Closure under Boolean combinations within NB games without bootstrapping

Determine whether the closure under Boolean combinations needed for the Prover–Adversary game NB can be implemented directly at the game level while preserving logarithmic-round winning strategies, without relying on bootstrapping via the inference system LNDT or other auxiliary constructions.

Background

To show equivalence between game-based and inference-based proof systems, it is typically necessary to handle Boolean combinations (especially negation) within the language of queries or objects. For deterministic branching programs this is straightforward, but for non-deterministic branching programs (NBPs) it is substantially more delicate and relates to Immerman–Szelepcsényi-style complementation.

In this paper, the authors chose to expand the query language of their games to explicitly include Boolean combinations, thereby simplifying the translation from proofs to strategies. They note, however, that achieving such closure directly within the game framework while maintaining the required logarithmic bound on the number of rounds may not be straightforward without additional bootstrapping.