P. Koiran. Puissance de calcul des r'eseaux de neurones artificiels. PhD thesis, Ecole Normale Sup'erieure of Lyon, 1993  Home/Search   Document Not in Database   Summary   Related Articles   Check

....precisely its computational power as the computational power of analog recurrent neural networks [23] Then, several notions of simulation are introduced and compared. First section is ended by a study of the computational power of iterations of piecewise linear functions: we extend the results of [14, 12] and prove that the computational power of one to one piecewise linear functions is exactly the computational power of analog automata. Section two is devoted to continuous dynamical systems. A general framework is first given in order to consider the continuous systems as computational machines. ....

....systems. We will distinguish the notions of off line computations (the input is encoded in the initial configuration) and on line computations (the input is given bit after bit, during the evolution of the system) The definitions in this section and in the following section are derived from [14, 12]. Definition 2.2 (Off line system) An off line system is a 5 tuple S = Q; ffi; OE; A; R) where ffl (Q; ffi ) is a transition system without input. ffl OE : f0; 1g Q is an encoding function. ffl A; R ae Q are subsets of Q, such that A R = called the accepting and rejecting sets. ....

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P. Koiran. Puissance de calcul des r'eseaux de neurones artificiels. PhD thesis, Ecole Normale Sup'erieure of Lyon, 1993

.... for Scientific Research (Belgium) Hierarchy of Dynamical Systems 2 studied from both viewpoints, which gives important results in physics, mathematics and computer science [7, 8, 9, 10, 11, 12, 13, 14] There are also many papers and works about universality of these different kinds of systems [15, 2, 16, 17]. To compare automata and dynamical systems more precisely, we need to unify their respective frameworks. As we have said before, several authors have proposed new bridges between them but it has not been completely established yet. In this paper, we present a generalized version of dynamical ....

....different dynamical systems. If a dynamical is able to simulate another dynamical system, the former is more powerful than the latter. Of course, it can happen that the latter is also able to simulate the former. Then, both systems are equivalent. Several results exist on simulation (among others, [16, 12, 22, 23, 24, 13, 25, 26]) We consider here the most simple definition of simulation we have found in the literature. Definition 7.1 Simulation Let (X; f) and (Y; g) be two dynamical systems. Then f simulates g iff there exists an encoding function OE : Y X and a decoding function : X Y , such that 9m 0 : 8y 2 ....

P. Koiran. Puissance de Calcul des R'eseaux de Neurones Artificiels. PhD thesis, Laboratoire de l'Informatique du Parall'elisme, E.N.S.Lyon, 1993.

.... and grammars [24, 23, 26, 44] Cellular automata [42, 12] can be studied from both viewpoints, which gives important results in physics, mathematics and computer science [7, 11, 13, 14, 21, 23, 27, 40] There are also many papers and works about universality of these different kinds of systems [4, 20, 22]. Recently, some papers have shown models strictly more powerful than Turing machines. This does not contradict the Church Turing Thesis because these models are based on analog or real computations. Neural networks [14, 36] classical discrete time low dimensional maps [5] or continuous time ....

....steps is independent from the input considered. If A simulates B, then A is more powerful than B but this is only a weak result. If one is able to show the converse, both systems are said to be equivalent. In this case, we have a stronger result. Several results exist on simulation (among others, [22, 23, 25, 26, 28, 27, 32, 33]) We consider here the most simple definition of simulation we have found in the literature. Definition 7.1 Simulation Let (X; f) and (Y; g) be two dynamical systems. Then f simulates g iff there exists an encoding function OE : Y X and a decoding function : X Y , such that 9m 0 : 8y 2 ....

P. Koiran. Puissance de Calcul des R'eseaux de Neurones Artificiels. PhD thesis, Laboratoire de l'Informatique du Parall'elisme, E.N.S.Lyon, 1993.

....of analog recurrent neural networks [25] Then, several notions of simulation are introduced and compared. These notions are derived and adapted from [3, 5, 9, 13] First section is ended by a study of the computational power of iterations of piecewise linear functions: we extend the results of [13, 14, 16] and prove that the computational power of one to one piecewise linear functions is exactly the computational power of analog automata. Section two is devoted to continuous dynamical systems. A general framework is first given in order to consider continuous systems as computational machines. The ....

....systems. We will distinguish the notions of off line computations (the input is encoded in the initial configuration) and online computations (the input is given bit after bit, during the evolution of the system) The definitions in this section and in the following section are derived from [13, 14, 16]. Definition 2.2 (Off line system) An off line system is a 5 tuple S = Q; ffi; OE; A; R) where ffl (Q; ffi ) is a transition system without input. ffl OE : f0; 1g Q is an encoding function. ffl A; R ae Q are subsets of Q, such that A R = called the accepting and rejecting sets. ....

[Article contains additional citation context not shown here]

Pascal Koiran. Puissance de Calcul des r'eseaux de neurones artificiels. PhD thesis, ' Ecole Normale Sup'erieure de Lyon, June 1993.

....components are in Un . In this paper, we construct two elementary functions: one in one dimension based on counter machines that simulates Turing machines with an exponential slowdown, and another in two dimensions that simulates TMs in real time. Preliminary versions of these results appeared in [5] and [10] 2 One dimension: Minsky machines and Collatz functions Recall [6] the classic 3x 1 problem. If f is the function on the integers f(x) ae x=2 (x even) 3x 1 (x odd) then, for all x, does there exist a t such that f t (x) 1 In dynamical systems terms, is all of N in the ....

P. Koiran, "Puissance de calcul des r'eseaux de neurones artificiels." Ph.D. Thesis, Ecole Normale Sup'erieure de Lyon, 1993.

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