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 toinvolve 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 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.