Notions of locality and their logical characterizations over finite models

Many known tools for proving expressibility bounds for first-order logic ar e based on one of several locality properties. In this paper we characteriz e the relationship between those notions of locality We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and o ne with open formulae. We prove that the Former implies Hanf's notion of lo cality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property which is one of the easiest tool s For proving expressibility bounds. These results apply beyond the first-o rder case. We use them to derive expressibility bounds for first-order logi c with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree.