Space Complexity in Polynomial Calculus
Bibliographic record
Abstract
During the last decade, an active line of research in proof complexity has been to study space complexity and time space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving. For the polynomial calculus proof system, the only previously known space lower bound is for CNF formulas of unbounded width in [Alekhnovich et al. '02], where the lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could refute any k-CNF formula in constant space. We prove several new results on space in polynomial calculus (PC) and in the extended proof system polynomial calculus resolution (PCR) studied in [Alekhnovich et al. '02]. (1) For PCR, we prove an Ω(n) space lower bound for a bitwise encoding of the functional pigeonhole principle with m pigeons and n holes. These formulas have width O(log n), and hence this is an exponential improvement over [Alekhnovich et al. '02] measured in the width of the formulas. (2) We then present another encoding of the pigeonhole principle that has constant width, and prove an Ω(n) space lower bound in PCR for these formulas as well. (3) We prove an Ω(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole principle formulas PHPmnwith m pigeons and n holes, and show that this is tight. (4) We prove that any k-CNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not known to be the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3-CNF in the canonical way.
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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 | 0.003 | 0.023 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.004 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.003 | 0.005 |
| Scholarly communication | 0.007 | 0.019 |
| Open science | 0.003 | 0.008 |
| Research integrity | 0.002 | 0.008 |
| Insufficient payload (model declined to judge) | 0.018 | 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".