A note on first-order projections and games

We show how the fact that there is a first-order projection from the problem transitive closure (TC) to some other problem Ω enables us to automatically deduce that a natural game problem, , whose instances are labelled instances of Ω, is complete for PSPACE (via log-space reductions). Our analysis is strongly dependent upon the reduction from TC to Ω being a logical projection in that it fails should the reduction be, for example, a log-space reduction or a quantifier-free first-order translation.