Model Enumeration of Two-Variable Logic with Quadratic Delay Complexity

We study the model enumeration problem of the function-free, finite domain fragment of first-order logic with two variables (FO²). Specifically, given an FO² sentence Γ and a positive integer n, how can one enumerate all the models of Γ over a domain of size n? In this paper, we devise a novel algorithm to address this problem. The delay complexity, the time required between producing two consecutive models, of our algorithm is quadratic in the given domain size n (up to logarithmic factors) when the sentence is fixed. This complexity is almost optimal since the interpretation of binary predicates in any model requires at least Ω(n²) bits to represent.