Faster decision of first-order graph properties

First-order logic captures a vast number of computational problems on graphs. We study the time complexity of deciding graph properties definable by first-order sentences in prenex normal form with k variables. The trivial algorithm for this problem runs in O(nk) time on n-node graphs (the big-O hides the dependence on k). Answering a question of Miklós Ajtai, we give the first algorithms running faster than the trivial algorithm, in the general case of arbitrary first-order sentences and arbitrary graphs. One algorithm runs in O(nk-3+ω) ≤ O(nk-0.627) time for all k ≥ 3, where ω < 2.373 is the n x n matrix multiplication exponent. By applying fast rectangular matrix multiplication, the algorithm can be improved further to run in nk-1+o(1) time, for all k ≥ 9. Finally, we observe that the exponent of k - 1 is optimal, under the popular hypothesis that CNF satisfiability with n variables and m clauses cannot be solved in (2 - ε)n · poly(m) time for some ε > 0.