Circuit Lower Bounds for MCSP from Local Pseudorandom Generators
Bibliographic record
Abstract
The Minimum Circuit Size Problem (MCSP) asks if a given truth table of a Boolean function f can be computed by a Boolean circuit of size at most θ, for a given parameter θ. We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a PRG is called local if its output bit strings, when viewed as the truth table of a Boolean function, can be computed by a Boolean circuit of small size. We get new and improved lower bounds for MCSP that almost match the best-known lower bounds against several circuit models. Specifically, we show that computing MCSP, on functions with a truth table of length N , requires • N 3− o (1) -size de Morgan formulas, improving the recent N 2− o (1) lower bound by Hirahara and Santhanam (CCC, 2017), • N 2− o (1) -size formulas over an arbitrary basis or general branching programs (no non-trivial lower bound was known for MCSP against these models), and • 2 Ω( N 1/( d +1.01)) -size depth- d AC 0 circuits, improving the (implicit, in their work) exponential size lower bound by Allender et al. (SICOMP, 2006). The AC 0 lower bound stated above matches the best-known AC 0 lower bound (for PARITY) up to a small additive constant in the depth. Also, for the special case of depth-2 circuits (i.e., CNFs or DNFs), we get an optimal lower bound of 2 Ω( N ) for MCSP.
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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 | 0.003 | 0.030 |
| Meta-epidemiology (narrow) | 0.003 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.004 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.005 | 0.016 |
| Open science | 0.005 | 0.007 |
| Research integrity | 0.003 | 0.011 |
| Insufficient payload (model declined to judge) | 0.026 | 0.004 |
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".