Embedded finite models, stability theory, and the impact of order

We extend bounds on the expressive power of first-order logic over finite structures and over ordered finite structures, by generalizing to the situation where the finite structures are embedded in an infinite structure M, where M satisfies some simple combinatorial properties studied in model-theoretic stability theory. We first consider first-order logic over finite structures embedded in a stable structure, and show that it has the same generic expressive power as first-order logic on unordered finite structures. It follows from this that having the additional structure of, for example, an abelian group or an equivalence relation, does not allow one to define any new generic queries. We also consider first-order logic over finite structures living within any model M that lacks the independence property and show that its expressive power is bounded by first-order logic over finite ordered structures. This latter result gives an enormous class of structures in which the expressive power of first-order logic is sharply limited; it shows that common queries such as parity and connectivity cannot be defined for finite structures living within structures from this huge class. It also gives a pure combinatorial property of an interpreted structure M that is sufficient to extend results on first-order logic on ordered structures to first-order logic on finite structures embedded in M