The size of a formula as a measure of complexity
  We introduce a refinement of the usual Ehrenfeucht-Fra\"{\i}ss\'e game. The new game will help us make finer distinctions than the traditional one. In particular, it can be used to measure the size formulas needed for expressing a given property. We will give two versions of the game: the first version characterizes the size of formulas in propositional logic, and the second version works for first-order predicate logic. 
Lauri Hella and Jouko V\"a\"an\"anen
math.LO cs.LO