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  "path": "/abs/2603.28031v1",
  "publishedAt": "2026-03-31T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Joseph M. Hellerstein"
  ],
  "textContent": "**Authors:** Joseph M. Hellerstein\n\nClassical 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 it is an orthogonal axis measuring the cost of committing. 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 couples determination and computational depth: enriching the commitment basis trades layers for step complexity, but the total sequential cost is bounded below. For circuit-encoded specifications, the resulting depth hierarchy captures the polynomial hierarchy ($Σ_{2k}^P$-complete for each fixed $k$, PSPACE-complete for unbounded $k$). Determination depth and computational depth are orthogonal: neither subsumes the other. Stable matching witnesses the independence sharply -- finding a stable matching is in P, yet every finite determination depth arises as the rotation-poset height of some instance. In the distributed setting, the framework recovers the Halpern--Moses common-knowledge impossibility.",
  "title": "On the Complexity of Determinations"
}