On the Computational Power of Timed Differentiable Petri Nets

Two kinds of well studied dynamical systems are state and time discrete ones (important in Computer Science) and state and time continuous ones (heavily considered in Automatic Control). Whereas in the discrete setting, the analysis of computational power of models has led to well-known hierarchies, the situation is more confused in the continuous case. A possible way to discriminate between these models is to state whether they can simulate Turing machine (or other equivalent models like counter machines). For instance, it is known that continuous systems described by ordinary differential equations (ODE) have this power. However, since the involved ODE is defined by overlapping local ODEs inside an infinite number of regions, this result has no significant application for differentiable models whose ODE is defined by an explicit representation. In this work, we considerably strengthen this result by showing that Time Differentiable Petri Nets (TDPN) can simulate Turing machines. Indeed the ODE ruling this model is particularly simple. First its expression is a linear expression enlarged with the “minimum” operator. Second, it can be decomposed into a finite number of linear ODE inside polyhedra. More precisely, we present different simulations in order to fulfill opposite requirements like robustness (allowing some perturbation of the simulation) and boundedness of the simulating net system. Then we establish that the simulation of two counter machines can be performed by a net with a constant number of places, i.e. whose dimension of associated ODE is constant. Afterwards, by modifying the simulation, we prove that marking coverability, submarking reachability and the existence of a steady-state are undecidable. Finally, we study the relations between TDPNs and time continuous Petri nets under infinite server semantics, a formalism with numerous results in control theory. We show that these models are equivalent and we analyse the complexity of the corresponding translations.