Final Report Summary - GALE (Games and Automata for Logic Extensions) The ERC project GALE has brought several substantial contributions to the field of automata theory of algorithmic logic. Generally speaking, this branch deals with the representation, the transformation, and the resolution of properties involving infinitely many infinite objects. Such tools are particularly well suited for verifying the correctness of systems taking into account the infinitely many possible executions it has, or the infinitely many inputs it may be used on. The project established in particular strong theoretical foundations for several non-conventional forms of automata, algebras or logics that are usable in such contexts. A major achievement was the development of the theory of regular cost functions, an extension of regular languages to a quantitative setting that enjoy many of the good properties of the classical cases. Regular cost functions have been developed over finite words, infinite words, finite trees, and partially infinite trees (a standard tool in the verification of hardware devices interacting with the environment for instance). The results can be used as a toolbox for deciding properties involving optimization such as bounding of fixpoints iterations, or constraining consumption of systems. The project has both developed the theory, potential applications in the field of databases or automata theory, and extensions.Other achievements, among others, involve the precise analysis of quantitative models of automata, that can be used for very accurately estimating the termination time of pieces of program, the fine understanding of logics over linear orders (that are good for representing time), or results on data languages that can be used to model some databases or system logs where many agents interact.
