1 Boolean Logic
Such simple things, And we make of them something so complex it defeats us, Almost.
—John Ashbery (b. 1927), American poet
Every digital device—be it a personal computer, a cellular telephone, or a network router—is based on a set of chips designed to store and process information. Al- though these chips come in different shapes and forms, they are all made from the same building blocks: Elementary logic gates. The gates can be physically imple- mented in many different materials and fabrication technologies, but their logical behavior is consistent across all computers. In this chapter we start out with one primitive logic gate—Nand—and build all the other logic gates from it. The result is a rather standard set of gates, which will be later used to construct our computer’s processing and storage chips. This will be done in chapters 2 and 3, respectively.
All the hardware chapters in the book, beginning with this one, have the same structure. Each chapter focuses on a well-defined task, designed to construct or inte- grate a certain family of chips. The prerequisite knowledge needed to approach this task is provided in a brief Background section. The next section provides a complete Specification of the chips’ abstractions, namely, the various services that they should deliver, one way or another. Having presented the what, a subsequent Implemen- tation section proposes guidelines and hints about how the chips can be actually implemented. A Perspective section rounds up the chapter with concluding com- ments about important topics that were left out from the discussion. Each chapter ends with a technical Project section. This section gives step-by-step instructions for actually building the chips on a personal computer, using the hardware simulator supplied with the book.
This being the first hardware chapter in the book, the Background section is somewhat lengthy, featuring a special section on hardware description and simulation tools.
8 Chapter 1
1.1 Background
This chapter focuses on the construction of a family of simple chips called Boolean gates. Since Boolean gates are physical implementations of Boolean functions, we start with a brief treatment of Boolean algebra. We then show how Boolean gates implementing simple Boolean functions can be interconnected to deliver the func- tionality of more complex chips. We conclude the background section with a descrip- tion of how hardware design is actually done in practice, using software simulation tools.
1.1.1 Boolean Algebra
Boolean algebra deals with Boolean (also called binary) values that are typically labeled true/false, 1/0, yes/no, on/off, and so forth. We will use 1 and 0. A Boolean function is a function that operates on binary inputs and returns binary outputs. Since computer hardware is based on the representation and manipulation of binary values, Boolean functions play a central role in the specification, construction, and optimization of hardware architectures. Hence, the ability to formulate and analyze Boolean functions is the first step toward constructing computer architectures.
Truth Table Representation The simplest way to specify a Boolean function is to enumerate all the possible values of the function’s input variables, along with the function’s output for each set of inputs. This is called the truth table representation of the function, illustrated in figure 1.1.
The first three columns of figure 1.1 enumerate all the possible binary values of the function’s variables. For each one of the 2 n possible tuples v1 . . . vn (here n 1⁄4 3), the lastcolumngivesthevalueof fðv1…vnÞ.
Boolean Expressions In addition to the truth table specification, a Boolean function can also be specified using Boolean operations over its input variables. The basic Boolean operators that are typically used are ‘‘And’’ (x And y is 1 exactly when both xand yare1)‘‘Or’’(xOr yis1exactlywheneitherxor yorbothare1),and‘‘Not’’ (Not x is 1 exactly when x is 0). We will use a common arithmetic-like notation for these operations: x y (or xy) means x And y, xþ y means x Or y, and x means Not x.
To illustrate, the function defined in figure 1.1 is equivalently given by the Boolean expression f ðx; y; zÞ 1⁄4 ðx þ yÞ z. For example, let us evaluate this expression on the
9 Boolean Logic xyz
000 001 010 011 100 101 110 111
Figure 1.1
f ðx; y; zÞ
0 0 1 0 1 0 1 0
Truth table representation of a Boolean function (example).
inputs x1⁄40, y1⁄41, z1⁄40 (third row in the table). Since y is 1, it follows that xþy1⁄41 and thus 101⁄4111⁄41. The complete verification of the equivalence between the expression and the truth table is achieved by evaluating the expression on each of the eight possible input combinations, verifying that it yields the same value listed in the table’s right column.
Canonical Representation As it turns out, every Boolean function can be expressed using at least one Boolean expression called the canonical representation. Starting with the function’s truth table, we focus on all the rows in which the function has value 1. For each such row, we construct a term created by And-ing together literals (variables or their negations) that fix the values of all the row’s inputs. For example, let us focus on the third row in figure 1.1, where the function’s value is 1. Since the variable values in this row are x 1⁄4 0, y 1⁄4 1, z 1⁄4 0, we construct the term xyz. Fol- lowing the same procedure, we construct the terms xyz and xyz for rows 5 and 7. Now, if we Or-together all these terms (for all the rows where the function has value 1), we get a Boolean expression that is equivalent to the given truth table. Thus the canonical representation of the Boolean function shown in figure 1.1 is f ðx; y; zÞ 1⁄4 xyz þ xyz þ xyz. This construction leads to an important conclusion: Every Boolean function, no matter how complex, can be expressed using three Boolean operators only: And, Or, and Not.
Two-Input Boolean Functions An inspection of figure 1.1 reveals that the number of Boolean functions that can be defined over n binary variables is 22n. For example, the sixteen Boolean functions spanned by two variables are listed in figure 1.2. These functions were constructed systematically, by enumerating all the possible 4-wise com- binations of binary values in the four right columns. Each function has a conventional
10 Chapter 1
Function
x
0011
y
0101
Constant 0 And
x And Not y x
Not x And y
y
Xor
Or
Nor Equivalence Not y
If y then x Not x
If x then y Nand Constant 1
0
xy xy x xy
y xyþxy xþy xþy xyþxy
y xþy x xþy xy 1
0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111
Figure 1.2 All the Boolean functions of two variables.
name that seeks to describe its underlying operation. Here are some examples: The name of the Nor function is shorthand for Not-Or: Take the Or of x and y, then negate the result. The Xor function—shorthand for ‘‘exclusive or’’—returns 1 when its two variables have opposing truth-values and 0 otherwise. Conversely, the Equivalence function returns 1 when the two variables have identical truth-values. The If-x-then-y function (also known as x ! y, or ‘‘x Implies y’’) returns 1 when x is 0 or when both x and y are 1. The other functions are self-explanatory.
The Nand function (as well as the Nor function) has an interesting theoretical property: Each one of the operations And, Or, and Not can be constructed from it, and it alone (e.g., x Or y 1⁄4 ðx Nand xÞ Nand ðy Nand yÞ. And since every Boolean function can be constructed from And, Or, and Not operations using the canonical representation method, it follows that every Boolean function can be constructed from Nand operations alone. This result has far-reaching practical implications: Once we have in our disposal a physical device that implements Nand, we can use many copies of this device (wired in a certain way) to implement in hardware any Boolean function.
11 Boolean Logic
1.1.2 Gate Logic
A gate is a physical device that implements a Boolean function. If a Boolean function f operates on n variables and returns m binary results (in all our examples so far, m was 1), the gate that implements f will have n input pins and m output pins. When we put some values v1 . . . vn in the gate’s input pins, the gate’s ‘‘logic’’—its internal structure—should compute and output f ðv1 . . . vnÞ. And just like complex Boolean functions can be expressed in terms of simpler functions, complex gates are com- posed from more elementary gates. The simplest gates of all are made from tiny switching devices, called transistors, wired in a certain topology designed to effect the
overall gate functionality.
Although most digital computers today use electricity to represent and transmit
binary data from one gate to another, any alternative technology permitting switch- ing and conducting capabilities can be employed. Indeed, during the last fifty years, researchers have built many hardware implementations of Boolean functions, includ- ing magnetic, optical, biological, hydraulic, and pneumatic mechanisms. Today, most gates are implemented as transistors etched in silicon, packaged as chips. In this book we use the words chip and gate interchangeably, tending to use the term gates for simple chips.
The availability of alternative switching technology options, on the one hand, and the observation that Boolean algebra can be used to abstract the behavior of any such technology, on the other, is extremely important. Basically, it implies that computer scientists don’t have to worry about physical things like electricity, circuits, switches, relays, and power supply. Instead, computer scientists can be content with the abstract notions of Boolean algebra and gate logic, trusting that someone else (the physicists and electrical engineers—bless their souls) will figure out how to actually realize them in hardware. Hence, a primitive gate (see figure 1.3) can be viewed as a black box device that implements an elementary logical operation in one way or another—we don’t care how. A hardware designer starts from such primitive gates and designs more complicated functionality by interconnecting them, leading to the construction of composite gates.
aa
b And out b Or out in Not out
Figure 1.3 Standard symbolic notation of some elementary logic gates.
12 Chapter 1
Gate interface
And out
Gate implementation
a b c
else out=0
a
b If a=b=c=1 then out=1 c
out
And
Figure 1.4 Composite implementation of a three-way And gate. The rectangle on the right defines the conceptual boundaries of the gate interface.
Primitive and Composite Gates Since all logic gates have the same input and out- put semantics (0’s and 1’s), they can be chained together, creating composite gates of arbitrary complexity. For example, suppose we are asked to implement the 3-way Boolean function Andða;b;cÞ. Using Boolean algebra, we can begin by observing that abc1⁄4ðabÞc, or, using prefix notation, Andða;b;cÞ1⁄4AndðAndða;bÞ;cÞ. Next, we can use this result to construct the composite gate depicted in figure 1.4.
The construction described in figure 1.4 is a simple example of gate logic, also called logic design. Simply put, logic design is the art of interconnecting gates in order to implement more complex functionality, leading to the notion of composite gates. Since composite gates are themselves realizations of (possibly complex) Boolean functions, their ‘‘outside appearance’’ (e.g., left side of figure 1.4) looks just like that of primitive gates. At the same time, their internal structure can be rather complex.
We see that any given logic gate can be viewed from two different perspectives: external and internal. The right-hand side of figure 1.4 gives the gate’s internal architecture, or implementation, whereas the left side shows only the gate interface, namely, the input and output pins that it exposes to the outside world. The former is relevant only to the gate designer, whereas the latter is the right level of detail for other designers who wish to use the gate as an abstract off-the-shelf component, without paying attention to its internal structure.
Let us consider another logic design example—that of a Xor gate. As discussed before, Xorða;bÞ is 1 exactly when either a is 1 and b is 0, or when a is 0 and b is 1. Said otherwise, Xorða; bÞ 1⁄4 OrðAndða; NotðbÞÞ; AndðNotðaÞ; bÞÞ. This definition leads to the logic design shown in figure 1.5.
Note that the gate interface is unique: There is only one way to describe it, and this is normally done using a truth table, a Boolean expression, or some verbal specifica-
And
13 Boolean Logic
a
out
a b
Figure 1.5
Xor out
And
And
Or
a b out
00 01 10 11
0 1 1 0
b
Xor gate, along with a possible implementation.
tion. This interface, however, can be realized using many different implementations, some of which will be better than others in terms of cost, speed, and simplicity. For example, the Xor function can be implemented using four, rather than five, And, Or, and Not gates. Thus, from a functional standpoint, the fundamental requirement of logic design is that the gate implementation will realize its stated interface, in one way or another. From an efficiency standpoint, the general rule is to try to do more with less, that is, use as few gates as possible.
To sum up, the art of logic design can be described as follows: Given a gate spec- ification (interface), find an efficient way to implement it using other gates that were already implemented. This, in a nutshell, is what we will do in the rest of this chapter.
1.1.3 Actual Hardware Construction
Having described the logic of composing complex gates from simpler ones, we are now in a position to discuss how gates are actually built. Let us start with an inten- tionally na ̈ıve example.
Suppose we open a chip fabrication shop in our home garage. Our first contract is to build a hundred Xor gates. Using the order’s downpayment, we purchase a sol- dering gun, a roll of copper wire, and three bins labeled ‘‘And gates,’’ ‘‘Or gates,’’ and ‘‘Not gates,’’ each containing many identical copies of these elementary logic gates. Each of these gates is sealed in a plastic casing that exposes some input and output pins, as well as a power supply plug. To get started, we pin figure 1.5 to our garage wall and proceed to realize it using our hardware. First, we take two And gates, two Not gates, and one Or gate, and mount them on a board according to the
14 Chapter 1
figure’s layout. Next, we connect the chips to one another by running copper wires among them and by soldering the wire ends to the respective input/output pins. Now, if we follow the gate diagram carefully, we will end up having three exposed wire ends. We then solder a pin to each one of these wire ends, seal the entire device (except for the three pins) in a plastic casing, and label it ‘‘Xor.’’ We can repeat this assembly process many times over. At the end of the day, we can store all the chips that we’ve built in a new bin and label it ‘‘Xor gates.’’ If we (or other people) are asked to construct some other chips in the future, we’ll be able to use these Xor gates as elementary building blocks, just as we used the And, Or, and Not gates before.
As the reader has probably sensed, the garage approach to chip production leaves much to be desired. For starters, there is no guarantee that the given chip diagram is correct. Although we can prove correctness in simple cases like Xor, we cannot do so in many realistically complex chips. Thus, we must settle for empirical testing: Build the chip, connect it to a power supply, activate and deactivate the input pins in vari- ous configurations, and hope that the chip outputs will agree with its specifications. If the chip fails to deliver the desired outputs, we will have to tinker with its physical structure—a rather messy affair. Further, even if we will come up with the right de- sign, replicating the chip assembly process many times over will be a time-consuming and error-prone affair. There must be a better way!
1.1.4 Hardware Description Language (HDL)
Today, hardware designers no longer build anything with their bare hands. Instead, they plan and optimize the chip architecture on a computer workstation, using structured modeling formalisms like Hardware Description Language, or HDL (also known as VHDL, where V stands for Virtual ). The designer specifies the chip struc- ture by writing an HDL program, which is then subjected to a rigorous battery of tests. These tests are carried out virtually, using computer simulation: A special software tool, called a hardware simulator, takes the HDL program as input and builds an image of the modeled chip in memory. Next, the designer can instruct the simulator to test the virtual chip on various sets of inputs, generating simulated chip outputs. The outputs can then be compared to the desired results, as mandated by the client who ordered the chip built.
In addition to testing the chip’s correctness, the hardware designer will typi- cally be interested in a variety of parameters such as speed of computation, energy consumption, and the overall cost implied by the chip design. All these param-
15 Boolean Logic
eters can be simulated and quantified by the hardware simulator, helping the de- signer optimize the design until the simulated chip delivers desired cost/performance levels.
Thus, using HDL, one can completely plan, debug, and optimize the entire chip before a single penny is spent on actual production. When the HDL program is deemed complete, that is, when the performance of the simulated chip satisfies the client who ordered it, the HDL program can become the blueprint from which many copies of the physical chip can be stamped in silicon. This final step in the chip life cycle—from an optimized HDL program to mass production—is typically out- sourced to companies that specialize in chip fabrication, using one switching tech- nology or another.
Example: Building a Xor Gate As we have seen in figures 1.2 and 1.5, one way to define exclusive or is Xorða; bÞ 1⁄4 OrðAndða; NotðbÞÞ; AndðNotðaÞ; bÞÞ. This logic can be expressed either graphically, as a gate diagram, or textually, as an HDL program. The latter program is written in the HDL variant used throughout this book, defined in appendix A. See figure 1.6 for the details.
Explanation An HDL definition of a chip consists of a header section and a parts section. The header section specifies the chip interface, namely the chip name and the names of its input and output pins. The parts section describes the names and topol- ogy of all the lower-level parts (other chips) from which this chip is constructed. Each part is represented by a statement that specifies the part name and the way it is con- nected to other parts in the design. Note that in order to write such statements legi- bly, the HDL programmer must have a complete documentation of the underlying parts’ interfaces. For example, figure 1.6 assumes that the input and output pins of the Not gate are labeled in and out, and those of And and Or are labeled a, b and out. This API-type information is not obvious, and one must have access to it before one can plug the chip parts into the present code.
Inter-part connections are described by creating and connecting internal pins, as needed. For example, consider the bottom of the gate diagram, where the output of a Not gate is piped into the input of a subsequent And gate. The HDL code describes this connection by the pair of statements Not(…,out=nota) and And(a=nota,…). The first statement creates an internal pin (outbound wire) named nota, feeding out into it. The second statement feeds the value of nota into the a input of an And gate. Note that pins may have an unlimited fan out. For example, in figure 1.6, each input is simultaneously fed into two gates. In gate
16
Chapter 1
in
out
notb
in out
nota
out
w1
out
w2
a b
a b
And
a b
Or
out
And
a
b
HDL program (Xor.hdl)
/* Xor (exclusive or) gate:
If a<>b out=1 else out=0. */
CHIP Xor {
Test script (Xor.tst)
load Xor.hdl, output-list a, b, seta0,setb0, eval, output; seta0,setb1, eval, output; seta1,setb0, eval, output; seta1,setb1, eval, output;
out;
out
Output file (Xor.out) a b out
000 011 101 110
IN a, b;
OUT out;
PARTS:
Not(in=a, out=nota);
Not(in=b, out=notb);
And(a=a, b=notb, out=w1);
And(a=nota, b=b, out=w2);
Or(a=w1, b=w2, out=out);
}
Figure 1.6 HDL implementation of a Xor gate.
diagrams, multiple connections are described using forks. In HDL, the existence of
forks is implied by the code.
Testing Rigorous quality assurance mandates that chips be tested in a specific, rep- licable, and well-documented fashion. With that in mind, hardware simulators are usually designed to run test scripts, written in some scripting language. For example, the test script in figure 1.6 is written in the scripting language understood by the hardware simulator supplied with the book. This scripting language is described fully in appendix B.
Let us give a brief description of the test script from figure 1.6. The first two lines of the test script instruct the simulator to load the Xor.hdl program and get ready to
17 Boolean Logic
print the values of selected variables. Next, the script lists a series of testing scenarios, designed to simulate the various contingencies under which the Xor chip will have to operate in ‘‘real-life’’ situations. In each scenario, the script instructs the simulator to bind the chip inputs to certain data values, compute the resulting output, and record the test results in a designated output file. In the case of simple gates like Xor, one can write an exhaustive test script that enumerates all the possible input values of the gate. The resulting output file (right side of figure 1.6) can then be viewed as a com- plete empirical proof that the chip is well designed. The luxury of such certitude is not feasible in more complex chips, as we will see later.
1.1.5 Hardware Simulation
Since HDL is a hardware construction language, the process of writing and debug- ging HDL programs is quite similar to software development. The main difference is that instead of writing code in a language like Java, we write it in HDL, and instead of using a compiler to translate and test the code, we use a hardware simulator. The hardware simulator is a computer program that knows how to parse and interpret HDL code, turn it into an executable representation, and test it according to the specifications of a given test script. There exist many commercial hardware simu- lators on the market, and these vary greatly in terms of cost, complexity, and ease of use. Together with this book we provide a simple (and free!) hardware simulator that is sufficiently powerful to support sophisticated hardware design projects. In particular, the simulator provides all the necessary tools for building, testing, and integrating all the chips presented in the book, leading to the construction of a general-purpose computer. Figure 1.7 illustrates a typical chip simulation session.
1.2 Specification
This section specifies a typical set of gates, each designed to carry out a common Boolean operation. These gates will be used in the chapters that follow to construct the full architecture of a typical modern computer. Our starting point is a single primitive Nand gate, from which all other gates will be derived recursively. Note that we provide only the gates’ specifications, or interfaces, delaying implementation details until a subsequent section. Readers who wish to construct the specified gates in HDL are encouraged to do so, referring to appendix A as needed. All the gates can be built and simulated on a personal computer, using the hardware simulator supplied with the book.
18 Chapter 1
simulator controls
test script
HDL program
current pin values
typical simulation step
output file
Figure 1.7 A screen shot of simulating an Xor chip on the hardware simulator. The simulator state is shown just after the test script has completed running. The pin values correspond to the last simulation step (a 1⁄4 b 1⁄4 1). Note that the output file generated by the simulation is con- sistent with the Xor truth table, indicating that the loaded HDL program delivers a correct Xor functionality. The compare file, not shown in the figure and typically specified by the chip’s client, has exactly the same structure and contents as that of the output file. The fact that the two files agree with each other is evident from the status message displayed at the bottom of the screen.
19 Boolean Logic
1.2.1 The Nand Gate
The starting point of our computer architecture is the Nand gate, from which all other gates and chips are built. The Nand gate is designed to compute the following Boolean function:
a b Nandða;bÞ
001 011 101 110
Throughout the book, we use ‘‘chip API boxes’’ to specify chips. For each chip, the API specifies the chip name, the names of its input and output pins, the function or operation that the chip effects, and an optional comment.
Chip name: Nand
Inputs: a, b
Outputs: out
Function: If a=b=1 then out=0 else out=1
Comment: This gate is considered primitive and thus there is
no need to implement it.
1.2.2 Basic Logic Gates
Some of the logic gates presented here are typically referred to as ‘‘elementary’’ or ‘‘basic.’’ At the same time, every one of them can be composed from Nand gates alone. Therefore, they need not be viewed as primitive.
Not The single-input Not gate, also known as ‘‘converter,’’ converts its input from 0 to 1 and vice versa. The gate API is as follows:
Chip name: Not
Inputs: in
Outputs: out
Function: If in=0 then out=1 else out=0.
20 Chapter 1
And
Or
The And function returns 1 when both its inputs are 1, and 0 otherwise.
The Or function returns 1 when at least one of its inputs is 1, and 0 otherwise.
Chip name: And
Inputs: a, b
Outputs: out
Function: If a=b=1 then out=1 else out=0.
Chip name: Or
Inputs: a, b
Outputs: out
Function: If a=b=0 then out=0 else out=1.
The Xor function, also known as ‘‘exclusive or,’’ returns 1 when its two inputs have opposing values, and 0 otherwise.
Multiplexor A multiplexor (figure 1.8) is a three-input gate that uses one of the inputs, called ‘‘selection bit,’’ to select and output one of the other two inputs, called ‘‘data bits.’’ Thus, a better name for this device might have been selector. The name multiplexor was adopted from communications systems, where similar devices are used to serialize (multiplex) several input signals over a single output wire.
Xor
Chip name: Xor
Inputs: a, b
Outputs: out
Function: If a=/b then out=1 else out=0.
Chip name: Mux
Inputs: a, b, sel
Outputs: out
Function: If sel=0 then out=a else out=b.
21 Boolean Logic
a b sel out sel out
00000a 01001b 1001
1101
001 0 0111 1010 1111
a
b
Mux out
sel
Figure 1.8 Multiplexor. The table at the top right is an abbreviated version of the truth table on the left.
sel a b
0 in0 1 0in
Figure 1.9 Demultiplexor.
a
b
in DMux
sel
Demultiplexor A demultiplexor (figure 1.9) performs the opposite function of a multiplexor: It takes a single input and channels it to one of two possible outputs according to a selector bit that specifies which output to chose.
1.2.3 Multi-Bit Versions of Basic Gates
Computer hardware is typically designed to operate on multi-bit arrays called ‘‘buses.’’ For example, a basic requirement of a 32-bit computer is to be able to compute (bit-wise) an And function on two given 32-bit buses. To implement this operation, we can build an array of 32 binary And gates, each operating separately
Chip name: DMux
Inputs: in, sel
Outputs: a, b
Function: If sel=0 then {a=in, b=0} else {a=0, b=in}.
22 Chapter 1
on a pair of bits. In order to enclose all this logic in one package, we can encapsulate the gates array in a single chip interface consisting of two 32-bit input buses and one 32-bit output bus.
This section describes a typical set of such multi-bit logic gates, as needed for the construction of a typical 16-bit computer. We note in passing that the architecture of n-bit logic gates is basically the same irrespective of n’s value.
When referring to individual bits in a bus, it is common to use an array syntax. For example, to refer to individual bits in a 16-bit bus named data, we use the no- tation data[0], data[1],. . ., data[15].
Multi-Bit Not An n-bit Not gate applies the Boolean operation Not to every one of the bits in its n-bit input bus:
Multi-Bit And An n-bit And gate applies the Boolean operation And to every one of the n bit-pairs arrayed in its two n-bit input buses:
Multi-Bit Or An n-bit Or gate applies the Boolean operation Or to every one of the n bit-pairs arrayed in its two n-bit input buses:
Chip name: Not16
Inputs: in[16] // a 16-bit pin Outputs: out[16]
Function: For i=0..15 out[i]=Not(in[i]).
Chip name: And16
Inputs: a[16], b[16]
Outputs: out[16]
Function: For i=0..15 out[i]=And(a[i],b[i]).
Chip name: Or16
Inputs: a[16], b[16]
Outputs: out[16]
Function: For i=0..15 out[i]=Or(a[i],b[i]).
23 Boolean Logic
Multi-Bit Multiplexor An n-bit multiplexor is exactly the same as the binary multi- plexor described in figure 1.8, except that the two inputs are each n-bit wide; the selector is a single bit.
Chip name: Mux16
Inputs: a[16], b[16], sel
Outputs: out[16]
Function: If sel=0 then for i=0..15 out[i]=a[i]
else for i=0..15 out[i]=b[i].
1.2.4 Multi-Way Versions of Basic Gates
Many 2-way logic gates that accept two inputs have natural generalization to multi- way variants that accept an arbitrary number of inputs. This section describes a set of multi-way gates that will be used subsequently in various chips in our computer architecture. Similar generalizations can be developed for other architectures, as needed.
Multi-Way Or An n-way Or gate outputs 1 when at least one of its n bit inputs is 1, and 0 otherwise. Here is the 8-way variant of this gate:
Multi-Way/Multi-Bit Multiplexor An m-way n-bit multiplexor selects one of m n- bit input buses and outputs it to a single n-bit output bus. The selection is speci- fied by a set of k control bits, where k 1⁄4 log2 m. Figure 1.10 depicts a typical example.
The computer platform that we develop in this book requires two variations of this chip: A 4-way 16-bit multiplexor and an 8-way 16-bit multiplexor:
Chip name: Or8Way
Inputs: in[8]
Outputs: out
Function: out=Or(in[0],in[1],…,in[7]).
24 Chapter 1
sel[1] sel[0] out
00a 01b 10c 1 1 d
a
b
c Mux d
sel[1] sel[0]
4-way
out
a,b,c,d and out are each 16-bit wide
Figure 1.10 4-way multiplexor. The width of the input and output buses may vary.
Chip name: Mux4Way16
Inputs: a[16], b[16], c[16], d[16], sel[2]
Outputs: out[16]
Function: If sel=00 then out=a else if sel=01 then out=b
else if sel=10 then out=c else if sel=11 then out=d Comment: The assignment operations mentioned above are all
16-bit. For example, “out=a” means “for i=0..15
out[i]=a[i]”.
Chip name: Mux8Way16
Inputs: a[16],b[16],c[16],d[16],e[16],f[16],g[16],h[16],
sel[3] Outputs: out[16]
Function: If sel=000 then out=a else if sel=001 then out=b else if sel=010 out=c … else if sel=111 then out=h
Comment: The assignment operations mentioned above are all 16-bit. For example, “out=a” means “for i=0..15 out[i]=a[i]”.
Multi-Way/Multi-Bit Demultiplexor An m-way n-bit demultiplexor (figure 1.11) channels a single n-bit input into one of m possible n-bit outputs. The selection is specified by a set of k control bits, where k 1⁄4 log2 m.
The specific computer platform that we will build requires two variations of this chip: A 4-way 1-bit demultiplexor and an 8-way 1-bit multiplexor, as follows.
25 Boolean Logic
sel[1] sel[0] a b c d
0 0 in 0 0 0 in 0 1 0in00
1 0 00in0
1 1 000in
Figure 1.11 4-way demultiplexor.
4-way DMux
sel[1] sel[0]
Chip name: DMux4Way
Inputs: in, sel[2]
Outputs: a, b, c, d
Function: If sel=00 then {a=in, b=c=d=0}
else if sel=01 then {b=in, a=c=d=0}
else if sel=10 then {c=in, a=b=d=0}
else if sel=11 then {d=in, a=b=c=0}.
Chip name: DMux8Way
Inputs: in, sel[3]
Outputs: a, b, c, d, e, f, g, h
Function: If sel=000 then {a=in, b=c=d=e=f=g=h=0}
else if sel=001 then {b=in, a=c=d=e=f=g=h=0}
else if sel=010 …
…
else if sel=111 then {h=in, a=b=c=d=e=f=g=0}.
1.3 Implementation
Similar to the role of axioms in mathematics, primitive gates provide a set of ele- mentary building blocks from which everything else can be built. Operationally, primitive gates have an ‘‘off-the-shelf’’ implementation that is supplied externally. Thus, they can be used in the construction of other gates and chips without worrying about their internal design. In the computer architecture that we are now beginning
26 Chapter 1
to build, we have chosen to base all the hardware on one primitive gate only: Nand. We now turn to outlining the first stage of this bottom-up hardware construction project, one gate at a time.
Our implementation guidelines are intentionally partial, since we want you to dis- cover the actual gate architectures yourself. We reiterate that each gate can be imple- mented in more than one way; the simpler the implementation, the better.
Not: The implementation of a unary Not gate from a binary Nand gate is simple. Tip: Think positive.
And: Once again, the gate implementation is simple. Tip: Think negative.
Or/Xor: These functions can be defined in terms of some of the Boolean functions implemented previously, using some simple Boolean manipulations. Thus, the re- spective gates can be built using previously built gates.
Multiplexor/Demultiplexor: Likewise, these gates can be built using previously built gates.
Multi-Bit Not/And/Or Gates: Since we already know how to implement the ele- mentary versions of these gates, the implementation of their n-ary versions is simply a matter of constructing arrays of n elementary gates, having each gate operate sep- arately on its bit inputs. This implementation task is rather boring, but it will carry its weight when these multi-bit gates are used in more complex chips, as described in subsequent chapters.
Multi-Bit Multiplexor: The implementation of an n-ary multiplexor is simply a matter of feeding the same selection bit to every one of n binary multiplexors. Again, a boring task resulting in a very useful chip.
Multi-Way Gates: Implementation tip: Think forks.
1.4 Perspective
This chapter described the first steps taken in an applied digital design project. In the next chapter we will build more complicated functionality using the gates built here.
27 Boolean Logic
1.5 Project
Although we have chosen to use Nand as our basic building block, other approaches are possible. For example, one can build a complete computer platform using Nor gates alone, or, alternatively, a combination of And, Or, and Not gates. These con- structive approaches to logic design are theoretically equivalent, just as all theorems in geometry can be founded on different sets of axioms as alternative points of de- parture. The theory and practice of such constructions are covered in standard text- books about digital design or logic design.
Throughout the chapter, we paid no attention to efficiency considerations such as the number of elementary gates used in constructing a composite gate or the number of wire crossovers implied by the design. Such considerations are critically important in practice, and a great deal of computer science and electrical engineering expertise focuses on optimizing them. Another issue we did not address at all is the physical implementation of gates and chips using the laws of physics, for example, the role of transistors embedded in silicon. There are of course several such implementation options, each having its own characteristics (speed, power requirements, production cost, etc.). Any nontrivial coverage of these issues requires some background in electronics and physics.
Objective Implement all the logic gates presented in the chapter. The only building blocks that you can use are primitive Nand gates and the composite gates that you will gradually build on top of them.
Resources The only tool that you need for this project is the hardware simulator supplied with the book. All the chips should be implemented in the HDL language specified in appendix A. For each one of the chips mentioned in the chapter, we provide a skeletal .hdl program (text file) with a missing implementation part. In addition, for each chip we provide a .tst script file that tells the hardware simulator how to test it, along with the correct output file that this script should generate, called .cmp or ‘‘compare file.’’ Your job is to complete the missing implementation parts of the supplied .hdl programs.
Contract When loaded into the hardware simulator, your chip design (modified .hdl program), tested on the supplied .tst file, should produce the outputs listed in the supplied .cmp file. If that is not the case, the simulator will let you know.
28 Chapter 1
Tips The Nand gate is considered primitive and thus there is no need to build it: Whenever you use Nand in one of your HDL programs, the simulator will auto- matically invoke its built-in tools/builtIn/Nand.hdl implementation. We rec- ommend implementing the other gates in this project in the order in which they appear in the chapter. However, since the builtIn directory features working ver- sions of all the chips described in the book, you can always use these chips without defining them first: The simulator will automatically use their built-in versions.
For example, consider the skeletal Mux.hdl program supplied in this project. Suppose that for one reason or another you did not complete this program’s im- plementation, but you still want to use Mux gates as internal parts in other chip designs. This is not a problem, thanks to the following convention. If our simula- tor fails to find a Mux.hdl file in the current directory, it automatically invokes a built-in Mux implementation, pre-supplied with the simulator’s software. This built- in implementation—a Java class stored in the builtIn directory—has the same in- terface and functionality as those of the Mux gate described in the book. Thus, if you want the simulator to ignore one or more of your chip implementations, simply move the corresponding .hdl files out of the current directory.
Steps We recommend proceeding in the following order:
0. The hardware simulator needed for this project is available in the tools direc-
tory of the book’s software suite.
1. Read appendix A, sections A1–A6 only.
2. Go through the hardware simulator tutorial, parts I, II, and III only.
3. Build and simulate all the chips specified in the projects/01 directory.