## Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy (1997)

Citations: | 26 - 6 self |

### BibTeX

@MISC{Bournez97achillesand,

author = {Olivier Bournez},

title = {Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 hyper-arithmetical hierarchy. Hence the reachability problem for rational PCD systems of dimension d = 2k + 3 (resp. d = 2k + 4), k 1, is hyper-arithmetical and is \Sigma ! k-complete (resp. \Sigma ! k +1 -complete).

### Citations

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(Show Context)
Citation Context ...f (R; c) 2 T estd\Gamma1 and d ? 2 then (R=xd ; c=xd ) 2 T estd . 3.2 RCT machines simulate two-stack pushdown automata and Turing machines We recall first what a two-stack pushdown automaton is (see =-=[11]-=-): assume alphabet \Sigma = f1; 2; : : : ; n \Sigma g is fixed. A stack is a word of \Sigma ! . The functions PUSH : \Sigma \Theta \Sigma ! ! \Sigma ! and POP : \Sigma ! ! \Sigma \Theta \Sigma ! are d... |

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Citation Context ...al numbers. It is a strict hierarchy and it satisfies the strict inclusions \Sigma ff ae \Sigma fi whenever ff ! fi. It can be related to the analytical hierarchy by \Delta 1 1 = [ fi \Sigma fi : see =-=[16]-=-. The idea of the construction of this hierarchy is the following: ffl \Sigma 1 is defined as the class of the recursively enumerable sets: that is to say \Sigma 1 is the class of the languages that a... |

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Citation Context ...models have been proposed in literature. In particular, in [2, 3, 4], the authors introduce Piecewise Constant Derivative systems (PCD systems), a sub-class of the so-called linear hybrid automata of =-=[1]-=-: such systems consist in partitioning the Euclidean space into a finite number of convex polyhedra such that the derivative within any region is constant. Hybrid systems can be considered either as c... |

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Citation Context ...n a continuous space with a continuous time: see [2, 3, 4, 9, 10]. Several theoretical computational models of machines working on a continuous space with a discrete time are known: in particular, in =-=[5]-=-, Blum, Shub and Smale introduce the real Turing machine (see [13] for an up-to-date survey). When PCD systems are considered as machines working on a continuous space with a discrete time their compu... |

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Citation Context ...ystems of dimension ds3 are equivalent to Turing machines or to a restriction of real Turing machines. The study of machines working on a continuous space with a continuous time is only beginning. In =-=[14]-=-, Moore proposes a recursion theory for computations on the reals in continuous time. When PCD systems are considered as machines working on continuous space with a continuous time no precise characte... |

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Citation Context ...constant. Hybrid systems can be considered either as computational machines working on a continuous space with a discrete time or as machines working on a continuous space with a continuous time: see =-=[2, 3, 4, 9, 10]-=-. Several theoretical computational models of machines working on a continuous space with a discrete time are known: in particular, in [5], Blum, Shub and Smale introduce the real Turing machine (see ... |

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29 | Achilles and the Tortoise climbing up the arithmetical hierarchy
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(Show Context)
Citation Context ...ontrol theory and computer science about hybrid systems. A hybrid system is a system that combines discrete and continuous dynamics. Several models have been proposed in literature. In particular, in =-=[2, 3, 4]-=-, the authors introduce Piecewise Constant Derivative systems (PCD systems), a sub-class of the so-called linear hybrid automata of [1]: such systems consist in partitioning the Euclidean space into a... |

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Citation Context ...M starting with stacks w1 ; w2 is a sequence of IDs (ID i ) i2N such that ID0 = (q0 ; w1 ; w2) and such that for all i 2 N ID i ` ID i+1 . It is well known that linear machines can simulate 2PDA (see =-=[12]-=-). In our context: Lemma 3.1 ([12]) Any two-stack pushdown automaton M can be simulated by a RCT machine M 0 of dimension 2 whose program is made only of linear machine instructions. Acronym RCT machi... |

26 | A Survey on Real Structural Complexity Theory
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Citation Context .... Several theoretical computational models of machines working on a continuous space with a discrete time are known: in particular, in [5], Blum, Shub and Smale introduce the real Turing machine (see =-=[13]-=- for an up-to-date survey). When PCD systems are considered as machines working on a continuous space with a discrete time their computational power is known: it is proved in [2, 3, 9] that PCD system... |

13 |
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(Show Context)
Citation Context ...the languages that are recursively enumerable in some fixed diagonalization of classes (\Sigma k i ) i . 2.3.2 Formal definition We give here the formal definitions. We use the classical notations of =-=[15, 16]-=-: in particular Wn (respectively:sW X n ) denotes the language recognized by the n th Turing machine (resp. by the n th Turing machine with oracle X); OE n (respectively OE X n ) denotes the function ... |

13 | A note on a P 6=NP result for a restricted class of real machines - Meer - 1992 |

7 | Some bounds on the computational power of piecewise constant derivative systems
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(Show Context)
Citation Context ...ecursive relations starting with a quantifier 9 and with 1 alternation. By lemma 5.4, L 2 \Sigma H(z k ) 2 = \Sigma ! k +1 . 2 We get immediately from theorem 4.2 and from proposition 5.5 for ks1 and =-=[6, 7, 8]-=- for k = 0: Theorem 5.1 Let k 0s0. ffl The languages that are semi-recognized by a PCD system of dimension 2k 0 + 3 in finite continuous time are precisely the languages of \Sigma ! k 0 . ffl The lang... |

3 |
On the computational power of hybrid and dynamical systems
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(Show Context)
Citation Context ...constant. Hybrid systems can be considered either as computational machines working on a continuous space with a discrete time or as machines working on a continuous space with a continuous time: see =-=[2, 3, 4, 9, 10]-=-. Several theoretical computational models of machines working on a continuous space with a discrete time are known: in particular, in [5], Blum, Shub and Smale introduce the real Turing machine (see ... |

2 |
Some bounds on the computational power of purely rational piecewise constant derivative systems
- Bournez
- 1997
(Show Context)
Citation Context ...mputational power was known: recently, Asarin and Maler [3] showed using Zeno's paradox, that every set of the arithmetical hierarchy can be recognized by a PCD system of finite dimension. We gave in =-=[7, 8]-=- a characterization of the computational power of a restricted class of PCD systems: the purely rational PCD systems. But no characterization was given for the general class of PCD systems. In this pa... |