A primer on Karnaugh maps
Simplification can actually reduce the number of logic gates you need. And Karnaugh maps are a good simplification tool.
Last time, I introduced the ideas behind symbolic logic, logic circuits, and so on. This month, I'll talk more about minimizing logic equations; but before we start, I'd like to make a couple of corrections.
Electronics Workbench
First, last month I mentioned the computer program, Electronics Workbench. I wondered out loud if the company that makes it was still around. A Google search reveals that they are (www.electronicsworkbench.com). Their current product, MultiSim, seems to fill the niche formerly occupied by their Windows product. Unfortunately, a sales rep told me that they now concentrate only on industrial customers and no longer sell individual licenses. For the record, I invited their sales manager to write a paragraph explaining their current product line, but got no reply. I guess that's because I'm not a corporation. Should you be interested in tinkering with electronics sims, better choose one of the others.
Second, I should explain where I'm going with this new focus on electronics. Some of you may find electronics in general and digital electronics in particular interesting, and there's certainly nothing wrong with that. Some of us still get that nostalgic glow when we smell rosin-core solder, so if I pique your interest, that's fine. But it's not my intention to turn you all into electronics wizards. Rather, my focus here is on the use of symbolic logic. It's my view that a little knowledge of the rules and techniques of symbolic logic will be a nice tool to add to your toolbox and will help you write better software.
NAND or NOR?
Finally, reader Hank Schultz e-mailed to chide me for using the terms NAND and NOR too loosely. At first, I thought Hank was being too critical. I distinctly remembered using symbols for Fairchild's RTL logic gates as both NAND and NOR gates.
Here's the deal. Last month, I gave you DeMorgan's theorem, which I'll reproduce here for convenience:
(1)
From this, it's easy to see that a logic gate designed to do the AND function can just as easily do the OR function; it's simply a matter of defining which voltage—high or low—you define as representing a 1 (true) or 0 (false). I showed you symbols for each usage, for both RTL and TTL logic families (though the former is now mostly useful only to historians).
By convention, any signal that has a little circle at its connection is considered to be inverted. With TTL NAND gates, the inputs work correctly if a high voltage is 1. The circle is on the output. I showed the circles on the input for the RTL gate.
Well, maybe I used the gates that way, but Fairchild never did. The URL www.wps.com/archives/solid-state-datasheets/Datasheets/Fairchild-uL900-914/1.JPG shows an original Fairchild data sheet, copyrighted 1966. Their diagram clearly shows the μL914 as a NOR gate, not a NAND. They were consistent in that depiction. Figure 1 shows the symbol they used, which is perfectly consistent with modern practice.
Figure 1: Equivalent logic
In my defense, consider Fairchild's own words: "The μL914 device is a dual two-input gate. Each gate performs the NAND/NOR logical function using RTL circuitry."
Ok, maybe they didn't use the NAND symbol for the 914, but they certainly agreed that one could use it as a NAND gate. And I did.
Minimal is good
Let's get back to logic minimization, with an example. X is a function of four variables, A through D. Remember, in this notation, "+" stands for "or," and "." stands for "and." Also remember, a bar over a symbol or expression stands for "not."
(2)
Looks pretty messy, doesn't it? Nevertheless, this is often the kind of equation one gets after considering what a certain logical behavior must be. Let's see if we can simplify the equation by factoring. We'll start by factoring out terms in A and (omitting the and-ing dots for simplicity):
(3)
Note that the term is repeated in the last set of parentheses. A term OR'd with itself is redundant; we clearly only need one of the terms. We write:
(4)
A little more factoring gives:
(5)
Now, C has to be either true or not true; it can't be something else:
(6)
And our equation becomes:
(7)
It's not obvious how we can simplify further, but perhaps we can. Let's try expanding the terms again, and factoring in a different way. We get:
(8)
Now, DeMorgan's theorem says:
(9)
The first factored term in Equation 8 becomes:
(10)
Sometimes it helps to multiply a term by 1, in the form of Equation 6:
(11)
Now we can combine the two terms that have D as a factor, to get:
(12)
In the first term, we have a thing— —or'd with its own complement. That's the same form as in Equation 6, so we know how to handle it:
(13)
Equation 12 now becomes:
(14)
When all else fails, try expanding again:
(15)
Now look at the first and fourth terms. They factor to give, simply, D:
(16)
At this point, we can see that we have one term involving D alone, plus a lot of terms involving . Time to factor again:
(17)
Inside the parentheses, we have one term involving A alone, plus a lot of terms involving . Factoring yet again gives:
(18)
There's that familiar form of Equation 6 again. Simplifying from here gives:
(19)
Are you kidding me? After all that work, the equation reduces to nothing but a single constant?
Yep, that's what it does. You might have guessed it's because I designed the equation that way. Even so, I wasn't faking the simplification; I needed every step to get to the final form.
Admittedly, this is an extreme case. But it serves to illustrate a very important point: simplifying logic equations can be important. If we were to implement Equation 2 directly, using hardware logic gates, we'd need 27 gates. After simplifying, we need none at all; only a single wire from the high (plus, true, 1) supply to X. Perhaps, if X feeds into some other equation, we'd get even further simplifications there.
Now you can see why my friend's computer program that simplified logic automatically was so valuable. Equation 2, as complicated as it is, is only one equation with four inputs. Real systems might have hundreds of such equations, with dozens of inputs. If you're blessed with so much intuition that you could look at Equation 2 and see immediately that its simplification is trivial, you'll go far. Most of us have trouble even finding the best simplification. I had trouble doing this myself, and I'm the one who wrote down the equation in the first place. One wrong turn in the process, and we might easily have missed the key step.
I mentioned last month that there's more than one way to represent a given logical relationship. The most fundamental one is the truth table, in which you write down the desired output for all possible sets of inputs. The other is the logic equation and a third is an electronic circuit mechanizing the relationship. In this particular example, the truth table would have been the one to use. The truth table would have shown 1s for every single set of inputs, so the logic's trivial nature would have been obvious.
We don't, however, usually get to choose the form of the problem as it's given to us. We can convert from one form to another, but the advantage of one form over the other is rarely so clear. What we need, in general, are techniques that lead us to minimizations that aren't so obvious.
The Karnaugh map
One such technique is called the Karnaugh map, and it's one of the neatest approaches you'll ever see. I remember when I first saw it in a GE manual, I felt as though I'd found the pot of gold at the end of the rainbow.
Here's how it works. First, for four inputs (the case where the Karnaugh map works best), make a 4x4 array as in Figure 2.
Figure 2: The basic Karnaugh map
Two things are noteworthy about this map. First, we've arranged the 16 possible values of the four inputs as a 4x4 array, with two bits encoding each row or column.
The second and key feature is the way we number the rows and columns. They aren't in binary sequence, as you might think. As you can see, they have the sequence 00, 01, 11, 10. Some of you may recognize this as a Gray code.
Why this particular sequence? Because the codes associated with any two adjacent rows or columns represent a change in only one variable. In a true binary counting code, sometimes several digits can change in a single step; for example, the next step after 0x1111 is 0x10000. Five output signals must change values simultaneously. In digital circuits, this can cause glitches if one gate delay is a bit faster or slower than another. The Gray code avoids the problem. It's commonly used in optical encoders.
Suppose the output value for two adjacent cells is the same. Since only one input variable is changed between the two cells, this tells us that the output doesn't depend on that input. It's a "don't care" for that output.
Figure 3: Looking for patterns
Look at Figure 3. Group X is true for inputs ABCD = 0100 and 1100. That means that it doesn't depend on A, and we can write:
(20)
Similarly, Group Y doesn't depend on B or C. Its value is:
(21)
Note that the groupings don't necessarily have to be in contiguous rows or columns. In Group Z, the group wraps around the edges of the map.
If we can group cells by twos, we eliminate one input. By fours, two inputs, and so on. If the cell associated with a given output is isolated, it depends on all four inputs, and no minimization is possible.
The Karnaugh map gives us a wonderful, graphical picture that lets us group the cells in a near-optimal fashion. In doing so, we minimize the representation. Neat, eh?
Figure 4: Equation 2, mapped
Now let's see how Equation 2 plots onto the Karnaugh map, as shown in Figure 4. To make it easier to see, I'll assign a different lowercase letter to each term of the equation:
(22)
In this case, the individual groups don't matter much. All that counts is the largest group we can identify, which is of course the entire array of 16 cells. The output X is true for all values of the inputs, so all four are don't-care variables.
As you can see, using a Karnaugh map lets us see the results and, usually, the optimal grouping at a glance. It might seem that drawing a bunch of graphs is a tedious way to minimize logic, but after a little practice, you get where you can draw these simple arrays fast and see the best groupings quickly. You have to admit, it's a better approach than slogging through Equations 2 through 19.
The decade counter
Next, I'd like to show you a classic problem that we used to think was really important—at least until we got chips that did all the work for us. It's the decade counter. In this problem, we have four inputs and four outputs, which happen to be the next state of the four inputs. I'll denote the current states by the upper case letters, A through D, and the output (next value) states, lowercase a through d. Table 1 gives the truth table.
Note that, although a 4-bit number can encode 16 unique values, there are only 10 rows in the truth table. The other six rows aren't needed because the counter should never be in those states. When we draw the Karnaugh maps for this truth table, the unused cells will show up as six "don't-care" states. Quite literally, we don't care what value is output for those cases, since the inputs should never occur.
I should note in passing that if we truly implement this design, we had danged well better make sure that the don't- care states really never, ever occur. This may require some care when things power up.
Figure 5: Looking for patterns in decade counters
For the decade counter, we'll need four Karnaugh maps as shown in figure 5, one for each output value. In the figures, note how I let the groupings spill over into the don't-care areas if it will let me make larger groups.
The logic equations are:
(23)
These are by no means the only possible choices of terms, but we have reasonable assurance that they are, if not optimal, at least nearly so. That's because we used the groupings of the Karnaugh map to get the largest groups we could.
An exercise for the student
As another example, I'll give you a problem and its solution. This one encodes the output of a decade counter, to drive a seven-segment display.
figure 6: The seven-segment display
The standard nomenclature for this display is shown in figure 6. It assigns the letters a through g to the segments. Given any combination of 4-bit numbers, our goal is to light the combination of the seven segments in such a way as to display the proper decimal digit.
The truth table is shown in Table 2. Note that, as in Table 1, I'm using A as the most significant bit of the binary number.
Table 2: Seven-segment decoder
A | B | C | D | a | b | c | d | e | f | g |
0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 |
0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 |
0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 |
0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 |
0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |
As I said, I was going to show you how to generate the equations for the decoder. In restrospect, though, I think it's much better to involve a little audience participation. You have the truth table, you have the technology. You can use the practice, so how about you solve the problem instead of me?
You're going to need seven different Karnaugh maps; one for every segment of the display. Remember, we're talking decade counter here, so don't try to generate the remaining hex digits. Because only 10 of the 16 states are going to occur, you can use the remaining six as don't-care states, as I did in Figures 5a through 5d.
One hint: look for similar patterns in each of the seven maps. If you can minimize the logic the same way on more than one map, it would lower the total gate count for a practical implementation.
Next month, I'll show you my solution and share a personal tale about how I tried to build my own displays. I'll also show you the more powerful method by Quine and McCuskey.
See you then.
Jack Crenshaw is a senior software engineer at Spectrum-Astro and the author of Math Toolkit for Real-Time Programming, from CMP Books. He holds a PhD in physics from Auburn University. E-mail him at jcrens@earthlink.net.
Reader response
K-maps are the most underutilized tools in the industry. You make a brief reference to Gray codes and problems with race conditions. It is worth pointing out that one of the most powerful uses of K-maps is the ability to resolve race conditions (particularly those caused by propagation delays in the complement of a signal) by adding redundant logic terms that "overlap" adjoining groups. It is almost impossible to identify and resolve these types of glitches with other reduction methods with which I'm familiar.
Pete Jungwirth
Firmware Engineer
Renaissance Learning
While Karnaugh maps (even multidimensional ones) were a good tool, we let the synthesis tool take of the logic minimization for us these days.
Mike Murphree
I found your site searching on the internet for Karnaugh maps, and I found it a worthy explanation to stop and read through. What I would like to know, is what is your solution for the above 8 segment display decoder. I'm in the process of figuring out how to make such a display with 16 rows to display the a, b, c, d, e,and f, too. I would be interested in seeing what you came up with as Karnaugh maps. Your site helped me alot, and I'd like to return to find other topics about decoders and circuits and such to learn about. Thanks for your input!
Suzanne Rogers
Never forget that K-maps don't always work! Whereas Boolean algebra can always find a minimum expression, K-maps cannot be expected to do so. Assuming that a technique that simplifies calculations will always work is one of the biggest errors one can make.
David
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