We investigate lifted inference on ordered domains with predecessor relations, where the elements of the domain respect a total (cyclic) order, and every element has a distinct (clockwise) predecessor. Previous work has explored this problem through weighted first-order model counting (WFOMC), which computes the weighted sum of models for a given first-order logic sentence over a finite domain. In WFOMC, the order constraint is typically encoded by the linear order axiom introducing a binary predicate in the sentence to impose a linear ordering on the domain elements. The immediate and second predecessor relations are then encoded by the linear order predicate. Although WFOMC with the linear order axiom is theoretically tractable, existing algorithms struggle with practical applications, particularly when the predecessor relations are involved. In this paper, we treat predecessor relations as a native part of the axiom and devise a novel algorithm that inherently supports these relations. The proposed algorithm not only provides an exponential speedup for the immediate and second predecessor relations, which are known to be tractable, but also handles the general k-th predecessor relations. The extensive experiments on lifted inference tasks and combinatorics math problems demonstrate the efficiency of our algorithm, achieving speedups of a full order of magnitude.

SUMMARY: Microbiome research is now moving beyond the compositional analysis of microbial taxa in a sample. Increasing evidence from large human microbiome studies suggests that functional consequences of changes in the intestinal microbiome may provide more power for studying their impact on inflammation and immune responses. Although 16S rRNA analysis is one of the most popular and a cost-effective method to profile the microbial compositions, marker-gene sequencing cannot provide direct information about the functional genes that are present in the genomes of community members. Bioinformatic tools have been developed to predict microbiome function with 16S rRNA gene data. Among them, PICRUSt2 (Phylogenetic Investigation of Communities by Reconstruction of Unobserved States) has become one of the most popular functional profile prediction tools, which generates community-wide pathway abundances. However, no state-of-art inference tools are available to test the differences in pathway abundances between comparison groups. We have developed ggpicrust2, an R package, for analyzing functional profiles derived from 16S rRNA sequencing. This powerful tool enables researchers to conduct extensive differential abundance analyses and generate visually appealing visualizations that effectively highlight functional signals. With ggpicrust2, users can obtain publishable results and gain deeper insights into the functional composition of their microbial communities. AVAILABILITY AND IMPLEMENTATION: The package is open-source under the MIT and file license and is available at CRAN and https://github.com/cafferychen777/ggpicrust2. Its shiny web is available at https://a95dps-caffery-chen.shinyapps.io/ggpicrust2_shiny/.