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 ($Σ_{2k}^P$-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.

翻译：经典复杂度理论衡量的是计算一个函数所需的成本，但许多计算任务需要在多个有效输出中承诺一个。我们引入了确定性深度——为选出单一有效输出所需的最少不可逆承诺的序列层数——并证明无论多少计算都无法消除这一成本。我们展示了一些关系型任务，其承诺是常数时间的表查找，但需要指数级并行宽度来补偿任何深度的减少。一个守恒定律表明，丰富承诺仅仅将确定性层重新标记为电路深度，从而保持总体的序列成本。对于电路编码的规范，由此产生的深度层级结构捕获了多项式层级（对于每个固定的$k$，为$Σ_{2k}^P$完全问题；对于无界$k$，为PSPACE完全问题）。在线设置中，确定性深度是完全不可约的：承诺层之间的无限计算无法减少其数量。