一句话：将加权 ω-受限单计数器自动机的行为刻画为混合代数系统解的分量，并在 B 或 N∞ 下给出混合上下文无关文法生成构造。
本文研究加权 ω-受限单计数器自动机（weighted ω-restricted one-counter automata）的行为刻画。设 S 为完备 star-omega 半环，Σ 为字母表。作者证明：对状态集为 {1,...,n} 的此类自动机 C，存在一个定义在完备半环-半模对 ((S<<Σ*>>)^{n×n}, (S<<Σ^ω>>)^n) 上的混合代数系统，使得 C 的行为 ||C|| 是该系统某个解的分量。进一步，当基半环为 B 或 N^∞ 时，作者构造出生成 ||C|| 的混合上下文无关文法。该构造推广了受限单计数器自动机的经典 triple construction，称为 triple-pair construction。结果属于加权 ω-语言与自动机理论，通过代数系统和文法构造证明，不涉及实验评估。
Let $S$ be a complete star-omega semiring and $\Sigma$ be an alphabet. For a weighted $\omega$-restricted one-counter automaton $\mathcal{C}$ with set of states $\{1, \dots, n\}$, $n \geq 1$, we show that there exists a mixed algebraic system over a complete semiring-semimodule pair ${((S \ll \Sigma^* \gg)^{n\times n}, (S \ll \Sigma^{\omega}\gg)^n)}$ such that the behavior $\Vert\mathcal{C} \Vert$ of $\mathcal{C}$ is a component of a solution of this system. In case the basic semiring is $\mathbb{B}$ or $\mathbb{N}^{\infty}$ we show that there exists a mixed context-free grammar that generates $\Vert\mathcal{C} \Vert$. The construction of the mixed context-free grammar from $\mathcal{C}$ is a generalization of the well-known triple construction in case of restricted one-counter automata and is called now triple-pair construction for $\omega$-restricted one-counter automata.