Chapter 3

Boolean Algebra and Digital Logic

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Chapter 3 Objectives

• Understand the relationship between Boolean logic and digital computer circuits.
• Learn how to design simple logic circuits.
• Understand how digital circuits work together to form complex computer systems.

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3.1 Introduction

• In the latter part of the nineteenth century, George Boole incensed philosophers and mathematicians alike when he suggested that logical thought could be represented through mathematical equations.
• How dare anyone suggest that human thought could be encapsulated and manipulated like an algebraic formula?
• Computers, as we know them today, are implementations of Boole’s Laws of Thought.
• John Atanasoff and Claude Shannon were among the first to see this connection.

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3.1 Introduction

• In the middle of the twentieth century, computers were commonly known as “thinking machines” and “electronic brains.”
• Many people were fearful of them.
• Nowadays, we rarely ponder the relationship between electronic digital computers and human logic. Computers are accepted as part of our lives.
• Many people, however, are still fearful of them.
• In this chapter, you will learn the simplicity that constitutes the essence of the machine.

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3.2 Boolean Algebra

• Boolean algebra is a mathematical system for the manipulation of variables that can have one of two values.
• In formal logic, these values are “true” and “false.”
• In digital systems, these values are “on” and “off,” 1 and 0, or “high” and “low.”
• Boolean expressions are created by performing operations on Boolean variables.
• Common Boolean operators include AND, OR, and NOT.

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3.2 Boolean Algebra

• A Boolean operator can be completely described using a truth table.
• The truth table for the Boolean operators AND and OR are shown at the right.
• The AND operator is also known as a Boolean product. The OR operator is the Boolean sum.

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3.2 Boolean Algebra

• The truth table for the Boolean NOT operator is shown at the right.
• The NOT operation is most often designated by an overbar. It is sometimes indicated by a prime mark ( ‘ ) or an “elbow” (^¬).

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3.2 Boolean Algebra

• A Boolean function has:
• At least one Boolean variable,
• At least one Boolean operator, and
• At least one input from the set {0,1}.
• It produces an output that is also a member of the set {0,1}.

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Now you know why the binary numbering system is so handy in digital systems.

3.2 Boolean Algebra

• The truth table for the Boolean function:

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is shown at the right.

• To make evaluation of the Boolean function easier, the truth table contains extra (shaded) columns to hold evaluations of subparts of the function.

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3.2 Boolean Algebra

• As with common arithmetic, Boolean operations have rules of precedence.
• The NOT operator has highest priority, followed by AND and then OR.
• This is how we chose the (shaded) function subparts in our table.

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3.2 Boolean Algebra

• Digital computers contain circuits that implement Boolean functions.
• The simpler that we can make a Boolean function, the smaller the circuit that will result.
• Simpler circuits are cheaper to build, consume less power, and run faster than complex circuits.
• With this in mind, we always want to reduce our Boolean functions to their simplest form.
• There are a number of Boolean identities that help us to do this.

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3.2 Boolean Algebra

• Most Boolean identities have an AND (product) form as well as an OR (sum) form. We give our identities using both forms. Our first group is rather intuitive:

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3.2 Boolean Algebra

• Our second group of Boolean identities should be familiar to you from your study of algebra:

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3.2 Boolean Algebra

• Our last group of Boolean identities are perhaps the most useful.
• If you have studied set theory or formal logic, these laws are also familiar to you.

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3.2 Boolean Algebra

• We can use Boolean identities to simplify the function:

as follows:

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3.2 Boolean Algebra

• Sometimes it is more economical to build a circuit using the complement of a function (and complementing its result) than it is to implement the function directly.
• DeMorgan’s law provides an easy way of finding the complement of a Boolean function.
• Recall DeMorgan’s law states:

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3.2 Boolean Algebra

• DeMorgan’s law can be extended to any number of variables.
• Replace each variable by its complement and change all ANDs to ORs and all ORs to ANDs.
• Thus, we find the the complement of:

​

is:

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3.2 Boolean Algebra

• Through our exercises in simplifying Boolean expressions, we see that there are numerous ways of stating the same Boolean expression.
• These “synonymous” forms are logically equivalent.
• Logically equivalent expressions have identical truth tables.
• In order to eliminate as much confusion as possible, designers express Boolean functions in standardized or canonical form.

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3.2 Boolean Algebra

• There are two canonical forms for Boolean expressions: sum-of-products and product-of-sums.
• Recall the Boolean product is the AND operation and the Boolean sum is the OR operation.
• In the sum-of-products form, ANDed variables are ORed together.
• For example:
• In the product-of-sums form, ORed variables are ANDed together:
• For example:

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3.2 Boolean Algebra

• It is easy to convert a function to sum-of-products form using its truth table.
• We are interested in the values of the variables that make the function true (=1).
• Using the truth table, we list the values of the variables that result in a true function value.
• Each group of variables is then ORed together.

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3.2 Boolean Algebra

• The sum-of-products form for our function is:

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We note that this function is not in simplest terms. Our aim is only to rewrite our function in canonical sum-of-products form.

3.3 Logic Gates

• We have looked at Boolean functions in abstract terms.
• In this section, we see that Boolean functions are implemented in digital computer circuits called gates.
• A gate is an electronic device that produces a result based on two or more input values.
• In reality, gates consist of one to six transistors, but digital designers think of them as a single unit.
• Integrated circuits contain collections of gates suited to a particular purpose.

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3.3 Logic Gates

• The three simplest gates are the AND, OR, and NOT gates.

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​

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• They correspond directly to their respective Boolean operations, as you can see by their truth tables.

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3.3 Logic Gates

• Another very useful gate is the exclusive OR (XOR) gate.
• The output of the XOR operation is true only when the values of the inputs differ.

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Note the special symbol ⊕ for the XOR operation.

3.3 Logic Gates

• NAND and NOR are two very important gates. Their symbols and truth tables are shown at the right.

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3.3 Logic Gates

• NAND and NOR are known as universal gates because they are inexpensive to manufacture and any Boolean function can be constructed using only NAND or only NOR gates.

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3.3 Logic Gates

• Gates can have multiple inputs and more than one output.
• A second output can be provided for the complement of the operation.
• We’ll see more of this later.

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3.4 Digital Components

• The main thing to remember is that combinations of gates implement Boolean functions.
• The circuit below implements the Boolean function:

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We simplify our Boolean expressions so that we can create simpler circuits.

3.5 Combinational Circuits

• We have designed a circuit that implements the Boolean function:

​

• This circuit is an example of a combinational logic circuit.
• Combinational logic circuits produce a specified output (almost) at the instant when input values are applied.
• In a later section, we will explore circuits where this is not the case.

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3.5 Combinational Circuits

• Combinational logic circuits give us many useful devices.
• One of the simplest is the half adder, which finds the sum of two bits.
• We can gain some insight as to the construction of a half adder by looking at its truth table, shown at the right.

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3.5 Combinational Circuits

• As we see, the sum can be found using the XOR operation and the carry using the AND operation.

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3.5 Combinational Circuits

• We can change our half adder into to a full adder by including gates for processing the carry bit.
• The truth table for a full adder is shown at the right.

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3.5 Combinational Circuits

• How can we change the half adder shown below to make it a full adder?

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3.5 Combinational Circuits

• Here’s our completed full adder.

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3.5 Combinational Circuits

• Just as we combined half adders to make a full adder, full adders can connected in series.
• The carry bit “ripples” from one adder to the next; hence, this configuration is called a ripple-carry adder.

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Today’s systems employ more efficient adders.

3.5 Combinational Circuits

• Decoders are another important type of combinational circuit.
• Among other things, they are useful in selecting a memory location according a binary value placed on the address lines of a memory bus.
• Address decoders with n inputs can select any of 2ⁿ locations.

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This is a block diagram for a decoder.

3.5 Combinational Circuits

• This is what a 2-to-4 decoder looks like on the inside.

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If x = 0 and y = 1, which output line is enabled?

3.5 Combinational Circuits

• A multiplexer does just the opposite of a decoder.
• It selects a single output from several inputs.
• The particular input chosen for output is determined by the value of the multiplexer’s control lines.
• To be able to select among n inputs, log₂n control lines are needed.

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This is a block diagram for a multiplexer.

3.5 Combinational Circuits

• This is what a 4-to-1 multiplexer looks like on the inside.

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If S₀ = 1 and S₁ = 0, which input is transferred to the output?

3.5 Combinational Circuits

• This shifter moves the bits of a nibble one position to the left or right.

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If S = 0, in which direction do the input bits shift?

3.6 Sequential Circuits

• Combinational logic circuits are perfect for situations when we require the immediate application of a Boolean function to a set of inputs.
• There are other times, however, when we need a circuit to change its value with consideration to its current state as well as its inputs.
• These circuits have to “remember” their current state.
• Sequential logic circuits provide this functionality for us.

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3.6 Sequential Circuits

• As the name implies, sequential logic circuits require a means by which events can be sequenced.
• State changes are controlled by clocks.
• A “clock” is a special circuit that sends electrical pulses through a circuit.
• Clocks produce electrical waveforms such as the one shown below.

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3.6 Sequential Circuits

• State changes occur in sequential circuits only when the clock ticks.
• Circuits can change state on the rising edge, falling edge, or when the clock pulse reaches its highest voltage.

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3.6 Sequential Circuits

• Circuits that change state on the rising edge, or falling edge of the clock pulse are called edge-triggered.
• Level-triggered circuits change state when the clock voltage reaches its highest or lowest level.

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3.6 Sequential Circuits

• To retain their state values, sequential circuits rely on feedback.
• Feedback in digital circuits occurs when an output is looped back to the input.
• A simple example of this concept is shown below.
• If Q is 0 it will always be 0, if it is 1, it will always be 1. Why?

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3.6 Sequential Circuits

• You can see how feedback works by examining the most basic sequential logic components, the SR flip-flop.
• The “SR” stands for set/reset.
• The internals of an SR flip-flop are shown below, along with its block diagram.

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3.6 Sequential Circuits

• The behavior of an SR flip-flop is described by a characteristic table.
• Q(t) means the value of the output at time t. Q(t+1) is the value of Q after the next clock pulse.

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3.6 Sequential Circuits

• The SR flip-flop actually has three inputs: S, R, and its current output, Q.
• Thus, we can construct a truth table for this circuit, as shown at the right.
• Notice the two undefined values. When both S and R are 1, the SR flip-flop is unstable.

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3.6 Sequential Circuits

• If we can be sure that the inputs to an SR flip-flop will never both be 1, we will never have an unstable circuit. This may not always be the case.
• The SR flip-flop can be modified to provide a stable state when both inputs are 1.

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• This modified flip-flop is called a JK flip-flop, shown at the right.

- The “JK” is in honor of

Jack Kilby.

3.6 Sequential Circuits

• At the right, we see how an SR flip-flop can be modified to create a JK flip-flop.
• The characteristic table indicates that the flip-flop is stable for all inputs.

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3.6 Sequential Circuits

• Another modification of the SR flip-flop is the D flip-flop, shown below with its characteristic table.
• You will notice that the output of the flip-flop remains the same during subsequent clock pulses. The output changes only when the value of D changes.

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3.6 Sequential Circuits

• The D flip-flop is the fundamental circuit of computer memory.
• D flip-flops are usually illustrated using the block diagram shown below.
• The characteristic table for the D flip-flop is shown at the right.

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3.6 Sequential Circuits

• The behavior of sequential circuits can be expressed using characteristic tables or finite state machines (FSMs).
• FSMs consist of a set of nodes that hold the states of the machine and a set of arcs that connect the states.
• Moore and Mealy machines are two types of FSMs that are equivalent.
• They differ only in how they express the outputs of the machine.
• Moore machines place outputs on each node, while Mealy machines present their outputs on the transitions.

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3.6 Sequential Circuits

• The behavior of a JK flop-flop is depicted below by a Moore machine (left) and a Mealy machine (right).

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3.6 Sequential Circuits

• Although the behavior of Moore and Mealy machines is identical, their implementations differ.

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This is our Moore machine.

3.6 Sequential Circuits

• Although the behavior of Moore and Mealy machines is identical, their implementations differ.

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This is our Mealy machine.

3.6 Sequential Circuits

• It is difficult to express the complexities of actual implementations using only Moore and Mealy machines.
• For one thing, they do not address the intricacies of timing very well.
• Secondly, it is often the case that an interaction of numerous signals is required to advance a machine from one state to the next.
• For these reasons, Christopher Clare invented the algorithmic state machine (ASM).

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The next slide illustrates the components of an ASM.

3.6 Sequential Circuits

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3.6 Sequential Circuits

• This is an ASM for a microwave oven.

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3.6 Sequential Circuits

• Sequential circuits are used anytime that we have a “stateful” application.
• A stateful application is one where the next state of the machine depends on the current state of the machine and the input.
• A stateful application requires both combinational and sequential logic.
• The following slides provide several examples of circuits that fall into this category.

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Can you think of others?

3.6 Sequential Circuits

• This illustration shows a 4-bit register consisting of D flip-flops. You will usually see its block diagram (below) instead.

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A larger memory configuration is shown on the next slide.

3.6 Sequential Circuits

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3.6 Sequential Circuits

• A binary counter is another example of a sequential circuit.
• The low-order bit is complemented at each clock pulse.
• Whenever it changes from 0 to 1, the next bit is complemented, and so on through the other flip-flops.

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3.6 Sequential Circuits

• Convolutional coding and decoding requires sequential circuits.
• One important convolutional code is the (2,1) convolutional code that underlies the PRML code that is briefly described at the end of Chapter 2.
• A (2, 1) convolutional code is so named because two symbols are output for every one symbol input.
• A convolutional encoder for PRML with its characteristic table is shown on the next slide.

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3.6 Sequential Circuits

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3.6 Sequential Circuits

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This is the Mealy machine for our encoder.

3.6 Sequential Circuits

• The fact that there is a limited set of possible state transitions in the encoding process is crucial to the error correcting capabilities of PRML.
• You can see by our Mealy machine for encoding that:

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F(1101 0010) = 11 01 01 00 10 11 11 10.

3.6 Sequential Circuits

• The decoding of our code is provided by inverting the inputs and outputs of the Mealy machine for the encoding process.
• You can see by our Mealy machine for decoding that:

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F(11 01 01 00 10 11 11 10) = 1101 0010

3.6 Sequential Circuits

• Yet another way of looking at the decoding process is through a lattice diagram.
• Here we have plotted the state transitions based on the input (top) and showing the output at the bottom for the string 00 10 11 11.

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F(00 10 11 11) = 1001

3.6 Sequential Circuits

• Suppose we receive the erroneous string: 10 10 11 11.
• Here we have plotted the accumulated errors based on the allowable transitions.
• The path of least error outputs 1001, thus 1001 is the string of maximum likelihood.

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F(00 10 11 11) = 1001

3.7 Designing Circuits

• We have seen digital circuits from two points of view: digital analysis and digital synthesis.
• Digital analysis explores the relationship between a circuits inputs and its outputs.
• Digital synthesis creates logic diagrams using the values specified in a truth table.
• Digital systems designers must also be mindful of the physical behaviors of circuits to include minute propagation delays that occur between the time when a circuit’s inputs are energized and when the output is accurate and stable.

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3.7 Designing Circuits

• Digital designers rely on specialized software to create efficient circuits.
• Thus, software is an enabler for the construction of better hardware.
• Of course, software is in reality a collection of algorithms that could just as well be implemented in hardware.
• Recall the Principle of Equivalence of Hardware and Software.

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3.7 Designing Circuits

• When we need to implement a simple, specialized algorithm and its execution speed must be as fast as possible, a hardware solution is often preferred.
• This is the idea behind embedded systems, which are small special-purpose computers that we find in many everyday things.
• Embedded systems require special programming that demands an understanding of the operation of digital circuits, the basics of which you have learned in this chapter.

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Chapter 3 Conclusion

• Computers are implementations of Boolean logic.
• Boolean functions are completely described by truth tables.
• Logic gates are small circuits that implement Boolean operators.
• The basic gates are AND, OR, and NOT.
• The XOR gate is very useful in parity checkers and adders.
• The “universal gates” are NOR, and NAND.

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Chapter 3 Conclusion

• Computer circuits consist of combinational logic circuits and sequential logic circuits.
• Combinational circuits produce outputs (almost) immediately when their inputs change.
• Sequential circuits require clocks to control their changes of state.
• The basic sequential circuit unit is the flip-flop: The behaviors of the SR, JK, and D flip-flops are the most important to know.

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Chapter 3 Conclusion

• The behavior of sequential circuits can be expressed using characteristic tables or through various finite state machines.
• Moore and Mealy machines are two finite state machines that model high-level circuit behavior.
• Algorithmic state machines are better than Moore and Mealy machines at expressing timing and complex signal interactions.
• Examples of sequential circuits include memory, counters, and Viterbi encoders and decoders.

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End of Chapter 3

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