Deterministic Reduction of Integer Nonsingular Linear System Solving to Matrix Multiplication
Bibliographic record
Abstract
We present a deterministic reduction to matrix multiplication for the problem of linear system solving: given as input a nonsingular A \in \Z^n \times n and b \in \Z^n \times 1 , compute A^-1 b. We give an algorithm that computes the minimal integer e such that all denominators of the entries in 2^eA^-1 are relatively prime to 2. Then, for a b that has entries with bitlength O(n) times as large as the bitlength of entries in A, we give an algorithm to produce the 2-adic expansion of 2^eA^-1 b up to a precision high enough such that A^-1 b over \Q can be recovered using rational number reconstruction. Both e and the 2-adic expansion can be computed in O(\MM(n,łog n + łog ||A||) \times (łog n) (łog n + łoglog ||A||)) bit operations. Here, ||A||= \max_ij |A_ij | and \MM(n,d) is the cost to multiply together, modulo 2^d, two n \times n integer matrices. Our approach is based on the previously known reductions of linear system solving to matrix multiplication which use randomization to find an integer lifting modulus X that is relatively prime to \det A. Here, we derandomize by first computing a permutation P, a unit upper triangular M, and a diagonal S with \det S a power of two, such that U := APMS^-1 is an integer matrix with 2 \perp \det U. This allows our modulus X to be chosen a power of 2.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".