Model Theory Makes Formulas Large

Abstract

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.The classical Łoś-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 Łoś-Tarski theorem have been proved for certain classes of finite structures, among them the class of finite acyclic structures and more generally classes of structures of bounded tree width. Our lower bound also applies to these variants.We further prove that a version of the Feferman-Vaught theorem based on a restriction by formula length necessarily entails a non-elementary blow-up in formula size.All these results are based on a similar technique of encoding large numbers by trees of small height in such a way that small formulas can speak about these numbers. 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.