On the Complexity of Determinations
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908940601655296 |
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| author | Hellerstein, Joseph M. |
| author_facet | Hellerstein, Joseph M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28031 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Complexity of Determinations Hellerstein, Joseph M. Computational Complexity F.1.3; F.2.2 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. |
| title | On the Complexity of Determinations |
| topic | Computational Complexity F.1.3; F.2.2 |
| url | https://arxiv.org/abs/2603.28031 |