GALE · Games and Automata for Logic Extensions

This proposal aims at generalising the central decidability results of Büchi and Rabin to more general settings, and understanding the consequences of those extensions both at theoretical and applicative levels.The original results of Büchi and Rabin state the decidability of monadic second-order logic (monadic logic for short) over infinite words and trees respectively. Those results are of such importance that Rabin's theorem is also called the `Mother of all decidability results'.The primary goal of this project is to demonstrate that it is possible to go significantly beyond the expressiveness of monadic logic, while retaining similar decidability results. We are considering extensions in two distinct directions. The first consists of enriching the logic with the ability to speak in a weak form about set cardinality. The second direction is an extension of monadic logic by topological capabilities. Those two branches form the core of the proposal.The second aspect of this proposal is the study of the `applicability' of this theory. Three tasks are devoted to this. The first task is to precisely determine the cost of using the more complex techniques we will be developing rather than using the classical theory. The second task will be devoted to the description and the study of weaker formalisms allowing better complexities, namely corresponding temporal logics. Finally, the last task will be devoted to the study of related model checking problems.The result of the completion of this program would be twofold. At a theoretical level, new deep results would be obtained, and new techniques in automata, logic and games would be developed for solving them. At a more applicative level, this program would define and validate new directions of research in the domain of the verification of open systems and related problems.