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Achilles and the Tortoise climbing up the hyperarithmetical hierarchy
, 1997
"... We pursue the study of the computational power of Piecewise Constant Derivative (PCD) systems started in [5, 6]. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous tim ..."
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Cited by 26 (6 self)
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We pursue the study of the computational power of Piecewise Constant Derivative (PCD) systems started in [5, 6]. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We prove that the languages recognized by rational PCD systems in dimension d = 2k + 3 (respectively: d = 2k + 4), k 0, in finite continuous time are precisely the languages of the ! k th (resp. ! k + 1 th ) level of the hyperarithmetical hierarchy. Hence the reachability problem for rational PCD systems of dimension d = 2k + 3 (resp. d = 2k + 4), k 1, is hyperarithmetical and is \Sigma ! kcomplete (resp. \Sigma ! k +1 complete).
On the computational power of timed differentiable Petri nets
 Formal Modeling and Analysis of Timed Systems, 4th Int. Conf. FORMATS 2006, volume 4202 of LNCS
, 2006
"... Abstract. Wellknown 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 the ..."
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Cited by 3 (1 self)
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Abstract. Wellknown 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 steadystate are undecidable for TDPNs. 1
Abstract geometrical computation: Turingcomputing ability and undecidability
, 2004
"... In the Cellular Automata (CA) literature, discrete lines inside (discrete) spacetime diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this ideali ..."
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Cited by 3 (1 self)
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In the Cellular Automata (CA) literature, discrete lines inside (discrete) spacetime diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this idealization: dimensionless signals are moving on a continuous space in continuous time generating Euclidean lines on (continuous) spacetime diagrams. Like CA, this model is parallel, synchronous, uniform in space and time, and uses local updating. The main difference is that space and time are continuous and not discrete (i.e. R instead of Z). In this article, the model is restricted to Q in order to remain inside Turingcomputation theory. We prove that our model can carry out any Turingcomputation through twocounter automata simulation and provide some undecidability results.
Achilles and the Tortoise Climbing Up the Arithmetical Hierarchy
 Journal of Computer and System Sciences
, 1995
"... In this paper we show how to construct for every set P of integers in the arithmetical hierarchy a dynamical system H with piecewiseconstant derivatives (PCD) such that deciding membership in P can be reduced to solving the reachability problem between two rational points for H. The ability of s ..."
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In this paper we show how to construct for every set P of integers in the arithmetical hierarchy a dynamical system H with piecewiseconstant derivatives (PCD) such that deciding membership in P can be reduced to solving the reachability problem between two rational points for H. The ability of such apparentlysimple dynamical systems, whose definition involves only rational parameters, to # A preliminary version of the paper appeared in P.S. Thiagarajan (Ed.), "Proc. FST/TCS'95", 471483, LNCS 1026, Springer, 1995. This research was supported in part by the European Community projects HYBRID ECUS043 and INTAS94697 as well as by Research Grants #93012884, 970100692 and 961596048 of Russian Foundation for Basic Research. Verimag is a joint laboratory of cnrs, ujf and inpg. Some of the results were obtained while the first author was a visiting professor at ensimag, inpg, Grenoble. 1 "solve" highly unsolvable problems is closely related to Zeno's paradox, namely the ability to pack infinitely many discrete steps in a bounded interval of time. 1
Author manuscript, published in "4th Conf. Computability in Europe (CiE~'08) (abstracts and extended abstracts of unpublished papers), Greece (2008)" Abstract geometrical computation
, 2010
"... beyond the ..."
unknown title
, 2010
"... (will be inserted by the editor) Abstract geometrical computation 3: black holes for classical and analog computating ..."
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(will be inserted by the editor) Abstract geometrical computation 3: black holes for classical and analog computating