An Extension of Threshold Logic

This paper presents an extension of the concept of linear separation to generate a wider class of Boolean functions. This generalization is affected by modifying the requirements for separation to allow both true and false assignments to reside on the same hyperplane (pseudoseparation) or within the same neighborhood of a hyperplane (marginal separation). Boolean functions which can be separated in either of these ways are called partially separable functions. This paper is divided into two parts. The first deals with pseudoseparable functions and the second deals with marginally separable functions. Some of the theoretical algebraic properties of partially separable functions are investigated, and the relationship between these and the corresponding properties for separable functions is explored. Necessary and sufflcient conditions for each type of separation are also discussed, A parameter which shows whether a given Boolean function is separable, pseudoseparable, or marginally separable is introduced.