On the computational power of dynamical systems and hybrid systems
Citations
785 citations
Cites background or methods from "On the computational power of dynam..."
...high-order neural nets,13 Bournez and Cosnard (1996) for simulation by analog automata, Garzon (1995) for simulation by cellular automata, Siegelmann and Sontag (1994) and Siegelmann (1998) for simulation by saturated linear systems that can involve arbitrary real (not necessarily rational) numbers, Branicky (1995b), Branicky (1995a), Ruohonen (1997) and Moore (1991) for simulation by di!erential equations, Moore (1990) for simulation by continuous-time physical devices in dimension three, and Koiran and Moore (1999) for simulation by analytic maps....
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...…systems), Littman, Dean and Kaelbling (1995), Rust (1996), Mundhenk, Goldsmith, Lusena and Allender (1997), (Markov decision theory), Alur et al. (1995) (hybrid systems), Bournez and Cosnard (1996), Moore (1998) (Turing machine as dynamical systems) and Orponen (1994) (recurrent neural networks)....
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...…by 13 In high-order neural nets, activations are combined using multiplication as opposed to just linear combinations. high-order neural nets,13 Bournez and Cosnard (1996) for simulation by analog automata, Garzon (1995) for simulation by cellular automata, Siegelmann and Sontag (1994) and…...
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242 citations
128 citations
Cites methods from "On the computational power of dynam..."
...…a new generation of hybrid computing has recently been explored using analogue computation with chaotic neuro-dynamics and digital computation through algorithm (Aihara 2003; Horio & Aihara 2008); see also Branicky (1995) and Bournez & Cosnard (1996) for the computational power of hybrid systems....
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90 citations
82 citations
Cites background from "On the computational power of dynam..."
...In traditional simulations of counter machines or Turing machines by dynamical systems, a machine configuration is encoded by a single point of the dynamical system’s state space [11, 16, 15, 14, 10, 6, 2]....
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References
13,779 citations
"On the computational power of dynam..." refers background in this paper
...It is well known [11] that there exist some languages L ⊂ {0, 1}+ which cannot be recognized by Turing machines....
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...Because a discrete two stack automaton is an analog two stack automaton, and since discrete two stack automata can simulate Turing machines [11], analog automata are able to simulate Turing machines....
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3,497 citations
"On the computational power of dynam..." refers background in this paper
...Since we know that a Turing machine can always be simulated, modulo a polynomial time overhead, by a reversible one (see for example: [7]), we claim that M , from second step, can be built reversible....
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2,091 citations
"On the computational power of dynam..." refers background in this paper
...formal definitions have been proposed in literature: see for example [1, 9, 21]....
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...We will need also the following definition [1]: Definition 4....
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...Alur and al. propose in [1] the following definition: Definition 4.3 (Hybrid System [1]) A hybrid system is made of 6-components: H = (Loc, V ar, Lab, Edg,Act, Inv) where: • Loc is a finite set of vertices called locations....
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...4 • A hybrid system H is time-deterministic [1] if for every l ∈ Loc and every v ∈ V , there is at most one function f ∈ Act(l) with f(0) = v....
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...3 (Hybrid System [1]) A hybrid system is made of 6-components:...
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1,260 citations
"On the computational power of dynam..." refers background in this paper
...See [1, 2] for the definitions of the following special cases of linear hybrid systems: discrete systems, finite-state systems, timed automata, multi-rate timed systems, n-rate time systems, integrator systems....
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...Some undecidability results are known [1, 2, 10, 12], but only a small number of papers have been devoted to the study of hybrid systems as computational models: the work of Asarin, Maler and Pnueli [3, 4, 5] about Piecewise Constant Derivative systems and the work of Branicky [9] about simulation capabilities of Ordinary Differential Equations can however be mentioned....
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