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Early development of ANN was limited by the revelation that the perceptron (the ANN of the time) could not implement a simple logical transformation, the exclusive or (xor). The logical capabilities of models can be examined by using fixed values to represent "true" and "false" -- usually 1 and 0, respectively. The exclusive or would then be the operation which produces a 1 when one or the other of two inputs is 1, and produces a 0 otherwise. Regardless of how the weights were set, a two-input perceptron could not implement this operation. This represented a fundamental limitation of the model for representing logical relationships. Later, it was realized that multilayer ANN models overcame this limitation.

The capability of MPR models to implement logical relationships was explored. It is apparent that any statement constructed with the logical operators "and", "or", "xor" or "not" can be easily represented using only linear interaction terms. These operators are modeled by the following:

 A and B A*B A or B A + B - A*B A xor B A + B - 2*A*B not A 1 - A

More complex logical statements are easily constructed by substituting the above expressions. For example, (A .and. B) .or. C is equivalent to A*B + C - A*B*C.

More importantly, a unique linear interaction model can be fitted to any truth table which can be constructed. For example, Table 1 shows a truth table constructed by the rule: Z = (B .xor. C) if .not. A, else Z = .not. (B .xor. C). Additional columns are shown for the computed values of A*B, A*C, B*C, and A*B*C.

TABLE 1. Truth table with three independent variables.

 A B C AB AC BC ABC Z 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1

Multilinear regression yields the first degree MPR model:

Z = A + B + C - 2AB - 2AC - 2BC + 4ABC.

A unique model can be found for any values placed in the column under Z. Increasing the degree n of the MPR model makes it possible to construct "truth tables" with more than two levels. For example, if variables were allowed to have three levels, say, -1, 0, and 1, a polynomial interaction model of degree 2 could completely describe the truth table.

From the point of view of continuous variables, this indicates that polynomial terms allow the model to discriminate independantly between, say, high, medium and low values of the variables. Linear models cannot treat these levels independantly.

The capability of MPR models to represent any logical relationship may be one of the most important indications of the strength of this type of model. It shows the power of the model to represent complex behaviors in an efficient representation which is easily determined from data. Copyright 2003 Simetrica LLC. All rights reserved. Simetrica Homepage | About Us | Contact | Products | Modeling Development Services | Client Case Studies | Email Feedback 