Nondeterministic seedless oritatami systems and hardness of testing their equivalence

The oritatami system (OS) is a model of computation by cotranscriptional folding, being inspired by the recent experimental success of RNA origami to self-assemble an RNA tile cotranscriptionally. The OSs implemented so far, including the binary counter and Turing machine simulator, are deterministic, that is, uniquely fold into one conformation, while nondeterminism is intrinsic in biomolecular folding. We introduce nondeterminism to the OS and propose a nondeterministic OS (NOS) that chooses an assignment of Boolean values nondeterministically and evaluates a logical formula on the assignment. This NOS is seedless in the sense that it does not require any initial conformation like the RNA origami. The NOS enables proving the coNP-hardness of deciding, given two NOSs, if there exists no conformation that one of them folds but the other does not.