A Common Recursive Form for Multiple Fundamental Arithmetic Operators and its Automated Synthesis
Bibliographic record
Abstract
The common recursive structure of a number of basic arithmetic operators is investigated. Previously, it was shown that fully combinational leading-digit N-bit detector circuits could be generated with a simple dyadic tree recursive structure having minimal complexity, regular and low fan-in and fan-out and log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (N) stages of delay for all outputs. Here, we show that this recursive structure is shared by many common arithmetic functions, including comparison, incrementation, decrementation, and fast parallel-prefix adder. This enables said functions to be described simply and parametrically in VHDL or Verilog using structural recursion. The commonality can also be leveraged to design multi-function circuits which are advantageous compared to separate operators chosen through multiplexers in, e.g., arithmetic-logic units. Preliminary synthesis results for both FPGA and digital CMOS are provided in order to characterize said benefits. This type of multifunction circuit could be employed in the design of fast, low-complexity arithmetic-logic units (ALUs) inside microprocessors, digital signal processors, or application-specific system-on-chip (SoC) designs.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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".