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.
Categories: cs.FL