Multi-linear Strategy Extraction for QBF Expansion Proofs via Local Soundness

In applications, QBF solvers are expected to not only decide whether a given formula is true or false but also return a solution in the form of a strategy. Determining whether strategies can be efficiently extracted from proof traces generated by QBF solvers is a fundamental research task. Most resolution-based proof systems are known to implicitly support polynomial-time strategy extraction through a simulation of the evaluation game associated with an input formula, but this approach introduces large constant factors and results in unwieldy circuit representations. In this work, we present an explicit polynomial-time strategy extraction algorithm for the \(\forall \hbox {-}\mathsf{Exp}\hbox {+}\mathsf{Res}\) proof system. This system is used by expansion-based solvers that implement counterexample-guided abstraction refinement (CEGAR), currently one of the most effective QBF solving paradigms. Our argument relies on a Curry-Howard style correspondence between strategies and \(\forall \hbox {-}\mathsf{Exp}\hbox {+}\mathsf{Res}\) derivations, where each strategy realizes an invariant obtained from an annotated clause derived in the proof system.