Weighted model counting beyond two-variable logic.

It was recently shown by van den Broeck at al. that the symmetric weighted first-order model counting problem (WFOMC) for sentences of two-variable logic FO2 is in polynomial time, while it is #P-1-complete for some FO3-sentences. We extend the result for FO2 in two independent directions: to sentences of the form phi Lambda for all x there exists(=1)y psi(x, y) with phi and psi formulated in FO2 and to sentences of the uniform one-dimensional fragment U-1 of FO, a recently introduced extension of two-variable logic with the capacity to deal with relation symbols of all arities. We note that the former generalizes the extension of FO2 with a functional relation symbol. We also identify a complete classification of first-order prefix classes according to whether WFOMC is in polynomial time or #P-1-complete.