发表机构
American University of the Middle East（中东美国大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明由有限自动机描述且被固定边缘操作保持的关系，其约束满足可在多项式时间求解，并给出图状自动机的多项式识别方法，且边界最优。
AI 中文摘要
有限自动机可以描述元数无界的、比其描述指数级更大的关系。我们证明，只要这些关系的长度切片被有限域上的一个公共固定边缘操作所保持，那么对于这类关系的约束满足问题可以在多项式时间内求解。该算法计算完整解关系及其投影的紧凑表示。其主要组成部分是将非确定性有限自动机多项式时间编译为少数子幂算法所需的叉见证和小投影。在Mal'tsev情形下，当域和操作表作为输入提供时，直接证明也是多项式的，从而回答了自动约束满足的Mal'tsev可处理性问题。我们还刻画了一族既无Mal'tsev项也无近一致项的3-边缘代数的所有不变关系。它们的范式将布尔活动约束与仿射值空间相结合，并产生可从NFA或任意生成器构造的规范二次位表示。对于图状自动机，这些结果给出了无图宽度限制的多项式时间识别、有效的边界组合以及有限图关系的比较。二次边界界在最坏情况下是最优的。一个固定的三状态例子将多项式时间识别与困难的精确计数区分开来。
英文摘要
Finite automata can describe relations of unbounded arity that are exponentially larger than their descriptions. We prove that constraint satisfaction for such relations is solvable in polynomial time whenever their length slices are preserved by a common fixed edge operation on a finite domain. The algorithm computes compact representations of the complete solution relation and its projections. Its main ingredient is a polynomial-time compilation of nondeterministic finite automata into the fork witnesses and small projections required by the few-subpowers algorithm. In the Mal'tsev case, a direct proof is polynomial also when the domain and operation table are supplied as input, answering the Mal'tsev tractability question for automatic constraint satisfaction. We also characterize all invariant relations of a family of 3-edge algebras with neither Mal'tsev nor near-unanimity terms. Their normal forms combine Boolean activity constraints with affine value spaces and yield canonical quadratic-bit representations constructible from NFAs or arbitrary generators. For graphoid automata, these results give polynomial-time recognition without a graph-width restriction, effective boundary composition, and comparison of finite graph relations. The quadratic boundary bounds are optimal in the worst case. A fixed three-state example separates polynomial-time recognition from hard exact counting.
Comments23 pages, 2 figures. Reproducibility package: https://doi.org/10.5281/zenodo.22648283
