A Novel Approach to the Collatz Conjecture with Petri Nets

The Collatz conjecture is a famous unsolved problem in mathematics, known for its deceptively simple rules that generate complex, unpredictable behaviour. It can be efficiently modelled using a Petri net that represents its inverse graph, where each place corresponds to an integer and each transition encodes an inverse rule. The net, constructed up to a bound n, reveals the tree-like structure of predecessors and highlights properties such as recurrence, reachability, and liveness. Token flows simulate possible trajectories towards 1. This formal approach enables the investigation of the problem through discrete event systems theory and opens perspectives for parametric or inductive extensions beyond the bounded domain. The model proposed provides a structured framework for visualising and analysing the inverse dynamics of the conjecture. Some key numerical results highlight the challenges of working within a finite domain: for nmax=1000, the constructed Petri net comprises 1000 places and 667 transitions, including 417 source nodes (no predecessors), 333 sink nodes (no successors), and 218 isolated orphans, i.e., nodes only reachable via Div2 transitions with no incoming 3n+1 edge.