A Comparison of Map-Based Methods for Handling Type-2 and Type-3 Problems of Digital Circuit Design

With the advent of digital computers, several prominent problems of digital circuit design emerged. A
particular elementary class of these problems, (called Type-2 problems) can be divided into two
subclasses depending on whether an honest translator is possible or a sneaky translator is warranted.
The case of an honest translator is simply an inverse problem of logic, in which knowledge of the
vectorial function Z(X) is utilised to produce its inverse vectorial function X(Z). Though an old method
of solving type-2 problems was known almost half a century ago, two modern map-based methods
are now possible, namely the method of Boolean-equation solving and the method of input-domain
constraining. The paper aims to expose and illustrate these two novel methods, with stress on
comparing them together and demonstrating their superiority to (as well as an agreement with) the old
conventional method. This purpose is achieved by way of three typical classical examples for which
conventional solutions are somewhat tedious and cumbersome, while modern solutions are simple
and insightful. Throughout these examples, the Karnaugh map is effectively utilised, either in its
conventional version or in its variable-entered version. The Boolean-equation-solving method seems
to involve certain unwarranted steps that might be possibly skipped. However, its map-based variant
is an effective method for handling a related class of digital-design problems called Type-3 problems.
An example of a Type-3 problem is given to show how this method resolves and circumvents a certain
discrepancy that conventional techniques fell short of handling completely. The present study
exposed, illustrated, and compared the two methods of Boolean-equation solving and input-domain
constraining, which are novel methods for handling Type-2 problems of digital circuit design. Three
typical classical examples are presented, for which known conventional methods of solution are
somewhat tedious and cumbersome, while the map-based methods of solution presented herein are
simple and insightful. Throughout these examples, the Karnaugh map is effectively utilised, either in
its conventional version or in its variable-entered version. When used with Type-2 problems, the
Boolean-equation-solving method seems to involve certain unwarranted steps that might be possibly
skipped. However, its map-based variant is an effective method for handling a related class of digitaldesign
problems called Type-3 problems. An example of a Type-3 problem is given to show how this
method resolves and circumvents a certain discrepancy that conventional techniques fell short of
handling completely.