The Design of Maple's Sum-of-Products and POLY Data Structures for Representing Mathematical Objects
Bibliographic record
Abstract
The principal data structure Maple uses to represent polynomials and general mathematical expressions involving functions like sin x, e 2x , y'(x),(n/k) etc., is known to the Maple developers as the sum-of-products data structure. Gaston Gonnet, as the primary author of the Maple kernel, designed and implemented this data structure in the early 1980s. As part of the process of simplifying a mathematical formula, he represented every Maple object and every sub-object uniquely in memory. This makes testing for equality of expressions very fast. In this article, on occasion of Gonnet's retirement, we present details of his design, its pros and cons, and changes we and others have made to it over the years. One of the cons of the sum-of-products data structure is it is not as efficient at multiplying multivariate polynomials as other special purpose computer algebra systems. We describe a new data structure called POLY that we added to Maple 17 (released in 2013) to improve performance for polynomials in Maple, and recent work done for Maple 18 (released in 2014).
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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 | 0.005 | 0.019 |
| Meta-epidemiology (narrow) | 0.001 | 0.002 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.007 | 0.014 |
| Open science | 0.004 | 0.007 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.008 | 0.008 |
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".