Formally Certified Approximate Model Counting
Bibliographic record
Abstract
Abstract Approximate model counting is the task of approximating the number of solutions to an input Boolean formula. The state-of-the-art approximate model counter for formulas in conjunctive normal form (CNF), $$\textsf{ApproxMC}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ApproxMC</mml:mi> </mml:math> , provides a scalable means of obtaining model counts with probably approximately correct (PAC)-style guarantees. Nevertheless, the validity of $$\textsf{ApproxMC}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ApproxMC</mml:mi> </mml:math> ’s approximation relies on a careful theoretical analysis of its randomized algorithm and the correctness of its highly optimized implementation, especially the latter’s stateful interactions with an incremental CNF satisfiability solver capable of natively handling parity (XOR) constraints. We present the first certification framework for approximate model counting with formally verified guarantees on the quality of its output approximation. Our approach combines: (i) a static , once-off, formal proof of the algorithm’s PAC guarantee in the Isabelle/HOL proof assistant; and (ii) dynamic , per-run, verification of $$\textsf{ApproxMC}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ApproxMC</mml:mi> </mml:math> ’s calls to an external CNF-XOR solver using proof certificates. We detail our general approach to establish a rigorous connection between these two parts of the verification, including our blueprint for turning the formalized, randomized algorithm into a verified proof checker, and our design of proof certificates for both $$\textsf{ApproxMC}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ApproxMC</mml:mi> </mml:math> and its internal CNF-XOR solving steps. Experimentally, we show that certificate generation adds little overhead to an approximate counter implementation, and that our certificate checker is able to fully certify $$84.7\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>84.7</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> of instances with generated certificates when given the same time and memory limits as the counter.
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How this classification was reachedexpand
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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.004 | 0.002 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".