Logics Capturing Local Properties

Abstract

Well-known theorems of Hanf’s and Gaifman’s establishing locality of first-order definable 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 characterizations of local properties behind Hanf’s and Gaifman’s theorems. We first deal with an infinitary logic with counting terms and quantifiers, that is known to capture Hanf-locality on structures of bounded valence. We show that testing isomorphism of neighborhoods can be added to it without violating Hanf-locality, while increasing its expressive power. We then show that adding local second-order quantification to it captures precisely all Hanf-local properties. To capture Gaifman-locality, one must also add a (potentially infinite) case statement. We further show that the hierarchy based on the number of variants in the case statement is strict.

Part of this work was done while visiting INRIA.