In most modern video games, character behavior is scripted; no matter how many times the player exploits a weakness, that weakness is never repaired. Yet if game characters could learn through interacting with the player, behavior could improve as the game is played, keeping it interesting. This paper introduces the real-time NeuroEvolution of Augmenting Topologies (rtNEAT) method for evolving increasingly complex artificial neural networks in real time, as a game is being played. The rtNEAT method allows agents to change and improve during the game. In fact, rtNEAT makes possible an entirely new genre of video games in which the player trains a team of agents through a series of customized exercises. To demonstrate this concept, the NeuroEvolving Robotic Operatives (NERO) game was built based on rtNEAT. In NERO, the player trains a team of virtual robots for combat against other players ’ teams. This paper describes results from this novel application of machine learning, and demonstrates that rtNEAT makes possible video games like NERO where agents evolve and adapt in real time. In the future, rtNEAT may allow new kinds of educational and training applications through interactive and adapting games. 1

We present a model of computation with ordinary differential equations (ODEs) which converge to attractors that are interpreted as the output of a computation. We introduce a measure of complexity for exponentially convergent ODEs, enabling an algorithmic analysis of continuous time flows and their comparison with discrete algorithms. We define polynomial and logarithmic continuous time complexity classes and show that an ODE which solves the maximum network flow problem has polynomial time complexity. We also analyze a simple flow that solves the Maximum problem in logarithmic time. We conjecture that a subclass of the continuous P is equivalent to the classical P. 2001 Elsevier Science (USA) Key Words: theory of analog computation; dynamical systems.

This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess.

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References and Further Readings Aserinsky, E., and N. Kleitman. (1953). Regularly occurring periods of eye motility and concomitant phenomena during sleep. Science 118: 273--274. Foulkes, D. (1985). Dreaming: A Cognitive-Psychological Analysis. Mahwah, NJ: Erlbaum. Freud, S. (1900). The Interpretation of Dreams. Trans. J. Strachey. New York: Basic Books. Hobson, J. A. (1988). The Dreaming Brain. New York: Basic Books. Hobson, J. A. (1990). Activation, input source, and modulation: A neurocognitive model of the state of the brain-mind. In R. Bootzin, J. Kihlstrom, and D. Schacter, Eds., Sleep and Cognition. Washington, DC: American Psychological Association, pp. 25--40. Hobson, J. A. (1994). The Chemistry of Conscious States. Boston: Little Brown. Hobson, J. A., and R. W. McCarley. (1977). The brain as a dreamstate generator: An activation-synthesis hypothesis of the dream

Non-self-similar sets defined by a decision procedure are numerically investigated by introducing the notion of inaccessibility to (ideal) decision procedure, that is connected with undecidability. A halting set of a universal Turing machine (UTM), the Mandelbrot set and a riddled basin are mainly investigated as non-self-similar sets with a decision procedure. By encoding a symbol sequence to a point in a Euclidean space, a halting set of a UTM is shown to be geometrically represented as a non-self-similar set, having different patterns and different fine structures on arbitrarily small scales. The boundary dimension of this set is shown to be equal to the space dimension, implying that the ideal decision procedure is inaccessible in the presence of error. This property is shown to be invariant under application of “fractal ” code transformations. Thus, a characterization of undecidability is given by the inaccessibility to the ideal decision procedure and its invariance against the code transformations. It is also shown that the distribution of halting time of the UTM, decays with a power law (or slower), and that this characteristic is also unchanged under code transformation. The Mandelbrot set is shown to have these features including the invariance against the code transformation, in common, and is connected with undecidable sets. In contrast, although a riddled basin, as a geometric representation of a certain context-free language, has the boundary dimension equal to the space dimension and a power law halting time distribution, these properties are not invariant against the code transformation. Thus, the riddled basin is ranked as middle between an ordinary fractal and a halting set of a UTM or the Mandelbrot set. Last, we