||In a recently launched research program for developing logic as a formal theory of (interactive) computability, several very interesting logics have been introduced and axiomatized. These fragments of the larger Computability Logic aim not only to describe what can be computed, but also provide a mechanism for extracting computational algorithms from proofs. Among the most expressive and fundamental of these is CL4, known to be (constructively) sound and complete with respect to the underlying computational semantics. Furthermore, ∀, ∃-free fragment of CL4 not containing blind quantifiers was shown to be decidable in polynomial space. The present work extends this result and proves that this fragment is, in fact, PSPACE-complete.