Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates
Bibliographic record
Abstract
The class FORMULA[s]∘G consists of Boolean functions computable by size-sDe Morgan formulas whose leaves are any Boolean functions from a class G. We givelower boundsand (SAT, Learning, andpseudorandom generators(PRGs))algorithmsfor FORMULA[n1.99]∘G, for classes G of functions withlow communication complexity. Let R(k)G be the maximumk-party number-on-forehead randomized communication complexity of a function in G. Among other results, we show the following: • The Generalized Inner Product function GIPkncannot be computed in FORMULA[s]° G on more than 1/2+ε fraction of inputs for s=o(n2/k⋅4k⋅R(k)(G)⋅log(n/ε)⋅log(1/ε))2). This significantly extends the lower bounds against bipartite formulas obtained by [62]. As a corollary, we get an average-case lower bound for GIPknagainst FORMULA[n1.99]∘PTFk−1, i.e., sub-quadratic-size De Morgan formulas with degree-k-1)PTF(polynomial threshold function) gates at the bottom. Previously, it was open whether a super-linear lower bound holds for AND of PTFs. • There is a PRG of seed length n/2+O(s⋅R(2)(G)⋅log(s/ε)⋅log(1/ε)) that ε-fools FORMULA[s]∘G. For the special case of FORMULA[s]∘LTF, i.e., size-sformulas withLTF(linear threshold function) gates at the bottom, we get the better seed length O(n1/2⋅s1/4⋅log(n)⋅log(n/ε)). In particular, this provides the first non-trivial PRG (with seed length o(n)) for intersections ofnhalfspaces in the regime where ε≤1/n, complementing a recent result of [45]. • There exists a randomized 2n-t#SAT algorithm for FORMULA[s]∘G, where t=Ω(n\√s⋅log2(s)⋅R(2)(G))/1/2. In particular, this implies a nontrivial #SAT algorithm for FORMULA[n1.99]∘LTF. • The Minimum Circuit Size Problem is not in FORMULA[n1.99]∘XOR; thereby making progress on hardness magnification, in connection with results from [14, 46]. On the algorithmic side, we show that the concept class FORMULA[n1.99]∘XOR can be PAC-learned in time 2O(n/log n).
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.004 | 0.024 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.004 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.007 | 0.015 |
| Open science | 0.005 | 0.005 |
| Research integrity | 0.003 | 0.008 |
| Insufficient payload (model declined to judge) | 0.012 | 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".