The Limitations and Power of NP-Oracle-Based Functional Synthesis Techniques
Bibliographic record
Abstract
Given a Boolean relational specification between inputs and outputs, the problem of functional synthesis is to construct a function that maps each assignment of the input to an assignment of the output such that each tuple of input and output assignments meets the specification. The past decade has witnessed significant improvement in the scalability of functional synthesis tools, allowing them to handle problems with tens of thousands of variables. A common ingredient in these approaches is their reliance on SAT solvers, thereby exploiting the breakthrough advances in SAT solving over the past three decades. While the recent techniques have been shown to perform well in practice, there is little theoretical understanding of the limitations and power of these approaches. The primary contribution of this work is to initiate a systematic theoretical investigation into the power of functional synthesis approaches that rely on NP oracles. We first show that even when small Skolem functions exist, naive bit-by-bit learning approaches fail due to the relational nature of specifications. We establish fundamental limitations of interpolation-based approaches, proving that even when small Skolem functions exist, resolution-based interpolation must produce exponential-size circuits. We prove that access to an NP oracle is inherently necessary for efficient synthesis. Our main technical result shows that it is possible to use NP oracles to synthesize small Skolem functions in time polynomial in the size of the specification and the size of the smallest sufficient set of witnesses, establishing positive results for a broad class of relational specifications.
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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 | 0.012 | 0.043 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.004 | 0.014 |
| Open science | 0.005 | 0.004 |
| Research integrity | 0.002 | 0.007 |
| Insufficient payload (model declined to judge) | 0.008 | 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".