Succinct Definitions in the First Order Theory of Graphs II: No Quantifier Alternation

We say that a first order sentence A defines a digraph G if A is true on G but false on any digraph non-isomorphic to G. Let Da(G) (resp. La(G)) denote the minimum quantifier rank (resp. length) of a such sentence in which negations occur only in front of atomic subformulas and any sequence of nested quantifiers has at most a quantifier alternations. We define the succinctness function qa(n) to be the minimum Da(G) over all digraphs on n vertices. In the preceding paper we proved that q3(n) cannot be bounded from below by any computable nondecreasing function growing to the infinity. We also showed that q0(n) ≤ 2 log ∗ n + O(1) for infinitely many n, where log n equals the minimum number of iterations of the binary logarithm sufficient to lower n to 1 or below. Here we prove a lower bound q0(n) ≥ log ∗ n − log log n − O(1) for all n. We derive it from two facts — a relationship Da(G) ≥ (1−o(1)) log ∗ La(G) and the known finite model property of the Bernays-Schönfinkel class of formulas.