Solving Combinatorial Counting Problems with Weighted First-Order Model Counting

Combinatorial counting problems pervade artificial intelligence, statistics, and discrete mathematics. Whether the task is enumerating subsets, multisets, permutations, partitions, or compositions under structural and arithmetic constraints, solving it remains a stubbornly manual exercise. Closed-form derivations are powerful but brittle, while naive encodings to propositional model counting or constraint satisfaction destroy the exchangeability that makes counting tractable in the first place. We present Cofola (COmbinatorial counting LAnguage with First-Order logic), a typed declarative language whose primitives are the combinatorial objects that recur in everyday counting questions, including sets, bags, tuples, sequences, circles, partitions, and compositions, together with natural relational and arithmetic constraints over them. A denotational semantics maps every Cofola program to a well-defined combinatorial counting problem, and a three-phase compilation pipeline (preprocessing, decomposition, and symmetry-preserving encoding) reduces this problem to a weighted first-order model counting (WFOMC) instance augmented with coefficient-extraction constraints. To stay inside known domain-liftable fragments whenever possible, the encoding groups indistinguishable entities, breaks the symmetry of unordered groupings lexicographically, and encodes sequences and circles via order axioms. On a suite of representative combinatorial counting problems, ranging from textbook math problems to multi-object scenarios that the closest prior framework cannot express, Cofola produces concise specifications and a uniform solving pipeline that is practical end-to-end.

On the Expressive Power and Limitations of Multi-Layer SSMs

We study the expressive power and limitations of multi-layer state-space models (SSMs). First, we show that multi-layer SSMs face fundamental limitations in compositional tasks, revealing an inherent gap between SSMs and streaming models. Then, we examine the role of chain-of-thought (CoT), showing that offline CoT does not fundamentally increase the expressiveness, while online CoT can substantially increase its power. Indeed, with online CoT, multi-layer SSMs become equivalent in power to streaming algorithms. Finally, we investigate the tradeoff between width and precision, showing that these resources are not interchangeable in the base model, but admit a clean equivalence once online CoT is allowed. Overall, our results offer a unified perspective on how depth, finite precision, and CoT shape the power and limits of SSMs.

Scene Graph-guided SegCaptioning Transformer with Fine-grained Alignment for Controllable Video Segmentation and Captioning

Recent advancements in multimodal large models have significantly bridged the representation gap between diverse modalities, catalyzing the evolution of video multimodal interpretation, which enhances users' understanding of video content by generating correlated modalities. However, most existing video multimodal interpretation methods primarily concentrate on global comprehension with limited user interaction. To address this, we propose a novel task, Controllable Video Segmentation and Captioning (SegCaptioning), which empowers users to provide specific prompts, such as a bounding box around an object of interest, to simultaneously generate correlated masks and captions that precisely embody user intent. An innovative framework Scene Graph-guided Fine-grained SegCaptioning Transformer (SG-FSCFormer) is designed that integrates a Prompt-guided Temporal Graph Former to effectively captures and represents user intent through an adaptive prompt adaptor, ensuring that the generated content well aligns with the user's requirements. Furthermore, our model introduces a Fine-grained Mask-linguistic Decoder to collaboratively predict high-quality caption-mask pairs using a Multi-entity Contrastive loss, as well as provide fine-grained alignment between each mask and its corresponding caption tokens, thereby enhancing users' comprehension of videos. Comprehensive experiments conducted on two benchmark datasets demonstrate that SG-FSCFormer achieves remarkable performance, effectively capturing user intent and generating precise multimodal outputs tailored to user specifications. Our code is available at https://github.com/XuZhang1211/SG-FSCFormer.

Tractable Weighted First-Order Model Counting with Bounded Treewidth Binary Evidence

The Weighted First-Order Model Counting Problem (WFOMC) asks to compute the weighted sum of models of a given first-order logic sentence over a given domain. Conditioning WFOMC on evidence -- fixing the truth values of a set of ground literals -- has been shown impossible in time polynomial in the domain size (unless $\mathsf{\#P \subseteq FP}$) even for fragments of logic that are otherwise tractable for WFOMC without evidence. In this work, we address the barrier by restricting the binary evidence to the case where the underlying Gaifman graph has bounded treewidth. We present a polynomial-time algorithm in the domain size for computing WFOMC for the two-variable fragments $\text{FO}^2$ and $\text{C}^2$ conditioned on such binary evidence. Furthermore, we show the applicability of our algorithm in combinatorial problems by solving the stable seating arrangement problem on bounded-treewidth graphs of bounded degree, which was an open problem. We also conducted experiments to show the scalability of our algorithm compared to the existing model counting solvers.

AngleRoCL: Angle-Robust Concept Learning for Physically View-Invariant T2I Adversarial Patches

Cutting-edge works have demonstrated that text-to-image (T2I) diffusion models can generate adversarial patches that mislead state-of-the-art object detectors in the physical world, revealing detectors' vulnerabilities and risks. However, these methods neglect the T2I patches' attack effectiveness when observed from different views in the physical world (i.e., angle robustness of the T2I adversarial patches). In this paper, we study the angle robustness of T2I adversarial patches comprehensively, revealing their angle-robust issues, demonstrating that texts affect the angle robustness of generated patches significantly, and task-specific linguistic instructions fail to enhance the angle robustness. Motivated by the studies, we introduce Angle-Robust Concept Learning (AngleRoCL), a simple and flexible approach that learns a generalizable concept (i.e., text embeddings in implementation) representing the capability of generating angle-robust patches. The learned concept can be incorporated into textual prompts and guides T2I models to generate patches with their attack effectiveness inherently resistant to viewpoint variations. Through extensive simulation and physical-world experiments on five SOTA detectors across multiple views, we demonstrate that AngleRoCL significantly enhances the angle robustness of T2I adversarial patches compared to baseline methods. Our patches maintain high attack success rates even under challenging viewing conditions, with over 50% average relative improvement in attack effectiveness across multiple angles. This research advances the understanding of physically angle-robust patches and provides insights into the relationship between textual concepts and physical properties in T2I-generated contents.

Model Enumeration of Two-Variable Logic with Quadratic Delay Complexity

We study the model enumeration problem of the function-free, finite domain fragment of first-order logic with two variables ($FO^2$). Specifically, given an $FO^2$ sentence $\Gamma$ and a positive integer $n$, how can one enumerate all the models of $\Gamma$ over a domain of size $n$? In this paper, we devise a novel algorithm to address this problem. The delay complexity, the time required between producing two consecutive models, of our algorithm is quadratic in the given domain size $n$ (up to logarithmic factors) when the sentence is fixed. This complexity is almost optimal since the interpretation of binary predicates in any model requires at least $\Omega(n^2)$ bits to represent.

Visibility-Uncertainty-guided 3D Gaussian Inpainting via Scene Conceptional Learning

3D Gaussian Splatting (3DGS) has emerged as a powerful and efficient 3D representation for novel view synthesis. This paper extends 3DGS capabilities to inpainting, where masked objects in a scene are replaced with new contents that blend seamlessly with the surroundings. Unlike 2D image inpainting, 3D Gaussian inpainting (3DGI) is challenging in effectively leveraging complementary visual and semantic cues from multiple input views, as occluded areas in one view may be visible in others. To address this, we propose a method that measures the visibility uncertainties of 3D points across different input views and uses them to guide 3DGI in utilizing complementary visual cues. We also employ uncertainties to learn a semantic concept of scene without the masked object and use a diffusion model to fill masked objects in input images based on the learned concept. Finally, we build a novel 3DGI framework, VISTA, by integrating VISibility-uncerTainty-guided 3DGI with scene conceptuAl learning. VISTA generates high-quality 3DGS models capable of synthesizing artifact-free and naturally inpainted novel views. Furthermore, our approach extends to handling dynamic distractors arising from temporal object changes, enhancing its versatility in diverse scene reconstruction scenarios. We demonstrate the superior performance of our method over state-of-the-art techniques using two challenging datasets: the SPIn-NeRF dataset, featuring 10 diverse static 3D inpainting scenes, and an underwater 3D inpainting dataset derived from UTB180, including fast-moving fish as inpainting targets.

An Efficient Unsupervised Framework for Convex Quadratic Programs via Deep Unrolling

Quadratic programs (QPs) arise in various domains such as machine learning, finance, and control. Recently, learning-enhanced primal-dual hybrid gradient (PDHG) methods have shown great potential in addressing large-scale linear programs; however, this approach has not been extended to QPs. In this work, we focus on unrolling "PDQP", a PDHG algorithm specialized for convex QPs. Specifically, we propose a neural network model called "PDQP-net" to learn optimal QP solutions. Theoretically, we demonstrate that a PDQP-net of polynomial size can align with the PDQP algorithm, returning optimal primal-dual solution pairs. We propose an unsupervised method that incorporates KKT conditions into the loss function. Unlike the standard learning-to-optimize framework that requires optimization solutions generated by solvers, our unsupervised method adjusts the network weights directly from the evaluation of the primal-dual gap. This method has two benefits over supervised learning: first, it helps generate better primal-dual gap since the primal-dual gap is in the objective function; second, it does not require solvers. We show that PDQP-net trained in this unsupervised manner can effectively approximate optimal QP solutions. Extensive numerical experiments confirm our findings, indicating that using PDQP-net predictions to warm-start PDQP can achieve up to 45% acceleration on QP instances. Moreover, it achieves 14% to 31% acceleration on out-of-distribution instances.

Bridging Weighted First Order Model Counting and Graph Polynomials

The Weighted First-Order Model Counting Problem (WFOMC) asks to compute the weighted sum of models of a given first-order logic sentence over a given domain. It can be solved in time polynomial in the domain size for sentences from the two-variable fragment with counting quantifiers, known as $C^2$. This polynomial-time complexity is also retained when extending $C^2$ by one of the following axioms: linear order axiom, tree axiom, forest axiom, directed acyclic graph axiom or connectedness axiom. An interesting question remains as to which other axioms can be added to the first-order sentences in this way. We provide a new perspective on this problem by associating WFOMC with graph polynomials. Using WFOMC, we define Weak Connectedness Polynomial and Strong Connectedness Polynomials for first-order logic sentences. It turns out that these polynomials have the following interesting properties. First, they can be computed in polynomial time in the domain size for sentences from $C^2$. Second, we can use them to solve WFOMC with all of the existing axioms known to be tractable as well as with new ones such as bipartiteness, strong connectedness, being a spanning subgraph, having $k$ connected components, etc. Third, the well-known Tutte polynomial can be recovered as a special case of the Weak Connectedness Polynomial, and the Strict and Non-Strict Directed Chromatic Polynomials can be recovered from the Strong Connectedness Polynomials, which allows us to show that these important graph polynomials can be computed in time polynomial in the number of vertices for any graph that can be encoded by a fixed $C^2$ sentence and a conjunction of an arbitrary number of ground unary literals.

Music Style Transfer With Diffusion Model

Previous studies on music style transfer have mainly focused on one-to-one style conversion, which is relatively limited. When considering the conversion between multiple styles, previous methods required designing multiple modes to disentangle the complex style of the music, resulting in large computational costs and slow audio generation. The existing music style transfer methods generate spectrograms with artifacts, leading to significant noise in the generated audio. To address these issues, this study proposes a music style transfer framework based on diffusion models (DM) and uses spectrogram-based methods to achieve multi-to-multi music style transfer. The GuideDiff method is used to restore spectrograms to high-fidelity audio, accelerating audio generation speed and reducing noise in the generated audio. Experimental results show that our model has good performance in multi-mode music style transfer compared to the baseline and can generate high-quality audio in real-time on consumer-grade GPUs.