# Exercises for Chaos Under Control

## Chapter 9: Neural Networks

43. The simple three cell automaton of Figure 9.32 can compute other logic functions provided the rules are changed for each. Supply simple, correct cellular automata rules for the functions

(a) Exclusive OR (either p or q [but not both])
p q either p or q
T T F
T F T
F T T
F F F

(b) AND
p q p and q
T T T
T F F
F T F
F F F

(c) OR NOT
p q p or not q
T T T
T F T
F T F
F F T
 Problems 44 through 48 refer to a neural net with this architecture

44.[N] Suppose the neural net is assigned weights wpr = 2.5, wps = -1.4, wqr = 1.2, wqs = 2.1, wro = -1.6, and wso = 1.4. Test the pair (p, q)/o = (T, T)/T. If necessary, use backpropagation to find weights which satisfy this input/output pair. The conversion rule is this: if the input of a neuron is < 0, then the activation level is 0, otherwise it is 1. Answer

45.[N] How well do the weights found in Exercise 44 compute the remaining entries in the IMPLIES table? Answer

Problems 46 through 48 refer to the neural net with weights wpr = 1.4, wps = -2.1, wqr = 3.2, wqs = 0.5, wro = -0.9, and wso = 1.6. The conversion rule is this: if the input of a neuron is ² 0, then the activation level is 0, otherwise it is 1.

46. Suppose both p and q have activation level 1. Use feed forward to compute the activation level at r. Answer

47. Suppose r, s, and o have activation level 1, and suppose the target value at o is 0.

(a) Compute the error at o, the error at r, and the error at s. Answer

(b) Use backpropagation to compute the new weights wro and wso. Answer

48. Suppose both p and q have activation level 0. Use feed forward to compute the activation level at o. (Think before you do arithmetic.) Answer