Information Theory, Inference, and Learning ... - Inference Group

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.520 42 — Hopfield NetworksLyapunov functionsExercise 42.11. [3 ] Erik’s puzzle. In a stripped-down version of Conway’s gameof life, cells are arranged on a square grid. Each cell is either alive ordead. Live cells do not die. Dead cells become alive if two or more oftheir immediate neighbours are alive. (Neighbours to north, south, eastand west.) What is the smallest number of live cells needed in orderthat these rules lead to an entire N × N square being alive?In a d-dimensional version of the same game, the rule is that if d neighboursare alive then you come to life. What is the smallest number oflive cells needed in order that an entire N × N × · · · × N hypercubebecomes alive? (And how should those live cells be arranged?)The southeast puzzle→→Figure 42.12. Erik’s dynamics.(a)✉✉✲❄(b)✲✉✉✲❄(c)✲✉✉✉✲❄(d)✲✉✉✉✉✲ . . . (z) ✲❡❡❡❡❡❡❡❡❡❡The southeast puzzle is played on a semi-infinite chess board, starting atits northwest (top left) corner. There are three rules:Figure 42.13. The southeastpuzzle.1. In the starting position, one piece is placed in the northwest-most square(figure 42.13a).2. It is not permitted for more than one piece to be on any given square.3. At each step, you remove one piece from the board, and replace it withtwo pieces, one in the square immediately to the east, and one in the thesquare immediately to the south, as illustrated in figure 42.13b. Everysuch step increases the number of pieces on the board by one.After move (b) has been made, either piece may be selected for the next move.Figure 42.13c shows the outcome of moving the lower piece. At the next move,either the lowest piece or the middle piece of the three may be selected; theuppermost piece may not be selected, since that would violate rule 2. At move(d) we have selected the middle piece. Now any of the pieces may be moved,except for the leftmost piece.Now, here is the puzzle:⊲ Exercise 42.12. [4, p.521] Is it possible to obtain a position in which all the tensquares closest to the northwest corner, marked in figure 42.13z, areempty?[Hint: this puzzle has a connection to data compression.]42.11 SolutionsSolution to exercise 42.3 (p.508). Take a binary feedback network with 2 neuronsand let w 12 = 1 and w 21 = −1. Then whenever neuron 1 is updated,it will match neuron 2, and whenever neuron 2 is updated, it will flip to theopposite state from neuron 1. There is no stable state.