marek@iuvax.cs.indiana.edu (Marek W Lugowski) (02/17/91)
All this talk of ants and my recent experiments with Computational Metabolism (C. Langton's _Artificial Life_, pp. 343-368), a tiling in motion, make me wonder about the utility of looking at ants through the prism of their final, or evolved, states. After all, these states are only fixed attractors with respect to the local conditions and the chaotic space. How much is there of interest, computationally, in such attractors? What can you learn from reaching them about any computation of interest, other than discrimination a la pattern recognition/categorization in the way of Widrow-Hoff or a steepest descent algorithm as found in simplest neural nets? That's boring... What I think is of interest is the path taken through the chaotic space. You may think of it as an informationally sensitive carrier wave, a la FM-modulation. There is enough richness specified in this path to encode a lot more than simple partition of the input space in a static recognition problem. For this you don't even need evolution to get started: I found out that for my tiles, which are no ants believe you me :), for different initial arrangements, under a particularly nasty set of rules that has the tiles leaping out of their little selves to wrap each species (color) around each other (you have to see the pictures...), I get stunningly different and beautifully chaotic paths to ...you guessed it, an attractor, one that, of course, I suspected anyway, since my rules don't change and I designed them for a square gworld and ran on a torus one. Now, if I could only learn how to modulate the path to effect an effective computation = controlled chaos... In summary, worry about paths in chaotic space and modulating them (and representing the process as a grammar, unfolding) instead of computing arrivals. Any thoughts, disagreements? -- Marek cc'ed to alife@iuvax.cs.indiana.edu, *the* mailing list :) (alife-request@iuvax for additions)
aboulang@bbn.com (Albert Boulanger) (02/19/91)
In article <1991Feb17.101444.17544@news.cs.indiana.edu> marek@iuvax.cs.indiana.edu (Marek W Lugowski) writes: What I think is of interest is the path taken through the chaotic space. You may think of it as an informationally sensitive carrier wave, a la FM-modulation. There is enough richness specified in this path to encode a lot more than simple partition of the input space in a static recognition problem. For this you don't even need evolution to get started: I found out that for my tiles, which are no ants believe you me :), for different initial arrangements, under a particularly nasty set of rules that has the tiles leaping out of their little selves to wrap each species (color) around each other (you have to see the pictures...), I get stunningly different and beautifully chaotic paths to ...you guessed it, an attractor, one that, of course, I suspected anyway, since my rules don't change and I designed them for a square gworld and ran on a torus one. Now, if I could only learn how to modulate the path to effect an effective computation = controlled chaos... Any thoughts, disagreements? Yes, two thoughts: *****************************1************************************ Trajectories instead of basins for computation: Asymmetric recurrent (Hopfield) neural nets have interesting (and trainable) trajectories. "Temporal Associations in Asymmetric Neural Networks", H Sompolinsky and I. Kanter, Phys Rev Lettr, Vol 57, No 22, 2861-2864 "Statistical Mechanics of Neural Networks", H. Sompolinsky, Physics Today, Dec. 1988, 70-80 "Hebbian Learning Reconsidered: Representation of Static and Dynamic Objects in Associative Neural Nets", A. Hertz, B. Sulzer, R. Kuhn, and J.L. Hemmen, Biol. Cybern., Vol 60, 1989, 457-467 Freeman and others have been proposing an the use of periodic attractors for associative memory: "Associative Memory in a Simple Model of Oscillating Cortex" Bill Baird, NIPS 2 Proceedings, D. Touretsky Ed, Morgan Kaufman. Backprop has been modified to work with hidden-layer networks with feedback connections and have the ability to learn phase-space trajectories. Many people have worked on this one: "Learning State Space Trajectories in Recurrent Neural Networks, Barak Pearlmutter, CMU Computer Science Report CMU-CS-88-191, Dec. 31, 1988 "Generalization of Back-Propagation to Recurrent Neural Networks" Fernando Pineda, Phys Rev Lettr, Vol 59, No 19, 2229-2232 Optical feedback with 4-wave mixing using photorefractive materials have interesting associative cycling behavior. *******************************2********************************* Chaos and computation Actually there are times that one wishes to use the ergodic properties of chaos in computation. This is a way of doing search. There is an annealing-like algorithm that makes use of this: "Chaotic Optimization and the Construction of Fractals: Solution of an Inverse Problem" Giorgio Mantica & Alan Sloan Complex Systems 3(1989), 37-62 Finally, here is some recent work by Crutchfield and Young in analyzing the pattern generation properties (using grammars) of system on the verge of chaos: "Computation at the Onset of Chaos", James Crutchfield and Karl Young, appearing in "Complexity, Entropy, and the Physics of Information", W. Zurek, ed., Addison-Wesley, 1989/ Harnessing chaos, Albert Boulanger aboulanger@bbn.com