Succinct definitions in the first order theory of graphs

We say that a first order sentence A defines a graph G if A is true on G but false on any graph non-isomorphic to G. Let L(G) (resp. D(G)) denote the minimum length (resp. quantifier rank) of a such sentence. We define the succinctness function s(n) (resp. its variant q(n)) to be the minimum L(G) (resp. D(G)) over all graphs on n vertices. We prove that s(n) and q(n) may be so small that for no general recursive function f we can have f(s(n)) ≥ n for all n. However, for the function q∗(n) = maxi≤n q(i), which is the least monotone nondecreasing function bounding q(n) from above, we have q∗(n) = (1 + o(1)) log n, where log n equals the minimum number of iterations of the binary logarithm sufficient to lower n to 1 or below. We show an upper bound q(n) < log n + 5 even under the restriction of the class of graphs to trees. Under this restriction, for q(n) we also have a matching lower bound. We show a relationship D(G) ≥ (1− o(1)) log L(G) and prove, using the upper bound for q(n), that this relationship is tight. For a non-negative integer a, let Da(G) and qa(n) denote the analogs of D(G) and q(n) for defining formulas in the negation normal form with at most a quantifier alternations in any sequence of nested quantifiers. We show a superrecursive gap between D0(G) and D3(G) and hence between D0(G) and D(G). Despite it, for q0(n) we still have a kind of log-star upper bound: q0(n) ≤ 2 log n + O(1) for infinitely many n. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 152133890. Web: http://www.math.cmu.edu/~pikhurko/ Courant Institute, New York University, New York, NY 10012. E-mail: [email protected] Dept. of Mechanics & Mathematics, Kyiv University, Ukraine. E-mail: [email protected]