Mining Circuit Lower Bound Proofs for Meta-Algorithms.
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Bibliographic record
Abstract
Abstract. We show that circuit lower bound proofs based on the method of random restrictions yield non-trivial compression algorithms for “easy ” Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an n-variate Boolean function f computable by some unknown small circuit from a known class of circuits, find in deterministic time poly(2n) a circuit C (no restriction on the type of C) computing f so that the size of C is less than the trivial circuit size 2n/n. We get non-trivial compression for functions computable by AC0 circuits, (de Morgan) formulas, and (read-once) branching programs of the size for which the lower bounds for the corresponding circuit class are known. These compression algorithms rely on the structural characterizations of “easy ” functions, which are useful both for proving circuit lower bounds and for designing “meta-algorithms ” (such as Circuit-SAT). For (de Morgan) formulas, such structural characterization is provided by the “shrinkage under random restrictions ” results by Subbotovskaya (1961) and H̊astad (1998), strengthened to the “high-probability ” ver-sion by Santhanam (2010), Impagliazzo, Meka & Zuckerman (2012b), and Komargodski & Raz (2013). We give a new, simple proof of the “high-probability ” version of the shrinkage result for (de Morgan) for-mulas, with improved parameters. We use this shrinkage result to get both compression and #SAT algorithms for (de Morgan) formulas of size about n2. We also use this shrinkage result to get an alternative proof of the result by Komargodski & Raz (2013) of the average-case lower bound against small (de Morgan) formulas. Finally, we show that the existence of any non-trivial compression al-gorithm for a circuit class C ⊆ P/poly would imply the circuit lower 2 Chen et al. bound NEXP 6 ⊆ C; a similar implication is independently proved also by Williams (2013). This complements the result by Williams (2010) that any non-trivial Circuit-SAT algorithm for a circuit class C would imply a superpolynomial lower bound against C for a language in NEXP.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it