Notions of Locality and their Logical Characterizations over Finite Models

Most proofs showing limitations of expressive power of first-order logic rely on Ehrenfeucht-Fraisse games. Playing the game often involves a nontrivial combinatorial argument, so it was proposed to find easier tools for proving expressivity bounds. Most of those known for first-order logic are based on its "locality", that is defined in different ways. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notion: One deals with sentences and one with open formulae. Each of these implies the bounded degree property, which is one of the easiest tools for proving expressivity bounds. These results apply beyond the first-order case. We use them to derive expressivity bounds for first-order logic with unary quantifiers and counting. Finally, we characterize these notions of locality on structures of small degree.