As explained above, the research objectives have been organized into three work plans. For the first task, many little negative results have been obtained. Hence, it is too early to disseminate the results that currently would have a minor impact on the state of the art, and it is not possible to predict the impact of future findings. For the second task, contributions have been provided for three classes of games: parity, mean payoff, and discounted payoff. The work "From Quasi-Dominions to Progress Measures" that will appear on "Computation and Automata", provides more insight on the use of efficient progress measure for solving parity games thanks to the integration of quasi dominions and progress measures. The algorithm described in this paper, despite the exponential complexity bound, proved to be the most efficient in practice, since as shown in the experimental section, to scales better than any known algorithm on games with a complex structure. To overcome the exponential worst-case behaviours, it has been developed an integration of the efficient quasi dominion technique called priority promotion and the quasi-polynomial bounded approach proposed by Parys. Such a work submitted to a journal in the work "Priority Promotion with Parysian Flair" describes the most efficient quasi-polynomial algorithm for solving parity games. On the theoretical side, the time complexity bound for parity games has been improved in the journal "Smaller Progress Measures and Separating Automata for Parity Games" that has been published on "Frontiers in Computer Science". In this journal a more succinct progress measure has been proposed, that led to a state space reduction, and then, to a better complexity bound. The journal "Solving Mean-Payoff Games via Quasi Dominions" refined a previous work presented at TACAS 2020, providing a better complexity study and including an experimental section that shows how the proposed approach is very effective and reduces the solution time of several orders of magnitude with respect to other known algorithms. A new approach for solving discounted payoff games has been developed and accepted at GandALF 2023. In "An Objective Improvement Approach to Solving Discounted Payoff Games" the classic strategy improvement approach, that is asymmetric and at every iteration changes the set of constraints, is revisited and turned into a new approach that is symmetric and works on the same set of constraints while changing the objective function. For the last task, the development of the first quantitative framework is still in progress. So far it has been implemented the full set of parity and mean payoff solvers. Once the first version of the framework will be available, it may have significant impact of the development of future algorithms, allowing on one hand to test the solver on worst case families of games and on the other hand to compare it with other solvers. This will provide to the researchers a tool to study the behaviour and to verify the practical efficiency of the different approaches. A side research work on automata theory investigated the good-for-games property of automata. The work "Semantic Flowers for Good-for-Games and Deterministic Automata" proposes alternative proofs of the expressive power of good-for-games automata by means of the application of the notion of "flowers".
