Abstract

NULL Convention Logic (NCL) is a symbolically complete logic which expresses process completely in terms of the logic itself and inherently and conveniently expresses asynchronous digital circuits. The traditional form of Boolean logic is not symbolically complete in the sense that it requires the participation of a fundamentally different form of expression-time in the form of the clock - which has to be very carefully coordinated with the logic part of the expression to completely and effectively express a process. We introduce NULL Convention Logic in relation to Boolean logic as a four value logic, and as a three value logic, and finally as two value logic - - quite different from traditional Boolean logic. We then show how systems can be constructed entirely in terms of NULL Convention Logic.

1. Introduction

NULL Convention Logic (NCL) [7] is derived directly from the Invocation Model of Process Expression. The Invocation Model is a conceptual model of general process expression in contrast to a model of computation. It transcends limiting mathematical notions of computation to provide a unifying conceptual framework which relates all forms of process expression from the simplest physical and chemical processes to the most complex natural and artificial processes. For instance, the processes of cell metabolism and of digital computers are characterized in terms of the same concepts and relationships. They simply occupy different places in a single expression space defined by the Invocation Model.

A central concept of the Invocation Model is the general notion of completeness. Just how fundamental the notion of completeness is will become clear in the course of this discussion. It will appear in many guises in different aspects of process expression, but always in service of providing necessary and sufficient conditions. We will begin with the notion of symbolic completeness of expression. Symbolically complete expression is complete in and of itself solely in terms of the relationships among symbol values in the expression. It integrates the expression of data transformation and what is generally viewed as the expression of control in a single symbolically determined expression, without appeal to any extra-symbolic expression such as a clock or a controller. Processes in nature are symbolically complete, resolving spontaneously and autonomously without appeal to any global time authorities or control sequencers. The artificial processes devised by humans can also be symbolically complete.

Traditional Boolean logic is not symbolically complete. Among its component elements, a traditional Boolean logic circuit exhibits time dependent relationships as well as symbolic-value-dependent relationships. The symbolic-value-dependent relationships depend on the interconnection of the logic gates and their truth tables. The time-dependent relationships depend on the propagation delay times of the component elements to express the validity of data values and the invalidity of data values. These two aspects of expression are independent of each other in that the time relationships can be expressed quite arbitrarily in relation to the expression of symbolic relationships and vice versa. These two quite independent and partial expressions must be carefully and explicitly coordinated to provide a complete correctly resolvable expression of a process. A carefully engineered Boolean logic circuit with its clock is a complete expression and can be made to work, but it is not a symbolically complete expression.