The shortest disjunctive normal form of a random Boolean function
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Bibliographic record
Abstract
Abstract This paper gives a new upper bound for the average length ℓ ( n ) of the shortest disjunctive normal form for a random Boolean function of n arguments, as well as new proofs of two old results related to this quantity. We consider a random Boolean function of n arguments to be uniformly distributed over all 2 such functions. (This is equivalent to considering each entry in the truth‐table to be 0 or 1 independently and with equal probabilities.) We measure the length of a disjunctive normal form by the number of terms. (Measuring it by the number of literals would simply introduce a factor of n into all our asymptotic results.) We give a short proof using martingales of Nigmatullin's result that almost all Boolean functions have the length of their shortest disjunctive normal form asymptotic to the average length ℓ ( n ). We also give a short information‐theoretic proof of Kuznetsov's lower bound ℓ ( n ) ≥ (1 + o (1)) 2 n /log n log log n . (Here log denotes the logarithm to base 2.) Our main result is a new upper bound ℓ ( n ) ≤ (1 + o (1)) H ( n ) 2 n /log n log log n , where H ( n ) is a function that oscillates between 1.38826 … and 1.54169 … . The best previous upper bound, due to Korshunov, had a similar form, but with a function oscillating between 1.581411 … and 2.621132 … . The main ideas in our new bound are (1) the use of Rödl's “nibble” technique for solving packing and covering problems, (2) the use of correlation inequalities due to Harris and Janson to bound the effects of weakly dependent random variables, and (3) the solution of an optimization problem that determines the sizes of “nibbles” and larger “bites” to be taken at various stages of the construction. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 161–186, 2003
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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 | 0.001 | 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 | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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