Is the Valiant-Vazirani Isolation Lemma Improvable?
Bibliographic record
Abstract
AbstractThe Valiant-Vazirani Isolation Lemma [TCS, vol. 47, pp. 85{93, 1986] provides an ecientprocedure for isolating a satisfying assignment of a given satis able circuit: given a Booleancircuit C on ninput variables, the procedure outputs a new circuit C 0 on the same ninputvariables with the property that the set of satisfying assignments for C 0 is a subset of thosefor C, and moreover, if C is satis able then C 0 has exactly one satisfying assignment. TheValiant-Vazirani procedure is randomized, and it produces a uniquely satis able circuit C 0 withprobability (1=n).Is it possible to have an ecient deterministic witness-isolating procedure? Or, at least, is itpossible to improve the success probability of a randomized procedure to (1)? We argue thatthe answer is likely ‘No’. More precisely, we prove that1. a non-uniform deterministic polynomial-time witness-isolating procedure exists if and onlyif NP P=poly, and2. if there is a randomized polynomial-time witness-isolating procedure with success proba-bility bigger than 2=3, then coNP NP=poly.Thus, an improved witness-isolating procedure would imply the collapse of the Polynomial-TimeHierarchy. Finally, we consider a black-box setting of witness isolation (generalizing the settingof the Valiant-Vazirani Isolation Lemma), and give the upper bound O(1=n) on the successprobability for a natural class of randomized witness-isolating procedures.
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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 | 0.010 | 0.053 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.009 |
| Scholarly communication | 0.004 | 0.010 |
| Open science | 0.004 | 0.007 |
| Research integrity | 0.003 | 0.009 |
| Insufficient payload (model declined to judge) | 0.009 | 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".