On the Complexity of Determinations

TL;DR

This paper introduces determination depth, a measure of the minimal sequential commitments needed to select a valid output, revealing fundamental complexity limits in computational tasks.

Contribution

It defines determination depth and demonstrates its invariance under computation, linking it to circuit complexity and the polynomial hierarchy.

Findings

Relational tasks with constant-time lookups require exponential width to reduce depth.

Enriching commitments relabels layers as circuit depth, preserving total cost.

Determination depth captures the polynomial hierarchy in circuit-encoded specifications.

Abstract

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.