Bibliographic record
Abstract
Nearly thirty years ago, it was shown that $Ω(\sqrt{n})$ registers are needed to solve obstruction-free consensus among $n$ processes. This lower bound was improved to $n$ registers in 2018, which exactly matches the best upper bound. The $Ω(\sqrt{n})$ space complexity lower bound actually applies to a class of objects called historyless objects, which includes registers, test-and-set objects, and readable swap objects. However, every known $n$-process obstruction-free consensus algorithm from historyless objects uses $Ω(n)$ objects. We give the first $Ω(n)$ space complexity lower bounds on consensus algorithms for two kinds of historyless objects. First, we show that any obstruction-free consensus algorithm from swap objects uses at least $n-1$ objects. More generally, we prove that any obstruction-free $k$-set agreement algorithm from swap objects uses at least $\lceil \frac{n}{k}\rceil - 1$ objects. This is the first non-constant lower bound on the space complexity of solving $k$-set agreement with swap objects when $k > 1$. We also present an obstruction-free $k$-set agreement algorithm from $n-k$ swap objects, exactly matching our lower bound when $k=1$. Second, we show that any obstruction-free binary consensus algorithm from readable swap objects with domain size $b$ uses at least $\frac{n-2}{3b+1}$ objects. Since any historyless object can be simulated by a readable swap object with the same domain, our results imply that any obstruction-free consensus algorithm from historyless objects with domain size $b$ uses at least $\frac{n-2}{3b+1}$ objects. For $b = 2$, we show a slightly better lower bound of $n-2$. The best known obstruction-free binary consensus algorithm from readable swap objects with domain size $2$ uses $2n-1$ objects, asymptotically matching our lower bound.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".