FloydNet: A Learning Paradigm for Global Relational Reasoning
Learning algorithmic computation often requires explicit relational intermediate states, yet many graph processors maintain their primary states on individual entities. We introduce \fnet and \textbf{Pivotal Attention} (PA), which maintain ordered pair states and update a target relation $(i,k)$ by attending over candidates formed from $(i,j)$ and $(j,k)$ for every pivot $j$. Motivated by the pair-and-pivot structure of Floyd--Warshall, PA learns relation composition and pivot weighting in parallel rather than executing its ordered min-plus recurrence. The \kfnet{k} framework extends this operation to ordered $k$-tuples, with Self-Attention and PA as its $k=1$ and $k=2$ cases at the attention-operation level. Under atomic tuple initialization and invariant readout, we show that \kfnet{k} is no more graph-discriminative than k-FWL; on BREC, each evaluated variant matches the success set of its corresponding WL reference. \fnet further achieves 96.64\% mean accuracy under the reported CLRS-30 protocol and a 99.8\% optimality rate with 10 samples on held-out non-metric TSP instances.