Weighted simple reset pushdown automata

Greibach Normal Form for $ω$-Algebraic Systems and Weighted Simple $ω$-Pushdown Automata

“…We only consider the algebraic series r with (r, ) = 0; cf. [6] for the other case. So we assume without loss of generality that r is the x 1 -component of the unique solution of the algebraic system (14) with variables x 1 , .…”

“…For the proof, we will need the corresponding result for finite words as an intermediate step. We have shown in [6], that for every algebraic series r (of finite words), there exists a simple reset pushdown automaton with behavior r. We recall the construction of the simple reset pushdown automata here for the convenience of the reader, as variants of these automata will be used in Section 7 for ω-algebraic series.…”

“…A special form, the reset pushdown matrices, is used for pushdown automata starting with an empty stack and allowing the automaton to push onto the empty stack. Here, we are interested in simple reset pushdown matrices, introduced in [6]. This simple form allows the automaton only to push one symbol, to pop one symbol or to ignore the stack.…”

“…As in [6], we now construct the simple reset pushdown automaton A m = (n + 1, Γ, I m , M, P), 1 ≤ m ≤ n, with r = A 1 as follows:…”

“…The following (main) theorem of [6] states that the behavior of the simple reset pushdown automata induced by the Greibach normal form (14) is the unique solution of the original algebraic system (14).…”