A Toolkit for Proving Limitations of the Expressive Power of Logics

Zero-one laws, Ehrenfeucht-Fraisse games, locality results, and logical reductions belong to the, by now, standard methods of Finite Model Theory, used for showing non-expressibility in certain logics (cf., e.g., the textbooks [1,2] or the entries in the Encyclopedia of Database Systems [3]). More recently, the close connections between logic and circuits, along with strong lower bound results obtained in circuit complexity, have led to new lower bounds on the expressiveness of logics (cf., e.g., [4,5,6,7]). In particular, [4] solved a long standing open question of Finite Finite Model Theory, asking about the strictness of the bounded variable hierarchy of first-order logic on finite ordered graphs.