On the Complexity of Determinations

Classical complexity theory measures the cost of computing a function, but many computational tasks require committing to one valid output among several. We introduce determination depth -- the minimum number of sequential layers of irrevocable commitments needed to select a single valid output -- and show that no amount of computation can eliminate this cost. We exhibit relational tasks whose commitments are constant-time table lookups yet require exponential parallel width to compensate for any reduction in depth. A conservation law shows that enriching commitments merely relabels determination layers as circuit depth, preserving the total sequential cost. For circuit-encoded specifications, the resulting depth hierarchy captures the polynomial hierarchy ($\Sigma_{2k}^P$-complete for each fixed $k$, PSPACE-complete for unbounded $k$). In the online setting, determination depth is fully irreducible: unlimited computation between commitment layers cannot reduce their number.