Gathering Teams of Deterministic Finite Automata on a Line
Bibliographic record
Abstract
Several mobile agents, modelled as deterministic finite automata, navigate in an infinite line in synchronous rounds. All agents start in the same round. In each round, an agent can move to one of the two neighboring nodes, or stay idle. Agents have distinct labels which are integers from the set {1,…,L}. They start in teams, each of which consists of x agents, for some fixed integer x. Agents in a team have the same starting node. The adversary decides the compositions of teams, and their starting nodes. Whenever an agent enters a node, it sees the entry port number and the states of all collocated agents; this information forms the input of the agent on the basis of which it transits to the next state and decides the current action. The aim is for all agents to gather at the same node and stop. Gathering is feasible, if this task can be accomplished for any decisions of the adversary, and its time is the worst-case number of rounds from the start till gathering. We consider the feasibility and time complexity of gathering teams of agents, and give a complete solution of this problem. It turns out that both feasibility and complexity of gathering depend on the crucial parameter x which is the size of teams. For the oriented line, gathering is impossible if x = 1, and it can be accomplished in time O(D), for x > 1, where D is the distance between the starting nodes of the most distant teams. This complexity is of course optimal. For the unoriented line, the situation is different. For x = 1, gathering is also impossible, but for x = 2, the optimal time of gathering is Θ(Dlog L), and for x ≥ 3 the optimal time of gathering is Θ(D). Solving the gathering problem for agents that are finite automata navigating in an infinite environment requires new methodological tools. Traditional gathering techniques in graphs are count driven: agents make decisions based on counting steps. Since distances between agents may be unbounded, agents have to count unbounded numbers of steps. When agents are finite automata, counting unbounded numbers of steps is impossible, hence we must use different methods. In all our gathering algorithms, changes of the agents' behavior are triggered not by counting steps but by events which are meetings between agents during which they interact. Hence our new technique is event driven. Designing the behavior of the agents based on meeting events, so as to guarantee gathering regardless of the adversary’s decisions is our main methodological contribution.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 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".