Model theory makes formulas large

Gaifman’s locality theorem states that every first-order sentence is equivalent to a local sentence. We show that there is no elementary bound on the length of the local sentence in terms of the original. Gaifman’s theorem is an essential ingredient in several algorithmic meta theorems for first order logic. Our result has direct implications for the running time of the algorithms. The classical Ło´s-Tarski theorem states that every first-order sentence preserved under extensions is equivalent to an existential sentence. We show that there is no elementary bound on the length of the existential sentence in terms of the original. Recently, variants of the Ło´s-Tarski theorem have been proved for certain classes of finite structures, among them the class of finite trees and more generally classes of structures of bounded tree width. Our lower bound also applies to these variants. The first-order theory of trees is decidable. We prove that there is no elementary decision algorithm. Notably, our lower bounds do not apply to restrictions of the results to structures of bounded degree. For such structures, we obtain elementary upper bounds in all cases. However, even there we can prove at least doubly exponential lower bounds.