Ehrenfeucht-Fraissé Goes Automatic for Real Addition

Various logical theories can be decided by automata-theoretic methods. Notable examples are Presburger arithmetic FO$(Z,+,<)$ and the linear arithmetic over the reals FO$(R,+,<)$, for which effective decision procedures can be built using automata. Despite the practical use of automata to decide logical theories, many research questions are still only partly answered in this area. One of these questions is the complexity of such decision procedures and the related question about the minimal size of the automata of the languages that can be described by formulas in the respective logic. In this paper, we establish a double exponential upper bound on the automata size for FO$(R,+,<)$ and an exponential upper bound for the discrete order over the integers FO$(Z,<)$. The proofs of these upper bounds are based on Ehrenfeucht-Fraiss{'\e} games. The application of this mathematical tool has a similar flavor as in computational complexity theory, where it can often be used to establish tight upper bounds of the decision problem for logical theories.