Modern generative models excel at producing realistic samples--images, text, molecules-- but often lack guarantees about structural properties. A protein generator may produce plausible sequences that violate stability constraints; a network topology generator may create graphs that fail connectivity requirements; a molecule generator may output structures violating chemical valence rules. Standard GAN discriminators provide a global "real vs. fake" signal, but they cannot pinpoint which specific structural constraint failed or guarantee that generated samples satisfy formal specifications. Meanwhile, mathematical logic has developed precise tools for reasoning about structural properties. Ehrenfeucht-Fraïssé (EF) games [1, 2] characterize when two structures are indistinguishable by logical formulas up to a given complexity (quantifier depth k). First-order (FO) and monadic second-order (MSO) logics can express rich structural properties--connectivity, bipartiteness, planarity, acyclicity--that are crucial in applications but invisible to standard discriminators.

Accurate semantic segmentation of terrestrial laser scanning (TLS) point clouds is limited by costly manual annotation. We propose a semi-automated, uncertainty-aware pipeline that integrates spherical projection, feature enrichment, ensemble learning, and targeted annotation to reduce labeling effort, while sustaining high accuracy. Our approach projects 3D points to a 2D spherical grid, enriches pixels with multi-source features, and trains an ensemble of segmentation networks to produce pseudo-labels and uncertainty maps, the latter guiding annotation of ambiguous regions. The 2D outputs are back-projected to 3D, yielding densely annotated point clouds supported by a three-tier visualization suite (2D feature maps, 3D colorized point clouds, and compact virtual spheres) for rapid triage and reviewer guidance. Using this pipeline, we build Mangrove3D, a semantic segmentation TLS dataset for mangrove forests. We further evaluate data efficiency and feature importance to address two key questions: (1) how much annotated data are needed and (2) which features matter most. Results show that performance saturates after ~12 annotated scans, geometric features contribute the most, and compact nine-channel stacks capture nearly all discriminative power, with the mean Intersection over Union (mIoU) plateauing at around 0.76. Finally, we confirm the generalization of our feature-enrichment strategy through cross-dataset tests on ForestSemantic and Semantic3D. Our contributions include: (i) a robust, uncertainty-aware TLS annotation pipeline with visualization tools; (ii) the Mangrove3D dataset; and (iii) empirical guidance on data efficiency and feature importance, thus enabling scalable, high-quality segmentation of TLS point clouds for ecological monitoring and beyond. The dataset and processing scripts are publicly available at https://fz-rit.github.io/through-the-lidars-eye/.

The paper proposes a method for recovering scene geometry from a single RGB image. This method uses a dense CRF with terms that enforce consistency between point-wise depth and normal estimates, using regularizers based on classification of planarity and presence of depth boundaries. Each of these estimates (depth, normal, planarity, edges) comes from a separate network proposed for each task in prior work. In addition to the geometry-terms in proposed DCRF-based model, the paper's contributions include using multiple passes through the depth and normal networks with dropout to derive'confidence' values of these metrics, and joint training to fine tune the depth and normal networks. While significantly engineered for its specific application domain, the paper does demonstrate a successful example of inference with a regularized objective, where different terms are predicted from trained neural networks.

We pose the identified classes of problems within the general framework of Weighted Constraint Satisfaction Problems (WCSPs), reformulated as minimum weighted vertex cover problems. We examine the Constraint Composite Graphs (CCGs) associated with these WCSPs and provide simple arguments for establishing their tractability. We construct simple - almost trivial - bipartite graph representations for submodular cost functions, and reformulate these WCSPs as max-flow problems on bipartite graphs. By doing this, we achieve better time complexities than state-of-the-art algorithms. We also use CCGs to exploit planarity in variable interaction graphs, and provide algorithms with significantly improved time complexities for classes of submodular constraints. Moreover, our framework for exploiting planarity is not limited to submodular constraints. Our work confirms the usefulness of studying CCGs associated with combinatorial problems modeled as WCSPs.

Jacob Holm was flipping through proofs from an October 2019 research paper he and colleague Eva Rotenberg--an associate professor in the department of applied mathematics and computer science at the Technical University of Denmark--had published online, when he discovered their findings had unwittingly given away a solution to a centuries-old graph problem. Holm, an assistant professor of computer science at the University of Copenhagen, was relieved no one had caught the solution first. "It was a real'Eureka!' moment," he says. Holm and Rotenberg were trying to find a shortcut for determining whether a graph is "planar"--that is, if it could be drawn flat on a surface without any of its lines crossing each other (flat drawings of a graph are also called "embeddings"). "Putting it very bluntly, we formally quantified why something is a terrible drawing." To mathematicians, a graph often looks different than what most of us are taught in school.