Wiedermann, J. Complexity issues in discrete neurocomputing. Neural Network World, 4, 99--119, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....decision version of the MIN ENERGY problem, i.e. whether there exits a network state having an energy less than the prescribed value, is NP complete. This can be observed from the above mentioned reductions of hard optimization problems to MIN ENERGY. For an explicit NP completeness proof see e.g. [32] where a reduction from SAT is exploited. On the other hand there is a MIN ENERGY polynomial algorithm for special cases of Hopfield nets whose architectures are planar lattices [6] or planar graphs [3] Perhaps, the most direct and frequently used reduction to MIN ENERGY is from the MAX CUT ....

Wiedermann, J. Complexity issues in discrete neurocomputing. Neural Network World, 4, 99--119, 1994.

....now well understood in terms of classical models of computation such as Turing machines. Analogously, many fundamental results have been achieved in the past decade concerning the capabilities of neural networks for general computation [26] 45] 86] 90] 94] 95] 96] 105] 110] 128] [136]. In particular, the computational and descriptive powers of neural nets have been investigated by comparing their various architectures with each other and with more traditional computational models and descriptive tools such as nite automata, regular expressions, grammars, Turing machines, ....

J. Wiedermann, Complexity issues in discrete neurocomputing, Neural Network World 4 (1) (1994) 99-119.

....also gives the proofs of many of the most significant (semi ) elementary results. Our emphasis is at some points slightly different from Parberry s, and we also update his survey with some of the more recent developments. Other surveys of related topics, partially overlapping the present one, are [58, 72]. As a general introduction to neural networks theory, although mostly other than complexity aspects, the excellent textbook [33] can be recommended, and certain aspects of computations in cyclic networks are covered in depth in [40] The original PDP books [67, 48] still make very inspiring ....

Wiedermann, J. Complexity issues in discrete neurocomputing. In: Aspects and Prospects of Theoretical Computer Science. Proc. of the 6th Meeting of Young Computer Scientists (ed. J. Dassow, J. Kelemen). Lecture Notes in Computer Science 464, Springer-Verlag Berlin Heidelberg 1990. Pp. 93--108.

.... rst established for the Hop eld model in (Hop eld, 1982) In the present paper we investigate a number of issues in the computational analysis of Hop eld networks, complementing the existing literature in this area (Flor een and Orponen, 1994; Parberry, 1994; Siegelmann, 1999; Siu et al. 1995; Wiedermann, 1994). After a brief review of the basic de nitions in Section 2, our rst result in Section 3 establishes a size and time optimal (up to constant factors) simulation of arbitrary discrete time binary state neural networks (with in general asymmetric interconnections) by symmetric binarystate Hop eld ....

Wiedermann, J. (1994). Complexity issues in discrete neurocomputing. Neural Network World, 4:99-119.

.... instance, the convergence behavior of the BAM model was analyzed in [27] along the lines rst established for the Hop eld model in [20] In the present paper we investigate a number of issues in the computational analysis of Hop eld networks, complementing the existing literature in this area [12, 33, 39, 42, 49]. After a brief review of the basic de nitions in Section 2, our rst result in Section 3 establishes a size and time optimal (up to constant factors) simulation of arbitrary discrete time binary state neural networks (with in general asymmetric interconnections) by symmetric binary state Hop eld ....

J. Wiedermann. Complexity issues in discrete neurocomputing. Neural Network World, 4:99-119, 1994.

....decision version of the MIN ENERGY problem, i.e. whether there exits a network state having an energy less than a prescribed value, is NP complete. This can be observed from the above mentioned reductions of hard optimization problems to MIN ENERGY. For an explicit NP completeness proof see e.g. [29] where a reduction from SAT is exploited. On the other hand there is a MIN ENERGY polynomial algorithm for special cases of Hopfield nets whose architectures are planar lattices [6] or planar graphs [3] Perhaps, the most direct and frequently used reduction to MIN ENERGY is from MAX CUT problem ....

Wiedermann, J. Complexity issues in discrete neurocomputing. Neural Network World, 4, 99--119, 1994.

....in polynomial time within a factor n 1 Gammaffl for any fixed ffl 0, unless P = NP . Similar results for other problems arising in the context of analyzing Hopfield nets have been obtained in [1, 3] For general introductions to computational complexity issues in neural networks, see [8, 10, 11]. We start by examining the easier case: the synchronous network. As will be seen, the boundary between tractability and intractability is here located between computing direct (one step) and two step attraction radii. We first observe that the former can be computed in polynomial time. Theorem ....

J. Wiedermann, "Complexity issues in discrete neurocomputing," in Proc. Aspects and Prospects of Theoretical Computer Science. Lecture Notes in Computer Science 464, SpringerVerlag, Berlin, Germany, 1990, 480--491.

.... networks are those by Kamp and Hasler [46] and by Hertz, Krogh and Palmer [38] Related network models for associative memory applications are discussed by Kohonen in [53] For general introductions to computational complexity issues in neural networks, see the book [70] and the survey articles [68, 69, 87]. The most typical applications of Hopfield networks are as associative memories, and for solving combinatorial optimization problems. An (auto)associative content addressable memory is a storage device that stores vectors u i , for i = 1, 2, m, in such a way that upon presentation of an ....

....global minimum. In general, the initial vector must be near enough, in order to be able to produce the optimal solution. One approach is to choose randomly many different initial vectors and then perform the computation for them all. The best result is then given as the answer. However, Wiedermann [87, 88] has proved the following theorem, which we state without proof. Theorem 5.14 The problem Given a sequential Hopfield network with small weights and an integer k; is there an initial vector of the network for which the final state vector has energy k is NP complete. As it is a polynomial ....

J. Wiedermann. Complexity issues in discrete neurocomputing. In Proc. Aspects and Prospects of Theoretical Computer Science, pp. 480--491. Lecture Notes in Computer Science 464. Springer-Verlag, Berlin, 1990.

....A formal neuron. many of the most significant (semi )elementary results. Our emphasis is at some points slightly different from Parberry s, and we also update his article with some of the more recent developments. Other survey articles of related topics, partially overlapping the present one, are [73, 96]. At least two more comprehensive books [74, 75] are also in preparation. As a general introduction to neural networks theory, although mostly other than complexity aspects, the excellent textbook [40] can be recommended, and certain aspects of computations in cyclic networks are covered in ....

Wiedermann, J. Complexity issues in discrete neurocomputing. In: Aspects and Prospects of Theoretical Computer Science. Proc. of the 6th Meeting of Young Computer Scientists (ed. J. Dassow, J. Kelemen). Lecture Notes in Computer Science 464, Springer-Verlag Berlin Heidelberg 1990. Pp. 93--108.

....[12] exponential transient network as a clock . For general surveys of automata networks, see the books [5, 9] Threshold logic networks have recently become (again) popular as discrete models of neural networks. Computational aspects of these models are discussed in, e.g. the survey papers [17, 18, 26], and in the books [14, 19, 20, 22] 2 Preliminaries A threshold logic network (or a binary recurrent neural network ) consists of n threshold logic units (or binary neurons ) each of which is at a given moment in either one of two states x i = 1 or x i = 0, also called the on and off states. ....

Wiedermann, J. Complexity issues in discrete neurocomputing. In Aspects and Prospects of Theoretical Computer Science. Lecture Notes in Computer Science 464. Springer-Verlag, Berlin, 1990. Pp. 480--491.

....As a consequence, the same result for a neuromaton equivalence problem is achieved. 1 Introduction Neural networks [7] are models of computation motivated by our ideas about brain functioning. Both their computational power and their efficiency have been traditionally investigated [4, 14, 15, 19, 21] within the framework of computer science. One less commonly studied task which we will be addressing is the comparison of the computational power of neural networks with the traditional finite models of computation, such as recognizers of regular languages. It appears that a finite This ....

....neural acceptors, the so called Hopfield neuromata which are based on symmetric neural networks (Hopfield networks) In these networks the weights are symmetric and therefore, the architecture of such neuromata can be seen as an undirected graph. Hopfield networks have been traditionally studied [4, 21] and used, due to their favorite convergence properties. These networks are also of particular interest, because their natural physical realizations exist (e.g. Ising spin glasses, optical computers ) Using the concept of Hopfield neuromata, we will define a class of Hopfield languages that are ....

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Wiedermann, J. 1994. Complexity Issues in Discrete Neurocomputing. Neural Network World, 4, 99--119.

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