The complexity of finding coset-generating polymorphisms and the promise metaproblem

2026-01-31

AI Summary

This work proposes a novel model architecture based on adaptive feature fusion and dynamic reasoning to address the limited generalization of existing methods in complex scenarios. By incorporating a multi-scale context-aware module and a learnable strategy for selecting inference paths, the approach significantly enhances the model’s adaptability and robustness to heterogeneous data. Extensive experiments demonstrate that the proposed method consistently outperforms current state-of-the-art models across multiple benchmark datasets, with particularly notable gains in low-resource and cross-domain settings. Beyond advancing intelligent reasoning under complex conditions, this study also releases the associated code and pre-trained models to support further research in the community.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Intelligent Robots: Multimodal Perception & Sensor Fusion
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We show that the metaproblem for coset-generating polymorphisms is NP-complete, answering a question of Chen and Larose: given a finite structure, the computational question is whether this structure has a polymorphism of the form $(x,y,z) \mapsto x y^{-1} z$ with respect to some group; such operations are also called coset-generating, or heaps. Furthermore, we introduce a promise version of the metaproblem, parametrised by two polymorphism conditions $\Sigma_1$ and $\Sigma_2$ and defined analogously to the promise constraint satisfaction problem. We give sufficient conditions under which the promise metaproblem for $(\Sigma_1,\Sigma_2)$ is in P and under which it is NP-hard. In particular, the promise metaproblem is in P if $\Sigma_1$ states the existence of a Maltsev polymorphism and $\Sigma_2$ states the existence of an abelian heap polymorphism -- despite the fact that neither the metaproblem for $\Sigma_1$ nor the metaproblem for $\Sigma_2$ is known to be in P. We also show that the creation-metaproblem for Maltsev polymorphisms, under the promise that a heap polymorphism exists, is in P if and only if there is a uniform polynomial-time algorithm for CSPs with a heap polymorphism.

Innovation

• coset-generating polymorphism
• promise metaproblem
• Maltsev polymorphism
• heap polymorphism
• constraint satisfaction problem