C++ Neural Networks and Fuzzy Logic: Preface

C++ Neural Networks and Fuzzy Logic

by Valluru B. Rao

MTBooks, IDG Books Worldwide, Inc.

ISBN: 1558515526   Pub Date: 06/01/95

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Tic−Tac−Toe Anyone?

David Fogel describes evolutionary general problem solving and uses the familiar game of Tic−Tac−Toe as

an example. The idea is to come up with optimal strategies in playing this game. The first player’s marker is

an X, and the second player’s marker is an O. Whoever gets three of his or her markers in a row or a column

or a diagonal before the other player does, wins. Shrewd players manage a draw position, if their equally

shrewd opponent thwarts their attempts to win. A draw position is one where neither player has three of his or

her markers in a row, or a column, or a diagonal.

The board can be described by a vector of nine components, each of which is a three−valued number. Imagine

the squares of the board for the game as taken in sequence row by row from top to bottom. Allow a 1 to show

the presence of an X in that square, a 0 to indicate a blank there, and a −1 to correspond to an O. This is an

example of a coding for the status of the board. For example, (−1, 0, 1, 0, −1, 0, 1, 1, −1) is a winning position

for the second player, because it corresponds to the board looking as below.

 O       X

     O

 X   X   O

A neural network for this problem will have an input layer with nine neurons, as each input pattern has nine

components. There would be some hidden layers. But the example is with one hidden layer. The output layer

also contains nine neurons, so that one cycle of operation of the network shows what the best configuration of

the board is to be, given a particular input. Of course, during this cycle of operation, all that needs to be

determined is which blank space, indicated by a 0 in the input, should be changed to 1, if strategy is being

worked for player 1. None of the 1’s and −1’s is to be changed.

In this particular example, the neural network architecture itself is dynamic. The network expands or contracts

according to some rules, which are described next.

Fogel describes the network as an evolving network in the sense that the number of neurons in the hidden

layer changed with a probability of 0.5. A node was equally likely to be added or deleted. Since the number of

unmarked squares dwindles after each play, this kind of approach with varying numbers of neurons in the

network seems to be reasonable, and interesting.

The initial set of weights is random values between −0.5 and 0.5, inclusive, according to a uniform

distribution. Bias and threshold values also come from this distribution. The sigmoid function:

1/(1 + e

−x

)

is used also, to determine the outputs.

Weights and biases were changed during the network operation training cycles. Thus, the network had a

learning phase. (You will read more on learning in Chapter 6.) This network is adaptive, since it changes its

C++ Neural Networks and Fuzzy Logic:Preface

Tic−Tac−Toe Anyone?

75

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architecture. Other forms of adaptation in neural networks are in changing parameter values for a fixed

architecture. (See Chapter 6.) The results of the experiment Fogel describes show that you need nine neurons

in the hidden layer also, for your network to be the best for this problem. They also purged any strategy that

was likely to lose.

Fogel’s emphasis is on the evolutionary aspect of an adaptive process or experiment. Our interest in this

example is primarily due to the fact that an adaptive neural network is used.

The choice of Tic−Tac−Toe, while being a simple and all too familiar game, is in the genre of much more

complicated games. These games ask a player to place a marker in some position in a given array, and as

players take turns doing so, some criterion determines if it is a draw, or who won. Unlike in Tic−Tac−Toe, the

criterion by which one wins may not be known to the players.