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On the Computational Power of Timed Differentiable Petri Nets

  • Serge Haddad
  • Laura Recalde
  • Manuel Silva
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4202)

Abstract

Well-known hierarchies discriminate between the computational power of discrete time and space dynamical systems. A contrario the situation is more confused for dynamical systems when time and space are continuous. A possible way to discriminate between these models is to state whether they can simulate Turing machine. For instance, it is known that continuous systems described by an ordinary differential equation (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 expressed by an explicit linear expression enlarged with the “minimum” operator. More precisely, we present two simulations of a two counter machine by a TDPN in order to fulfill opposite requirements: robustness and boundedness. These simulations are performed by nets whose dimension of associated ODEs is constant. At last, we prove that marking coverability, submarking reachability and the existence of a steady-state are undecidable for TDPNs.

Keywords

Turing Machine Hybrid Automaton Transition Pair Current Instruction Counter Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Serge Haddad
    • 1
  • Laura Recalde
    • 2
  • Manuel Silva
    • 2
  1. 1.LAMSADE-CNRS UMR 7024University Paris-DauphineFrance
  2. 2.GISEDUniversity ZaragossaSpain

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