Functional Boolean models for systems with continuous variables

Abstract

Logical models are used for the description and analysis of biological systems. The problem discussed in this paper is the representation by a functional Boolean (logical) model of a system which has continuously ranging variables. The system to be represented is a relation among input and output vectors. The Boolean model is a system constructed by (i) mapping of con tinuous system variables into model variables of discrete levels (i.e. quantization), and (ii) coding of discrete levels into Boolean two-level variables. As is discussed in the paper, the construction of a Boolean model for a functional continuous system by quantization may produce a Boolean model which is not functional. A method is described for overcoming this problem by an overlap of quantized regions. A variable in an over lap region can be mapped into either of two discrete levels. Coding of the overlap regions into Boolean variables is simplified by the use of Gray's code. Consistency (i.e. the existence of a functional Boolean model for a system constructed by quantization) is described for the case of overlap, and a theorem is presented to simplify the testing of a Boolean system for consistency.