Revisionist Simulations: A New Approach to Proving Space Lower Bounds
Bibliographic record
Abstract
Determining the space complexity of $x$-obstruction-free $k$-set agreement for $x\leq k$ is an open problem. In $x$-obstruction-free protocols, processes are required to return in executions where at most $x$ processes take steps. The best known upper bound on the number of registers needed to solve this problem among $n>k$ processes is $n-k+x$ registers. No general lower bound better than $2$ was known. We prove that any $x$-obstruction-free protocol solving $k$-set agreement among $n>k$ processes uses at least $\lfloor(n-x)/(k+1-x)\rfloor+1$ registers. Our main tool is a simulation that serves as a reduction from the impossibility of deterministic wait-free $k$-set agreement: if a protocol uses fewer registers, then it is possible for $k+1$ processes to simulate the protocol and deterministically solve $k$-set agreement in a wait-free manner, which is impossible. A critical component of the simulation is the ability of simulating processes to revise the past of simulated processes. We introduce a new augmented snapshot object, which facilitates this. We also prove that any space lower bound on the number of registers used by obstruction-free protocols applies to protocols that satisfy nondeterministic solo termination. Hence, our lower bound of $\lfloor(n-1)/k\rfloor+1$ for the obstruction-free ($x=1$) case also holds for randomized wait-free free protocols. In particular, this gives a tight lower bound of exactly $n$ registers for solving obstruction-free and randomized wait-free consensus. Finally, our new techniques can be applied to get a space lower of $\lfloor n/2\rfloor+1$ for $ε$-approximate agreement, for sufficiently small $ε$. It requires participating processes to return values within $ε$ of each other. The best known upper bounds are $\lceil\log(1/ε)\rceil$ and $n$, while no general lower bounds were known.
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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 | 0.010 | 0.052 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.002 | 0.007 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.003 | 0.010 |
| Scholarly communication | 0.007 | 0.026 |
| Open science | 0.006 | 0.016 |
| Research integrity | 0.003 | 0.015 |
| Insufficient payload (model declined to judge) | 0.013 | 0.002 |
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".