Getting faster results, with more powerful, lower power consuming components and fewer interconnections by n-state switching

The drive of new technology is to provide more and better capabilities, hopefully at lower cost.  Especially in transmission, processing and storage of data there is an insatiable demand for resources.  Any time it is declared that resources are abundant and no purpose can be found for available resources an application appears that pushes resources again to the limit.  Error free wireless transmission, storing and processing of large high definition multi-media files simultaneously for a multitude of users is starting to become the norm nowadays, stretching available resources to the limit and requiring ever more powerful technology.  On top of that most data will be transmitted wirelessly.  The use of multi-state logic or n-state switching can contribute substantially to bettermuch faster processing at moderate clock speeds.

Multi-state switching and useful switching functions

The concept behind multi-state logic or n-state switching is simple.  Instead of having a signal able to assume one of 2 states such as in binary logic, in n-state switching a signal may assume one of more than 2 states, for instance 4 states.

It is often automatically assumed that the states of an n-state switching solution are dependent. That means that states are related and mistakes can be made in determining which states are present. Such a requirement of dependency of states is actually not a necessary requirement and working with independent states may provide significant advantages. However for describing switching functions the distinction between dependent and independent states is not important.

In binary logic the relation between two input signals and one output signal is described by a truth table of one of 16 possible binary switching functions.  Some of these binary functions are trivial, such as always 0 or always 1. Other functions, such as non-commutative functions are not very popular. Among the "popular" and best known binary functions are the AND, NAND, OR, NOR, XOR and EQUAL functions.

In 4-valued logic the relation between two 4-valued input signals and one 4-valued output signal is described by one of over 4 billion possible 4-valued switching functions.

Thus n-state switching of 2 input/one output functions has more functions than binary functions. N-state functions act upon a greater number of states of the same number of input signals.  It can thus deal in a much more efficient way with signals than binary logic.  This aspect may reduce power consumption, number of interconnections and increase processed information per clock cycle.

One of the challenges of course is to select and apply in a useful way the right n-state switching functions. Some of these functions are actually fairly well known, such as n-valued adders and multipliers.  Most of these functions are used in Galois Field arithmetic.

A special class of useful n-valued functions with n > 2 is formed by reversible n-valued functions.  This class of functions is especially useful in Linear Feedback Shift Register or LFSR circuits.  LFSR circuits are widely used in the generation of sequences and in coding and decoding applications.  This class of useful functions may be compared to the binary functions XOR and EQUAL.

An example

An example of a truth table a self-reversing 4-valued 2-inputs/single output is shown in the following table.

This table can be read as follows: a variable A can have a value 0, 1, 2 or 3 and is represented by a column under A in the table.  A variable B can also have a value 0, 1, 2 or 3 and is represented by a row to the right of B in the table. A value C is determined by selecting a value for A (or a column) and a value for B (or a row).  The value for C is determined by the element where column A and row B cross. It is easy to see that for instance when A = 2 and B = 2 then C =0.  The above function is of course the addition over GF(4).

Existing and beneficial n-state technology

While n-state technology may appear to be exotic, it is not nearly as exotic as people think. All of us apply 10-valued logic when doing arithmetic.  We memorize standard radix-10 addition tables and a 10-valued carry table when we do additions.  Most of us have also memorized standard radix-10 multiplication tables including the related carry tables.

The advantage of n-valued logic in arithmetic is well recognized and is for instance applied in table driven arithmetical computers.  The focus of radix-n addition is mostly on reducing the number of symbols that has to be processed. However speed of processing is also a very important factor.  Addition of for instance 4 and 5 in radix-10 takes processing of two symbols into one symbol (9).  Addition of the numbers in a binary way requires addition of [0 1 0 0] = 4 and[ 0 1 0 1] = 5 into [1 0 0 1] = 9, which includes processing a carry digit.  One can say that n-valued logic functions have a higher information processing capability than binary logic functions and do not require as many steps as lower valued logic.

Multiple Valued Logic exists as an academic discipline and is not unlike mathematical logic.  However most of it is very theoretical and is to n-state switching as presented by Ternarylogic as mathematical logic is to binary switching logic.

Finite Fields

The broadest and most applied form of n-state logic is in Finite Field or Galois Field mathematics.  The relations between n-valued symbols in a Galois Field (GF(n)) can be expressed by n-state truth tables. Usually a value of n is used wherein n is a multiple of 2.  Because of this n-valued logic tables can be realized by standard binary logic functions. For instance a 4-valued symbol can be represented by a 2-bits word.  The 4-valued truth table of the earlier shown 4-valued function can be realized by applying the binary XOR function to corresponding bits representing two 4-valued symbols.

N-state logic is widely used in error correcting coding on for instance music CDs and DVDs. However by applying the arithmetical approach the solutions are relatively complex and time and resource consuming. By using a real n-state approach one can create much easier to understand and faster coders and decoders.

Transmission codes

Not really n-state switching, but a step towards n-state logic is the field of n-valued transmission codes.  These are well established and widely applied.  The basic approach herein is to take a signal, provide it with a value and represent the value in digital n-valued symbols.  More common in this is to take a number of bits, which is a binary word representing a decimal value, and replace the word with a single symbol of a discrete value.  Rather than transmitting a series of bits only a single n-valued symbol has to be transmitted.  For instance high-speed DSL uses a 2B1Q code, replacing two bits by one 4-valued (or quaternary) symbol. A discrete value of a signal herein can be represented by the amplitude of a signal, but also by a phase of a signal or a position of a pulse.

Further reading

Ternarylogic LLC has created an introductory website for the field of n-state switching.  Please visit www.nstatelogic.com.