Succinct definitions in the first order theory of graphs

Oleg Pikhurko, Joel Spencer, Oleg Verbitsky

Research output: Contribution to journalArticlepeer-review


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 such a 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 bo 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 nondecreasing function bounding q(n) from above we have q* = (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 restriciton of the class of graphs to trees. Under this restriction, for q(n) we laso 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 this, for q0(n) we still have a kind of log-star upper bound: q0(n) ≤ 2 log* n + O(1) for definitely many n.

Original languageEnglish (US)
Pages (from-to)74-109
Number of pages36
JournalAnnals of Pure and Applied Logic
Issue number1-3
StatePublished - May 2006


  • Definability
  • Finite graphs
  • First order logic
  • Turing machine simulation

ASJC Scopus subject areas

  • Logic


Dive into the research topics of 'Succinct definitions in the first order theory of graphs'. Together they form a unique fingerprint.

Cite this