Extension Variables in QBF Resolution

We investigate two QBF resolution systems that use extension variables: weak extended Q-resolution, where the extension variables are quantified at the innermost level, and extended Q-resolution, where the extension variables can be placed inside the quantifier prefix. These systems have been considered previously by (Jussila et al. 2007), who give experimental evidence that extended Q-resolution is stronger than weak extended Q-resolution. Here we prove an exponential separation between the two systems, thereby confirming the conjecture of (Jussila et al. 2007). Conceptually, this separation relies on showing strategy extraction for weak extended Q-resolution by boundeddepth circuits. In contrast, we show that this strong strategy extraction result fails in extended Q-resolution.