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
本文研究图状自动机的求值方程，证明经典图状方程等价于特殊交换Frobenius方程，并给出半环上矩阵求值相等的统一定义及骨律成立的条件。
AI 中文摘要
图状自动机通过组合状态词之间的关系来对图求值。我们考察使得这种求值独立于所选图表达式的方程，以及当关系被交换半环上的矩阵替代时会发生什么变化。我们证明经典图状方程等价于特殊交换Frobenius方程加上所需的对称律。已知的cospan和超图表示随后描述了结构表达式和带标签表达式的相等性，其中骨律精确删除孤立的内部顶点。我们使用已有的划分表示来比较这种相等性与求值矩阵的相等性。对于任何非空有限状态集$Q$和交换半环$K$，我们给出两个结构表达式具有相同矩阵的统一定义。该定义确定哪些孤立顶点计数被区分，以及何时一个消失的标量使得来自不同边界划分的矩阵相等。对于典范结构，骨律恰好当$|Q|\cdot1_K=1_K$时成立。保持所有关系复合是一个单独的要求，等价于$|Q|\geq2$时的加法幂等性。然后我们考察由阿贝尔群的不交并给出的$Q$上已知关系结构的特征矩阵。它们的Frobenius方程在每个交换半环上成立，而特殊性取决于每个群的阶，骨律取决于群的个数。这些条件指定了当权重被分配给图状自动机时哪些方程仍然有效。
英文摘要
Graphoid automata evaluate graphs by composing relations between words of states. We examine the equations that make this evaluation independent of the chosen graph expression, and what changes when relations are replaced by matrices over a commutative semiring. We prove that the classical graphoid equations are equivalent to the special commutative Frobenius equations together with the required symmetry laws. The known cospan and hypergraph presentations then describe equality of structural and labelled expressions, with the bone law deleting precisely the isolated internal vertices. We use established partition representations to compare this equality with equality of evaluated matrices. For any nonempty finite state set $Q$ and commutative semiring $K$, we give a uniform criterion for two structural expressions to have the same matrix. The criterion determines which isolated-vertex counts are distinguished and when a vanishing scalar makes matrices from different boundary partitions equal. For the canonical structure, the bone law holds precisely when $|Q|\cdot1_K=1_K$. Preserving all relational compositions is a separate requirement, equivalent to additive idempotency for $|Q|\geq2$. We then examine the characteristic matrices of the known relational structures on $Q$ given by disjoint unions of Abelian groups. Their Frobenius equations hold over every commutative semiring, whereas specialness depends on the order of each group and the bone law depends on the number of groups. These conditions specify which equations remain valid when weights are assigned to a graphoid automaton.
Comments29 pages
