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Considerations of Completeness in the
Expression of Combinational Processes
Karl M. Fant
Theseus Research, Inc.
2524 Fairbrook Drive
Mountain View, CA 94040

Scott A. Brandt
UC Santa Cruz
Baskin Engineering 251
Santa Cruz, CA 95064


This paper explores the application of notions of completeness to the consideration of process expression. It is argued that current approaches to characterizing process expression embody elements of arbitrary sufficiency that hinders a full appreciation of the essential nature of process expression. Considering completeness relationships can eliminate these elements of arbitrary sufficiency and facilitate a more direct appreciation of the essential nature of process expression.

Specifically, the expression of combinational processes is considered. It is shown that a range of different forms of combinational expression can be related through concepts of varying degrees of completeness of logical determinability. Considerations of completeness also lead to the practical 2 value NULL Convention Logic [reference 2] which enables the practical design of fully logically determined systems.


Traditionally, in mathematics and in computer science, a complete process expression, one that is sufficient to resolve, is expressed as a coordination of multiple partial expressions, each with a different conceptual basis. Each partial expression is expressionally insufficient in itself, in that It is not resolvable on its own terms and is not expandable to resolvability on its own terms. These multiple forms of partial expression must be coordinated by additional expression which will be called meta-coordination.

A familiar example of the coordination of different forms of expression is clocked Boolean logic, which includes an expression in terms of Boolean logic and an expression in terms of time. The logic expresses the data transformation portion of the process but cannot express when a process begins and when a process ends. This is expressed in the form of periodic duration boundaries, typically, by a regularly pulsing clock signal controlling storage elements. Each logical data transformation process begins and ends on the duration boundary. These two different forms of expression must be coordinated such that all logic expressions complete their resolution well within each duration period. This meta-coordination is not inherent in either the expression of the logic or of the duration. Its basis is the time behavior of a specific implementation of the logic. The complete expression consists of a partial expression of logical relationships, a partial expression of time relationships and a partial meta-coordination expression that relates the logical and time expressions for some specific implementation. This will be referred to as multiple-form complete expression because the complete expression consists of multiple coordinated forms of individually insufficient partial expressions each with a different conceptual basis.

Another familiar example is the notion of the algorithm which postulates a symbol system that can be manipulated by a trained human or by an appropriately designed machine. The symbol system does not embody the expression of its own manipulation and a human or a machine does not inherently embody the symbol system. The training or the designing are the meta-coordinations coordinating the symbol system with, respectively, the human expression or the machine expression.

meta-coordination is an ad-hoc contribution of arbitrary adequacy that compensates for the primary expressional inadequacy of the individual forms. Any expression can be made adequately complete as a multiple form expression. One can always appeal to a sufficient number of incomplete forms of expression and then properly coordinate them to make a resolvable whole. In this sense, meta-coordination provides universality and generality of expression. It can always be applied and it can always be made to work. This is fine if what one wants to achieve is a working expression of a process but if one wants to understand the essential nature of process expression the presence of meta-coordination is a fundamental flaw that undermines any such effort. Appeal to arbitrary sufficiency reveals nothing about essential necessity.

NULL Convention Logic

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