Weighted omega-Restricted One Counter Automata

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.