Bibliographic record
Abstract
$ \newcommand{\cclass}[1]{{\textsf{#1}}} \newcommand{\poly}{\mathop{\mathrm{poly}}} \newcommand{\F}{\mathbb{F}} \newcommand{\AC}{\cclass{AC}} $ We give the first separation between the power of formulas and circuits in the $\AC^0[\oplus]$ basis (unbounded fan-in AND, OR, NOT and MOD$_2$ gates). We show that there exist $\poly(n)$-size depth-$d$ circuits that are not equivalent to any depth-$d$ formula of size $n^{o(d)}$ for all $d \le O({\log(n)}/{\log\log(n)})$. This result is obtained by a combination of new lower and upper bounds for Approximate Majorities, the class of Boolean functions $\{0,1\}^n \to \{0,1\}$ that agree with the Majority function on a $3/4$ fraction of the inputs. $\AC^0[\oplus]$ formula lower bound. We show that every depth-$d$ $\AC^0[\oplus]$ formula of size $s$ has a $1/4$-error polynomial approximation over $\F_2$ of degree $O((1/d)\log s)^{d-1}$. This strengthens a classic $O(\log s)^{d-1}$ degree approximation for circuits due to Razborov (1987). Since any polynomial that approximates the Majority function has degree $\Omega(\sqrt n)$, this result implies an $\exp(\Omega(dn^{1/2(d-1)}))$ lower bound on the depth-$d$ $\AC^0[\oplus]$ formula size of all Approximate Majority functions for all $d \le O(\log n)$. Monotone $\AC^0$ circuit upper bound. For all $d \le O({\log(n)}/{\log\log(n)})$, we give a randomized construction of depth-$d$ monotone $\AC^0$ circuits (without NOT or MOD$_2$ gates) of size $\exp(O(n^{1/2(d-1)}))$ that compute an Approximate Majority function. This strengthens a construction of formulas of size $\exp(O(dn^{1/2(d-1)}))$ due to Amano (2009). --------------------------- A preliminary version of this paper appeared in the Proc. of the 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017).
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.036 | 0.010 |
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".