Is the Valiant-Vazirani Isolation Lemma Improvable?
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
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.
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.
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 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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