Büchi VASS Recognise $$\mathbf {\Sigma }^{1}_{1}$$-complete $${\omega }$$-languages

Abstract

This short note exhibits an example of a Σ 1 1 -complete language that can be recognised by a one blind counter Büchi automaton (or equivalently a Büchi VASS with only one place).

On the High Complexity of Petri Nets $$\omega $$-Languages

“…Theorem 23 (Skrzypczak [34]). There exists a Σ 1 1 -complete ω-language accepted by a 1-blind-counter automaton.…”

“…On the other side, Finkel and Skrzypczak proved in [18] that there exist Σ 0 3 -complete, hence non Δ 0 3 , ω-languages accepted by non-deterministic onepartially-blind-counter Büchi automata. The existence of a Σ 1 1 -complete, hence non Borel, ω-language accepted by a Petri net was independently proved by Finkel and Skrzypczak in [17,34]. Moreover, Skrzypczak has proved in [34] that one blind counter is sufficient.…”

“…The existence of a Σ 1 1 -complete, hence non Borel, ω-language accepted by a Petri net was independently proved by Finkel and Skrzypczak in [17,34]. Moreover, Skrzypczak has proved in [34] that one blind counter is sufficient. In this paper, we fill the gap between Σ 0 3 and Σ 1…”