On counting logics and local properties

The expressive power of first-order logic over finite structures is limited in two ways: it lacks a recursion mechanism, and it cannot count. Overcoming the first limitation has been a subject of extensive study. A number of fixpoint logics have been introduced, and shown to be subsumed by an infinitary logic L_∞ω^ω . This logic is easier to analyze than fixpoint logics, and it still lacks counting power, as it has a 0-1 law. On the counting side, there is no analog of L_∞ω^ω. There are a number of logics with counting power, usually introduced via generalized quantifiers. Most known expressivity bounds are based on the fact that counting extensions of first-order logic preserve the locality properties. This paper has three main goals. First, we introduce a new logic L_∞ω*(C) that plays the same role for counting as L_∞ω^ω does for recursion-it subsumes a number of extensions of first-order logic with counting, and has nice properties that make it easy to study. Second, we give a simple direct proof that L_∞ω*(C) expresses only local properties: those that depend on the properties of small neighborhoods, but cannot grasp a structure as a whole. This is a general way of saying that a logic lacks a recursion mechanism. Third, we consider a finer analysis of locality of counting logics. In particular, we address the question of how local a logic is, that is, how big are those neighborhoods that local properties depend on. We get a uniform answer for a variety of logics between first-order and L_∞ω*(C). This is done by introducing a new form of locality that captures the tightest condition that the duplicator needs to maintain in order to win a game. We use this technique to give bounds on outputs of L_∞ω*(C)-definable queries. We also specialize some of the results for structures of small degree