Logics capturing local properties
ACM Transactions on Computational Logic
Well-known theorems of Hanf's and Gaifman's establishing locality of rst-order de nable properties have been used in many applications. These theorems were recently generalized to other logics, which led to new applications in descriptive complexity and database theory. However, a logical characterization of local properties that correspond to Hanf's and Gaifman's theorems, is still lacking. Such a characterization only exists for structures of bounded valence. In this paper, we give logical
... we give logical characterizations of local properties behind Hanf's and Gaifman's theorems. We rst deal with an in nitary logic with counting terms and quanti ers, that is known to capture Hanflocality on structures of bounded valence. We show that testing isomorphism of neighborhoods can be added to it without violating Hanflocality, while increasing its expressive power. We then show that adding local second-order quanti cation to it captures precisely all Hanf-local properties. To capture Gaifman-locality, one must also add a (potentially in nite) case statement. We further show that the hierarchy based on the number of variants in the case statement is strict. ? Part of this work done while visiting INRIA.