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Chapter 1
Logic Design Section (will be
Section 13.7)
1.1 A Brief Introduction to Switching Theory
and Logic Design
Disclaimer : I’m still looking for a good application for drawing logic gates. The figures here are quite rough.
Early computers relied on many switches to perform the logical operations needed for computation. This was true as late as the 1970’s when early personal computers such as the Altair ( Figure 1.1.1) started to appear. Pioneering com- puter scientists such as Claude Shannon realized that the operation of these computers could be simplified by making use of an isomorphism between com- puter circuits and boolean algebra. The term Switching Theory was used at the time. Logical gates realized through increasingly smaller and smaller inte- grated circuits still perform the same functions as in early computers, but using purely electronic means. In this section, we give examples of some switching circuits. Soon afterward, we will transition to the more modern form of circuits that are studied in Logic Design , where gates replace switches. Out main goal is to give you an overview of how boolean functions corresponds to any such circuit. We will introduce the common system notation used in logic de- sign and show how it corresponds with the mathematical notation of Boolean algebras. Any computer scientist should be familiar with both systems.
Figure 1.1.1: The Altair Computer, an early PC, by Todd Dailey, Creative Commons
The simplest switching device is the on-off switch. If the switch is closed/ ON, current will pass through it; if it is open/OFF, current will not pass through it. If we designate ON by 1, and OFF by 0, we can describe electrical circuits containing switches by Boolean expressions with the variables repre- senting the variable states of switches or the variable bits passing through gates. The electronics involved in these switches take into account whether we are negating a switch or not. For electromagnetic switches, a magnet is used to control whether the switch is open or closed. The magnets themselves may be controlled by simple ON/OFF switches. There are two types of electromagnetic switches. One is normally open (OFF) when the magnet is not activated, but activating the magnet will close the circuit and the switch is then ON. A separate type of switch corresponds with a negated switch. For that type, the switch is closed when the magnet is not activated, and when the magnet is activated, the switch opens. We won’t be overly concerned with the details of these switches or the electronics corresponding to logical gates. We will simply assume they are available to plug into a circuit. For simplicity, we use the inversion symbol on a varible that labels a switch to indicate that it is a switch of the second type, as in Figure 1.1.3. Standby power generators that many people have in their homes use a transfer switch to connect the generator to the home power system. This switch is open (OFF) if there is power coming from the normal municipal power supply. It stays OFF because a magnet is keeping it open. When power is lost, the magnet is no longer activated, and the switch closes and is ON. So the transfer switch is a normally ON switch.
Figure 1.1.2: Representation of a normally OFF switch controlled by variable x 1
Figure 1.1.3: Representation of a normally ON switch controlled by variable x 1 The standard notation used for Boolean algebra operations in switching theory and logic design is + for join, instead of ∨; and · for meet, instead of
degree. For example, the OR gate, usually drawn as in Figure 1.1.7 imple- ments the logical OR function. This happens electronically, but is equivalent to Figure 1.1.5. The AND gate, which is equivalent to two sequential switches is shown in Figure 1.1.7.
Figure 1.1.7: An OR gate
Figure 1.1.8: An AND gate
The complementation process is represented in a gate diagram by an in- verter, as pictured in Figure 1.1.9.
Figure 1.1.9: Inverter, or NOT gate
When drawing more complex circuits, multiple AND’s or OR’s are some- times depicted using a more general gate drawing. For example if we want to depict an OR gate with three inputs that is ON as long as at least one input is ON, we would draw it as in Figure 1.1.10, although this would really be two binary gates, as in Figure 1.1.11. Both diagrams are realizing the boolean expression x 1 + x 2 + x 3. Strictly speaking, the gates in Figure 1.1.11 represent (x 1 +x 2 )+x 3 , but the associative law for join tells us that the grouping doesn’t matter.
Figure 1.1.10: Simple version of a ternary OR gate
Figure 1.1.11: A ternary OR gate created with binary OR gates In Figure 1.1.12, we show a few other commonly used gates, XOR, NAND, and NOR, which correspond to the boolean exressions x 1 ⊕ x 2 , x 1 · x 2 , and x 1 + x 2 , respectively.
Figure 1.1.
Let’s start with a logic circuit and see how the laws of boolean algebra can help us simplify it.
Example 1.1.13 Simplification of a circuit. Consider the circuit in Figure 1.1.14. As usual, we assume that three inputs enter on the left and the output exits on the right.
Figure 1.1.14: Initial gate diagram
If we trace the inputs through the gates we see that this circuit realizes the boolean function
f (x 1 , x 2 , x 3 ) = x 1 · x 2 · ((x 1 + x 2 ) + (x 1 + x 3 )).
We simplify the boolean expression that defines f , simplifying the circuit in so doing. You should be able to identify the laws of Boolean algebra that are used in each of the steps. See Exercise 1.1.1.
x 1 · x 2 · ((x 1 + x 2 ) + (x 1 + x 3 )) = x 1 · x 2 · (x 1 + x 2 + x 3 ) = x 1 · x 2 · x 1 + x 1 · x 2 · x 2 + x 1 · x 2 · x 3 = x 1 · x 2 + 0 · x 1 + x 3 · x 1 · x 2 = x 1 · x 2 + x 3 · x 1 · x 2 = x 1 · x 2 · (1 + x 3 ) = x 1 · x 2
Therefore, f (x 1 , x 2 , x 3 ) = x 1 · x 2 , which can be realized with the much simpler circuit in Figure 1.1.15, without using the input x 3.
Figure 1.1.18: A realization of the table of desired outputs.
Exercises
1. List the laws of boolean algebra that justify the steps in the simplification of the boolean function f (x 1 , x 2 , x 3 ) in Example 1.1.13. Some steps use more than one law. Answer.
(1) Associative, commutative, and idempotent laws.
(2) Distributive law.
(3) Idempotent and complement laws.
(4) Null and identity laws
(5) Distributive law.
(6) Null and identity laws.
2. Write the following Boolean expression in the notation of logic design.
(x 1 ∧ x 2 ) ∨ (x 1 ∧ x 2 ) ∨ (x 1 ∧ x 2 ).
Answer. (x 1 · x 2 ) + (x 1 · x 2 ) + (x 1 · x 2 ).
3. Find a further simplification of the boolean function in Example 1.1.16, and draw the corresponding gate diagram for the circuit that it realizes. Answer. A simpler boolean expression for the function is x 2 · (x 1 + x 3 ).
Figure 1.1.19: An even simpler circuit
4. Consider the switching circuit in Figure 1.1.20.
Figure 1.1.20: Can this circuit be simplifed?
(a) Draw the corresponding gate diagram for this circuit.
(b) Construct a table of outputs for each of the eight inputs to this circuit.
(c) Determine the minterm normal of the Boolean function based on the table.
(d) Simplify the circuit as much as possible.
5. Consider the circuit in Figure 1.1.21.
Figure 1.1.21: Can this circuit be simplifed?
(a) Trace the inputs though this circuit and determine the Boolean func- tion that it realizes.
(b) Construct a table of outputs for each of the eight inputs to this circuit.
(c) Find the minterm normal form of f.
(d) Draw the circuit based on the minterm normal form.
(e) Simplify the circuit algebraically and draw the resulting circuit.
6. Consider the Boolean function f (x 1 , x 2 , x 3 , x 4 ) = x 1 +(x 2 · (x 1 + x 4 ) + x 3 · (x 2 + x 4 )).
(a) Simplify f algebraically.
(b) Draw the gate diagram based on the simplified version of f.
7. Draw a logic circuit using only AND, OR and NOT gates that realizes an XOR gate.
