Weigthed omega-Restricted One Counter Automata

Let S be a complete star-omega semiring and Σ be an alphabet. For a weighted ω-restricted one counter automaton C with stateset {1, . . . ,n}, n ≥ 1, we show that there exists a mixed algebraic system over a complete semiringsemimodule pair ((S ≪ Σ∗ ≫)n×n, (S ≪ Σω ≫)n) such that the behavior ‖C‖ of C is a component of a solution of this system. In case the basic semiring is B or N∞ we show that there exists a mixed context-free grammar that generates ‖C‖. The construction of the mixed context-free grammar from 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 ω-restricted one counter automata.