# Karnaugh Mapping with Two Outputs?

Is it possible to create a Karnaugh Map when there are two outputs, the examples I have seen are all one output. If it is possible, what is the process to convert a truth table like this to a k map? If it is not possible, are there any processes like a k-map I can use to help simplify the truth table?

For Example:

A,B,C|OutA,OutB
0,0,0|0,0
0,0,1|0,1
0,1,0|0,1
0,1,1|1,0
1,0,0|0,1
1,0,1|1,0
1,1,0|1,0
1,1,1|1,1


I beleive that you will need to do two Karnaugh maps; one for each output. After you have done the two maps, you can see if there are any common terms that can be used in a solution for both outputs.

I have provided a reference (a) below to a method which describes means for the combining of single output maps to provide solutions with multiple outputs. Examples with 2 outputs and 3 outputs are given but the method is extensible indefinitely.

Here is a larger and possibly more complete treatment of the same chapter. More examples and maybe more text plus an exam questions example.

All this MAY be from this book

• Introduction to Logic Design Alan Marcovitz ISBN13: 9780073191645 ISBN10: 0073191647 Division: Higher Education Pub Date: MAR-09, Publish Status: In Print Pages: 656 Edition: 3 Price: A$172.95 / NZ$ 192(Incl. GST)

• Also - Introduction to Logic Design with CD Alan Marcovitz ISBN13: 9780071123990 ISBN10: 0071123997 Division: Higher Education Pub Date: AUG-01, Publish Status: Out of Print Edition: 1 Price: A$149.95 / NZ$ 175(Incl. GST)

Also referenced (b) is a method that claims to allow wholly method based derivation of 2 (or more?) output solutions.

It looks like (a) will do what you want without making your head spin too much, and that (b) may be superb if you can spend the effort wading through it. In addition (b) provides terms which are liable to produce useful search-engines leads - eg It sounds like, as well as investigating the method described here, PPIs, Quine-McCluskey, Petrick, MOPIs.

I suspect that you may be able to go a long way towards dealing with a 2 output solution by, for each output, using the other output as an input in a one output Karnaugh map. ie if you have inputs A B C and outputs y z you may get useful results by considering maps A B C y -> z & A B C z -> y. But, maybe not. I'll have to have a play with this. Failing that, the following methods appear to meet you need.

(a) Rule based method for obtaining multiple output solutions from multiple single output Karnaugh maps.

This is a free access chapter from a McGraw Hill book with an excellent treatment of Karnaugh maps including dealing with systems with multiple outputs. Unfortunately they make it difficult to determine the book's identity. Examples of 2 output and 3 output solutions are given.

(b) A purely map procedure for two-level multiple-output logic minimization

Here is a complete paper describing "A purely map procedure for two-level multiple-output logic minimization"

A (n attempted :-) ) de-densification of their main points yields their claims of:

• a class-tested pedagogical treatment of the problem of collective two-level multiple-output logic minimization through a brief exposition of a novel efﬁcient procedure for tackling this problem.

• This procedure is a purely map heuristic which generalizes the Karnaugh map (K-map) procedure used in single-output minimization.

• In fact, the K-map procedure can be viewed as a short-cut technique for obtaining the minimal sum of a switching function without deriving the set of all its prime implicants or its complete sum

• The procedure is a short-cut technique for obtaining a minimal collective cover without the need to determine the set of all MOPIs or even its subset of PPIs

• The procedure retains a pure map nature as it avoids any resort to algebraic or tabular techniques in the form of a presence function or cover matrix, respectively.

• It is of a good pedagogical value because of the pictorial insight it provides, and because it leads to a natural understanding of pertinent terminology such as MOPIs and PPIs.

• All it requires of students is some repeated application of the familiar map heuristic ...

• and the implementation of a simple algorithm which handles visual interactions between various maps.

• Maps are grouped at distinct levels of a Hasse diagram, which is conveniently drawn in a K-map layout so that any parent map is easily visualized as adjacent to all it

In addition, if you can wade through the language, it's useful to note what they claim to be providing an alternative to, as this describes others ways of solving the problem, and provides search terms for more investigation. Viz -

• In many texts (e.g. [15]) students are not burdened with any particular details of collective minimization. Instead, they are instructed to apply separate or individual minimization and then try to identify obvious common product terms so as to share these between the pertinent output functions. Other texts [7, 10] introduce students to the classical method of collective minimization through a map derivation of all paramount prime implicants (PPIs), followed by the construction of a Quine–McCluskey cover matrix and/or a presence or Petrick function for selecting a minimal subset of the set of all PPIs. This method requires further checking for irredundant connections if exact minimization is to be achieved. Some texts present a simpliﬁed version of the classical method in which the ﬁnal solution is generated directly from the set of multiple-output prime implicants (MOPIs) without reducing the MOPI set to its subset of PPIs [8, 12, 13]. In this paper

It sounds like, as well as investigating the method described here, PPIs, Quine, McCluskey, Petrick, MOPIs and Google may be your friends :-).