of the LTL realizability problem to a game whose winning condition is defined by a universal automaton on infinite words with a k-co-Büchi acceptance condition. This acceptance condition asks that runs visit at most k accepting states, so it implicitly defines a safety game. To obtain efficient algorithms from this construction, we need several additional ingredients. First, we study the structure of the underlying automata constructions, and we show that there exists a partial order that structures the state space of the underlying safety game. This partial order can be used to define an efficient antichain algorithm. Second, we show that the algorithm can be implemented in an incremental way by considering increasing values of k in the acceptance condition. Finally, we show that for large LTL formulas that are written as conjunctions of smaller formulas, we can solve the problem compositionally by first computing winning strategies for each conjunct that appears in the large formula. We report on the behavior of those algorithms on several benchmarks. We show that the compositional algorithms are able to handle LTL formulas that are several pages long.

We extend the quantitative synthesis framework by going beyond the worst-case. On the one hand, classical analysis of two-player games involves an adversary (modeling the environment of the system) which is purely antagonistic and asks for strict guarantees. On the other hand, stochastic models like Markov decision processes represent situations where the system is faced to a purely randomized environment: the aim is then to optimize the expected payoff, with no guarantee on individual outcomes. We introduce the beyond worst-case synthesis problem, which is to construct strategies that guarantee some quantitative requirement in the worst-case while providing a higher expected value against a particular stochastic model of the environment given as input. This problem is relevant to produce system controllers that provide nice expected performance in the everyday situation while ensuring a strict (but relaxed) performance threshold even in the event of very bad (while unlikely) circumstances. We study the beyond worst-case synthesis problem for two important quantitative settings: the meanpayoff and the shortest path. In both cases, we show how to decide the existence of finite-memory strategies satisfying the problem and how to synthesize one if one exists. We establish algorithms and we study complexity bounds and memory requirements.

The classical LTL synthesis problem is purely qualitative: the given LTL specification is realized or not by a reactive system. LTL is not expressive enough to formalize the correctness of reactive systems with respect to some quantitative aspects. This paper extends the qualitative LTL synthesis setting to a quantitative setting. The alphabet of actions is extended with a weight function ranging over the rational numbers. The value of an infinite word is the mean-payoff of the weights of its letters. The synthesis problem then amounts to automatically construct (if possible) a reactive system whose executions all satisfy a given LTL formula and have mean-payoff values greater than or equal to some given threshold. The latter problem is called LTL MP synthesis and the LTL MP realizability problem asks to check whether such a system exists. We first show that LTL MP realizability is not more difficult than LTL realizability: it is 2ExpTime-Complete. This is done by reduction to two-player mean-payoff parity games. While infinite memory strategies are required to realize LTL MP specifications in general, we show that -optimality can be obtained with finite memory strategies, for any > 0. To obtain an efficient algorithm in practice, we define a Safraless procedure to decide whether there exists a finite-memory strategy that realizes a given specification for some given threshold. This procedure is based on a reduction to two-player energy safety games which are in turn reduced to safety games. Finally, we show that those safety games can be solved efficiently by exploiting the structure of their state spaces and by using antichains as a symbolic data-structure. All our results extend to multi-dimensional weights. We have implemented an antichain-based procedure and we report on some promising experimental results.