On the Relationship Between Boolean and Fuzzy Cellular Automata
Bibliographic record
Abstract
Fuzzy cellular automata (FCA) are continuous cellular automata where the local rule is defined as the “fuzzification” of the local rule of a corresponding Boolean cellular automaton in disjunctive normal form. In this paper we are interested in the relationship between Boolean and fuzzy models and we analytically show, for the first time, the existence of a strong connection between them by focusing on two properties: density conservation and additivity. We begin by giving a probabilistic interpretation of our fuzzification which leads to two important results. First, it establishes an equivalence between convergent fuzzy CA and the mean field approximation on Boolean CA, an estimation of their asymptotic density. Second, we show that the density conservation property, extensively studied in the Boolean domain, is preserved in the fuzzy domain: a Boolean CA is density conserving if and only if the corresponding FCA is sum preserving. A similar result is established for another novel “spatial” density conservation property. Finally, we prove an interesting parallel between additivity of Boolean CA and oscillation of the corresponding fuzzy CA around its fixed point. In fact, we show that a Boolean CA has a certain form of additivity if and only if the behavior of the corresponding fuzzy CA around its fixed point coincides with the Boolean behavior. These connections between the Boolean and the fuzzy models are the first formal proofs of a relationship between them.
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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 | 0.002 | 0.014 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".