Pi2^P-completeness of universality and inclusion for deterministic limit Parikh automata

Prove that the universality and inclusion problems for deterministic limit Parikh automata on infinite words are Pi2^P-complete by establishing matching Pi2^P upper bounds to complement the Pi2^P-hardness results shown in the paper.

Background

The paper shows that deterministic limit Parikh automata (PA) are effectively closed under Boolean operations and that their decision problems, including universality and inclusion, are decidable. It further establishes Pi2^P-hardness for universality and inclusion in this model, but does not provide matching upper bounds.

To capture the exact complexity, the authors formulate a conjecture asserting Pi2^P-completeness for universality and inclusion. They also note earlier that the current lower bounds are not tight and explicitly state that whether these problems lie in Pi2^P remains open. This problem asks for proving the missing upper bounds to settle completeness.