Weighted model counting beyond two-variable logic

TL;DR

This paper extends the understanding of weighted first-order model counting (WFOMC) complexity beyond two-variable logic, identifying new tractable cases and classifying prefix classes by computational difficulty.

Contribution

It generalizes polynomial-time WFOMC results for FO2 to more complex logical fragments and provides a complete classification of prefix classes by complexity.

Findings

WFOMC remains polynomial for certain FO2 extensions.

WFOMC is Sharp-P_1 complete for some FO3 sentences.

A full classification of prefix classes by WFOMC complexity.

Abstract

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 Sharp-P_1 complete for some FO3-sentences. We extend the result for FO2 in two independent directions: to sentences of the form "phi and \forall\exists^=1 psi" with phi and psi in FO2, and to sentences formulated in the uniform one-dimensional fragment of FO, a recently introduced extension of two-variable logic with the capacity to deal with relation symbols of all arities. 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 Sharp-P_1 complete.