## On some Relations between Dynamical Systems and Transition Systems (1994)

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Venue: | In Proceedings of ICALP |

Citations: | 31 - 4 self |

### BibTeX

@INPROCEEDINGS{Asarin94onsome,

author = {Eugene Asarin and Oded Maler},

title = {On some Relations between Dynamical Systems and Transition Systems},

booktitle = {In Proceedings of ICALP},

year = {1994},

pages = {59--72},

publisher = {Springer-Verlag}

}

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### Abstract

. In this paper we define a precise notion of abstraction relation between continuous dynamical systems and discrete state-transition systems. Our main result states that every Turing Machine can be realized by a dynamical system with piecewise-constant derivatives in a 3-dimensional space and thus the reachability problem for such systems is undecidable for 3 dimensions. A decision procedure for 2-dimensional systems has been recently reported by Maler and Pnueli. On the other hand we show that some non-deterministic finite automata cannot be realized by any continuous dynamical system with less than 3 dimensions. 1 Introduction There has been recently an increasing interest in models of hybrid systems, i.e., systems that combine intercommunicating discrete and continuous components (see [9], [12], [3]). The introduction of these models is motivated by a real practical concern: more and more computers (discrete transition systems) are nowadays embedded within real-world control loops...

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Citation Context ...We say that A is deterministic if ffi is a function from Q to Q and call A an automaton when Q is finite. A run of such a system starting at some q 2 Q is a (finite or infinite) state sequence oe = oe=-=[1]-=-; oe[2]; : : : such that oe[1] = q and for every is1, (oe[i]; oe[i + 1]) 2 ffi . The set of all possible runs starting at q is denoted by L(A; q), and the set of possible runs starting at some q 2 Q 0... |

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Citation Context ... construction, on the other hand, uses continuous-time systems. There have been various undecidability results for other variants of hybrid systems with piecewise-constant derivatives (timed automata =-=[2]-=- or integration graphs [8]), but those were obtained in a richer model where a transition between regions is accompanied by a discrete change, and the trajectories are discontinuous. In [4] automata w... |

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150 |
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Citation Context ...es are discontinuous in this phase-space (the direction or the velocity goes through an abrupt change). Hence the equivalence between our PCD results and theirs is an optical illusion. Finally Putnam =-=[11]-=-, while attempting to prove the thesis every open physical system realizes every automaton, uses a notion of abstraction we find implausible. 14 Consider, for example a deterministic automaton without... |

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30 |
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Citation Context ...d automata [2] or integration graphs [8]), but those were obtained in a richer model where a transition between regions is accompanied by a discrete change, and the trajectories are discontinuous. In =-=[4]-=- automata were simulated (without a precise formal definition of this term) by smooth dynamical systems defined over a state-space of certain symmetric matrices. Those systems have high dimensionality... |

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Computability properties of low-dimensional dynamical systems
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Citation Context ...ion function of an automaton, defined over f0; 1g m , is realized by its continuous extension to [0; 1] m using arithmetical operations. Similarly stacks have been simulated by rational arithmetic in =-=[6]-=-. In these works, however, the simulating system is already defined over discretestime, i.e., using iterated maps of the form xn+1 = f(xn ). Our construction, on the other hand, uses continuous-time s... |

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Citation Context ...mical systems defined over a state-space of certain symmetric matrices. Those systems have high dimensionality that grows with the size of the automaton. The closest work to ours has been reported in =-=[13]-=- where stack machines were constructed from optical elements such as mirrors and lenses. These constructions were used to prove undecidability of the ray tracing problem. It should be noted, however, ... |

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Defining software by continuous, smooth functions
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Citation Context ... computed (given a rational initial condition) and yet their reachability problem is undecidable. There have been other works on simulation of transition systems by dynamical systems. For example, in =-=[7]-=- the boolean transition function of an automaton, defined over f0; 1g m , is realized by its continuous extension to [0; 1] m using arithmetical operations. Similarly stacks have been simulated by rat... |

2 |
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Citation Context ... computed (given a rational initial condition) and yet their reachability problem is undecidable. There have been other works on simulation of transition systems by dynamical systems. For example, in =-=[7]-=- the boolean transition function of an automaton, de ned over f0� 1g m , is realized by its continuous extension to [0� 1] m using arithmetical operations. Similarly stacks have been simulated by rati... |

1 |
Representation andReality
- Putnam
- 1988
(Show Context)
Citation Context ...es are discontinuous in this phase-space (the direction or the velocity goes through an abrupt change). Hence the equivalence between our PCD results and theirs is an optical illusion. Finally Putnam =-=[11]-=-, while attempting to prove the thesis every open physical system realizes every automaton, uses a notion of abstraction we nd implausible. D3 q2 D4sConsider, for example a deterministic automaton wit... |

1 |
The computability and complexity ofoptical beam tracing
- Reif, Tygar, et al.
- 1990
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Citation Context ...namical systems de ned over a state-space of certain symmetric matrices. Those systems have high dimensionality thatgrows with the size of the automaton. The closest work to ours has been reported in =-=[13]-=- where stack machines were constructed from optical elements such as mirrors and lenses. These constructions were used to prove undecidability oftheray tracing problem. It should be noted, however, th... |