Home         Authors   Papers   Year of conference   Themes   Organizations        To MES conference

Linear Synthesis of k-valued Digital Structures: Principle of Generalization
Authors
	Prokopenko N.N.
	Chernov N.I.
	Yugai V.Ya.
	Butyrlagin N.V.
Date of publication
	2016
Abstract
	The circuit microelectronic elements stock improvement measures are proposed to change the traditional body of mathematics of logic synthesis of digital IP-units (of Boolean algebra) with another - linear algebra. Linear algebra is an assemblage of the two mathematic structures: a field enabling to build up basis vectors of linear space with using of operations differing from the operations of linear space and a linear space with the standard set of operations of algebraic addition of vectors and multiplication of a vector by a number. Such a structure provides certain freedom in choosing operations of forming bases including the bases having the properties of functional elements. As this takes place, a series of useful properties is provided as follows: – freedom in choosing operations while forming bases for implementing functional elements makes it possible to take an optimal choice of circuits engineering solutions of the base functional elements within the chosen technology; – representation of a vector in the linear space through the basis of interest (i.e. of logic function through the descriptions chosen for representation of the basis functional elements) can be considered as a difference representation; in this case all the destabilizing factors affecting both of the difference components are substracted, resulting in their affect reduction and dynamic range widening of normal operation of the functional elements and IP-units taking everything into consideration; – while choosing a more appropriate (for example, current) representation of a signal the arithmetic operations of linear space on signals (addition, subtraction) are executed while wiring the elements, i. e. without using the hardware. – current representation allows to form a multi-valued signal in the form of a sum of the double-valued elements what determines a possibility to use the double-valued functional elements to implement multi-valued digital circuits through the application of wiring to the elements to ‘sum up’ the double-valued signals; – circuits engineering of the current digital circuits is, in fact, analogous what attracts widespread attention especially in creating analog-digital systems of the ‘system-on-chip’ type, since suggests producing analog and digital parts of ‘system’ within unified technological cycle; – while building up the computer-aided control systems one is often forced to process the current signals of multi-valued sensors and to provide logic transformations of current signals, for which current elements stock is ideally suited for: no necessary is in need in additional transformation of signals and appropriate increasing of the hardware costs spent, worsening of frequency characteristics, etc. A possibility of real logic synthesis and circuits engineering design of multi-valued elements stock on double-valued functional elements, i.e. transistors are an advantage of linear algebra alongside logic synthesis of the double-valued digital structures having the improved technical characteristics. A problem of synthesis of multi-valued elements stock is usually considered as a problem of generalizing the double-valued logic functions (operations, functionally complete systems, etc.) for multi-valued case. Some set of solutions in multi-valued logic corresponds to its solutions in double-valued logic. As this takes place, a choice of some solution determines not only circuits engineering peculiarities, but system engineering ones of implementation of multi-valued digital devices. One solution of the problem of generalizing double-valued logic to multi-valued one is possible even in Boolean algebra; however practical importance of this fact is too small: a development of the multi-valued elements stock in Boolean logic is impossible due to the absence of functional elements having amounts of various states which are greater than two. For example, linear algebra allows to carry out logic synthesis and circuits engineering design of multi-valued elements stock; this being so, one solution of the problem of generalizing double-valued logic to multi-valued one in linear algebra is of practical importance. The prime object of this work is a demonstration of some peculiarities of linear algebra in generalizing double-valued operations and structures to multi-valued case and also logic synthesis of multi-valued digital structures with using a way of generalizing put forward by us. In connection with the available possibilities of implementation of multi-valued digital IP-units some problems of their synthesis with using the principle of generalizing analog double-valued structures. An analysis of the study results reveals that various essential filling of this principle results in various variants of generalizing double-valued digital structures to multi-valued ones. The examples of the solution of the problem of generalizing double-valued digital to multi-valued ones are illustrated by the example of designing double- and three-valued linear one-digit adders. On grounds of comparison of the schematic circuits and algorithms of functioning of the adders it is demonstrated that the adders may be divided into prefull, full, and superfull ones depending on the number of inputs. The general linear expressions had been determined for logic synthesis of such adders of any valuedness. The time characteristics of the three-valued linear adder circuit operation demonstrate high efficiency of the obtained circuits engineering solution. The base concept of the linear logic synthesis of digital IP-modules that is put forward in this article combines analog approach to circuits engineering design of digital structures of processing the data of the analog-digital (multi-valued) signal representation. To a certain extent, this is a new implementation of the known principles of building up digital elements stock and techniques of data processing. The cited examples of the linear logic synthesis make a prediction about an appearance of new properties and methods of multi-valued processing of data the existence of which is impossible in double-valued logic by virtue of its degenerate nature. The results of the carried out studies demonstrate that while generalizing double-valued structures and techniques of data processing to multi-valued case it is necessary to follow a common known caution, considering that various variants of generalizing may result in cardinal distinctions in the principles of designing multi-valued digital devices.
Keywords
	logical synthesis, double-valued logic, multi-valued logic, functionally complete system, linear algebra, linear space, basis matrix, current mirror, linear circuitry.
Library reference
	Prokopenko N.N., Chernov N.I., Yugai V.Ya., Butyrlagin N.V. Linear Synthesis of k-valued Digital Structures: Principle of Generalization // Problems of Perspective Micro- and Nanoelectronic Systems Development - 2016. Proceedings / edited by A. Stempkovsky, Moscow, IPPM RAS, 2016. Part 1. P. 70-77.
URL of paper
	http://www.mes-conference.ru/data/year2016/pdf/D045.pdf