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# Faster lifting for two-variable logic using cell graphs

*Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence*, PMLR 161:1393-1402, 2021.

#### Abstract

We consider the weighted first-order model counting (WFOMC) task, a problem with important applications to inference and learning in structured graphical models. Bringing together earlier work [Van den Broeck et al., 2011, 2014], a formal proof was given by Beame et al. [2015] showing that the two-variable fragment of first-order logic, FO^2, is domain-liftable, meaning it admits an algorithm for WFOMC whose runtime is polynomial in the given domain size. However, applying this theoretical upper bound is often impractical for real-world problem instances. We show how to adapt their proof into a fast algorithm for lifted inference in FO^2, using only off-the-shelf tools for knowledge compilation, and several careful optimizations involving the cell graph of the input sentence, a novel construct we define that encodes the interactions between the cells of the sentence. Experimental results show that, despite our approach being largely orthogonal to that of Forclift [Van den Broeck et al., 2011], our algorithm often outperforms it, scaling to larger domain sizes on more complex input sentences.