Bibliographic record
Abstract
$ \newcommand{\QMAEXP}{{\sf{QMA_{EXP}}}} \newcommand{\BQEXP}{{\sf{BQEXP}}} \newcommand{\NEXP}{{\sf{NEXP}}} \newcommand{\EXP}{{\sf{EXP}}} $ We study the complexity of a class of problems involving satisfying constraints which remain the same under translations in one or more spatial directions. In this paper, we show hardness of a classical tiling problem on an $N \times N$ $2$-dimensional grid and a quantum problem involving finding the ground state energy of a $1$-dimensional quantum system of $N$ particles. In both cases, the only input is $N$, provided in binary. We show that the classical problem is $\NEXP$-complete and the quantum problem is $\QMAEXP$-complete. Thus, an algorithm for these problems which runs in time polynomial in $N$ (exponential in the input size) would imply that $\EXP = \NEXP$ or $\BQEXP = \QMAEXP$, respectively. Although tiling in general is already known to be $\NEXP$-complete, the usual approach is to require that either the set of tiles and their constraints or some varying boundary conditions be given as part of the input. In the problem considered here, these are fixed, constant-sized parameters of the problem. Instead, the problem instance is encoded solely in the size of the system. A preliminary version of this paper was posted on the arXiv in 2009. An extended abstract appeared in FOCS'09.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.020 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.010 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.015 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".