Content uploaded by Issam Damaj
Author content
All content in this area was uploaded by Issam Damaj on May 03, 2019
Content may be subject to copyright.
This is the peer reviewed version of the following article: [I. Damaj, Logic Design, in Wiley
Encyclopedia of Computer Science and Engineering, Benjamin Wah (Editor), Hoboken: John
Wiley & Sons, Inc, New Jersey, January 15, 2008. V 3 P 1753 – 1762.], which has been published
in final form at [https://doi.org/10.1002/9780470050118.ecse225]. This article may be used for
non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-
Archived Versions.
Logic Design
Issam W. Damaj, Dhofar University
Introduction
Over the years, digital electronic systems have progressed from vacuum-tube to complex
integrated circuits, some of which contain millions of transistors. Electronic circuits can be
separated into two groups, digital and analog circuits. Analog circuits operate on analog
quantities that are continuous in value and in time, while digital circuits operate on digital
quantities that are discrete in value and time (1).
Analog signals are continuous in time besides being continuous in value. Most measurable
quantities in nature are in analog form, for example, temperature. Measuring around the
hour temperature changes is continuous in value and time, where the temperature can take
any value at any instance of time with no limit on precision but on the capability of the
measurement tool. Fixing the measurement of temperature to one reading per an interval
of time and rounding the value recorded to the nearest integer will graph discrete values at
discrete intervals of time that easily could be coded into digital quantities. From the given
example, it is clear that an analog-by-nature quantity could be converted to digital by taking
discrete-valued samples at discrete intervals of time and then coding each sample. The
process of conversion is usually known as analog-to-digital conversion (A/D). The
opposite scenario of conversion is also valid and known as digital-to-analog conversion
(D/A). The representation of information in a digital form has many advantages over
analog representation in electronic systems. Digital data that is discrete in value, discrete
in time, and limited in precision could be efficiently stored, processed and transmitted.
Digital systems are said practically to be more noise immune as compared to analog
electronic systems due to the physical nature of analog signals. Accordingly, digital
systems are more reliable than their analog counterpart. Examples of analog and digital
systems are shown in Figure 1.
Analog Amplifier Speaker
Microphone
A Simple Analog System
Personal Digital Assistant and a Mobile Phone
Speaker
Microphone
A Digital System
Figure 1. A simple analog system and a digital system; the analog signal amplifies the input signal
using analog electronic components. The digital system can still include analog components like a
speaker and a microphone, the internal processing is digital.
A Bridge between Logic and Circuits
Digital electronic systems represent information in digits. The digits used in digital systems
are the 0 and 1 that belong to the binary mathematical number system. In logic, the 1 and
0 values correspond to True and False. In circuits, the True and False could be thought of
as High voltage and Low voltage. These correspondences set the relationships among logic
(True and False), binary mathematics (0 and 1), and circuits (High and Low).
Logic, in its basic shape, deals with reasoning that checks the validity of a certain
proposition - a proposition could be either True or False. The relationship among logic,
binary mathematics, and circuits enables a smooth transition of processes expressed in
propositional logic to binary mathematical functions and equations (Boolean algebra), and
to digital circuits. A great scientific wealth exists that strongly supports the relationships
among the three different branches of science that lead to the foundation of modern digital
hardware and logic design.
Boolean algebra uses three basic logic operations AND, OR, and NOT. The NOT operation
if joined with a proposition P works by negating it; for instance, if P is True then NOT P
is False and vice versa. The operations AND and OR should be used with two propositions,
for example, P and Q. The logic operation AND, if applied on P and Q would mean that P
AND Q is True only when both P and Q are True. Similarly, the logic operation OR, if
applied on P and Q, would mean that P OR Q is False only when P and Q are False. Truth
tables of the logic operators AND, OR, and NOT are shown in Figure 2.a. Figure 2.b shows
an alternative representation of the truth tables of AND, OR, and NOT in terms of 0s and
1s.
Combinational Logic Circuits
Digital circuits implement the logic operations AND, OR, and NOT as hardware elements
called “gates” that perform logic operations on binary inputs. The AND-gate performs an
AND operation, an OR-gate performs an OR operation, and an Inverter performs the
negation operation NOT. Figure 2.c shows the standard logic symbols for the three basic
operations. With analogy from electric circuits, the functionality of the AND and OR gates
are captured as shown in Figure 3. The actual internal circuitry of gates is built using
transistors; two different circuit implementations of inverters are shown in Figure 4.
Examples of AND, OR, NOT gates integrated circuits (ICs) are shown in Figure 5. Besides
the three essential logic operations, there are four other important operations - the NOR
(NOT-OR), NAND (NOT-AND), Exclusive-OR (XOR) and Exclusive-NOR (XNOR).
Input P
Input Q
Output:
P AND Q
Input P
Input Q
Output:
P OR Q
Input X
Output:
NOT P
False
False
False
False
False
False
False
True
False
True
False
False
True
True
True
False
True
False
False
True
False
True
True
True
True
True
True
True
(a)
Input P
Input Q
Output:
P AND Q
Input P
Input Q
Output:
P OR Q
Input X
Output:
NOT P
0
0
0
0
0
0
0
1
0
1
0
0
1
1
1
0
1
0
0
1
0
1
1
1
1
1
1
1
(b)
AND Gate OR Gate Inverter
0
1
00
0
00
1
0
1
1
1
0
1
1
1
0
0
0 1 0
01 1
0
1 1
1
(c)
Figure 2. (a) Truth tables for AND, OR, and Inverter. (b) Truth tables for AND, OR, and Inverter in
binary numbers, (c) Symbols for AND, OR, and Inverter with their operation.
X Y
X AND Y
X
YX OR Y
X
YX
Y
X AND Y X OR Y
Figure 3. A suggested analogy between AND and OR gates and electric circuits.
Input Output
+VDD 130 W1.6 kW4 kW
1 kW
+VCC
Output
Input
CMOS Inverter TTL Inverter
Figure 4. Complementary Metal-oxide Semiconductor (CMOS) and Transistor-Transistor Logic
(TTL) Inverters.
GND
Vcc
GND
Vcc
GND
Vcc
Figure 5. The 74LS21 (AND), 74LS32 (OR), 74LS04 (Inverter) TTL ICs.
A combinational logic circuit is usually created by combining gates together to implement
a certain logic function. A combinational circuit produces its result upon application of its
input(s). A logic function could be a combination of logic variables, such as A, B, C, etc.
Logic variables can take only the values 0 or 1. The created circuit could be implemented
using AND-OR-Inverter gate-structure or using other types of gates. Figure 6.a shows an
example combinational implementation of the following logic function F(A, B, C):
F(A, B, C) = ABC + A’BC + AB’C’
F(A, B, C) in this case could be described as a standard sum-of-products (SOP) function
according to the analogy that exists between OR and addition (+), and between AND and
product (.); the NOT operation is indicated by an Apostrophe “ ’ ” following the variable
name. Usually, standard representations are also referred to as canonical representations.
In an alternative formulation, consider the following function E(A,B,C) in a product-of-
sums (POS) form:
E(A, B, C) = (A + B’ + C).(A’ + B + C)( A + B + C’)
The canonical POS implementation is shown in Figure 6.b. Some other specifications
might require functions with a greater number of inputs and accordingly more complicated
design process.
A
B
C
F(A, B, C)
A
B
C
A
B
C
(a)
A
B
CE(A, B, C)
A
B
C
A
B
C
(b)
Figure 6. AND-OR-Inverter implementation of the function (a) SOP: F(A, B, C) = ABC + A’BC +
AB’C’. (b) POS: E(A, B, C) = (A + B’ + C).(A’ + B + C)( A + B + C’)
The complexity of a digital logic circuit that corresponds to a Boolean function is directly
related to the complexity of the base algebraic function. Boolean functions may be
simplified by several means. The simplification process that produces an expression with
the least number of terms with the least number of variables is usually called minimization.
The minimization has direct effects on reducing the cost of the implemented circuit and
sometimes enhancing its performance. The minimization (optimization) techniques range
from simple (manual) to complex (automated). An example of manual optimization
methods is the Karnough map (K-map).
K-maps
A K-map is like a truth table as it presents all the possible values of input variables and
their corresponding output. The main difference between K-maps and truth tables is in the
cell’s arrangement. In a K-map, Cells are arranged in a way so that simplification of a given
algebraic expression is simply a matter of properly grouping the cells. K-maps can be used
for expressions with different number of input variables; three, four, or five. In the
following examples, maps with only three and four variables are shown to stress the
principle. Methods for optimizing expressions with more than five variables can be found
in the literature. The Quine-McClusky is an example method that can accommodate several
variables larger than five (2).
A 3-variable K-map is an array of 8 (or 23) cells. Figure 7.a depicts the correspondence
between three inputs (A, B, and C) truth table and a K-map. The value of a given cell
represents the output at certain binary values of A, B, and C. in a similar way, a 4-variable
K-map is arranged as shown in Figure 7.b. K-maps could be used for expressions in either
POS or SOP forms. Cells in a K-map are arranged so that they satisfy the Adjacency
property; where only a single variable changes its value between adjacent cells. For
instance, the cell 000, that is the binary value of the term A’B’C’, is adjacent to cell 001
that corresponds to the term A’B’C. The cell 0011 (A’B’CD) is adjacent to the cell 0010
(A’B’C’D).
BC
A
00
01
11
10
0
0
0
1
0
1
0
1
1
1
Input A
Input B
Input C
Output F
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
1
1
1
0
1
1
1
1
1
A
C
B
(a)
CD
AB
00
01
11
10
00
01
11
10
(b)
Figure 7. (a) The correspondence between three inputs (A, B, and C) truth table and a K-map. (b) An
empty 4-variable K-map.
Minimizing SOP Expressions
The minimization of an algebraic Boolean function f has the following four key steps:
1. Evaluation.
2. Placement.
3. Grouping.
4. Derivation.
The minimization starts by evaluating each term in the function f and then placing a 1 in
the corresponding cell on the K-map. A term ABC in a function f(A, B, C) is evaluated to
111, another term AB’CD in a function g(A, B, C, D) is evaluated to 1011. An example of
evaluating and placing the following function f is shown in Figure 8.a:
f(A, B, C) = A’B’C’ + A’B’C + ABC’ + AB’C’
After placing the 1s on a K-map, grouping filled-with-1s cells is done according to the
following rules (See Figure 8.b):
• A group of adjacent filled-with-1s cells must contain a number of cells that belongs
to the set of powers of two (1, 2, 4, 8, or 16).
• A group should include the largest possible number of filled-with-1s cells.
• Each 1 on the K-map must be included in at least one group.
• Cells contained in a group could be shared within another group as long as
overlapping groups included non-common 1s.
After the grouping step, the derivation of minimized terms is done according to the
following rules:
• Each group containing 1s creates one product term.
• The created product term includes all variables that appear in only one form
(completed or uncomplemented) across all cells in a group.
After deriving terms, the minimized function is composed of their sum. An example
derivation is shown in Figure 8.b. Figure 9 presents the minimization of the following
function:
g(A, B, C, D) = AB’C’D’ + A’B’C’D’ + A’B’C’D’ + A’B’CD + AB’CD + A’B’CD’ +
A’BCD + ABCD + AB’CD’
BC
A
00
01
11
10
0
1
1
1
1
1
A
C
B
A’B’C’
AB’C’
A’B’C
ABC’
(a)
BC
A
00
01
11
10
0
1
1
1
1
1
A
C
B
fmin = A’B’ + AC’
(b)
Figure 8. (a) Terms evaluation of the function f(A, B, C) = A’B’C’ + A’B’C + ABC’ + AB’C’. (b)
Grouping and derivation.
CD
AB
00
01
11
10
00
1
1
1
1
01
1
11
1
10
1
1
1
gmin = A’B’ + CD + B’D’
Figure 9. minimization steps of the following function: g(A, B, C, D) = AB’C’D’ + A’B’C’D’ +
A’B’C’D’ + A’B’CD + AB’CD + A’B’CD’ + A’BCD + ABCD + AB’CD’
Combinational Logic Design
The basic combinational logic design steps could be summarized as follows:
1. Specification of the required circuit.
2. Formulation of the specification to derive algebraic equations.
3. Optimization (minimization) of the obtained equations
4. Implementation of the optimized equations using a suitable hardware (IC)
technology.
The above steps are usually joined with an essential verification procedure which ensures
the correctness and completeness of each design step.
As an example, consider the design and implementation of a 3 variables majority function.
The function F(A, B, C) will return a 1 (High or True) whenever the number of 1s in the
inputs is greater than or equal to the number of 0s.
The above specification could be reduced into a truth table as shown in Figure 7.a. The
terms that make the function F return a 1 are the terms F(0, 1, 1), F(1, 0, 1), F(1, 1, 0), or
F(1, 1, 1). This could be alternatively formulated as in the following equation:
F = A’BC + AB’C + ABC’ + ABC
Following the specification and the formulation, a K-map is used to obtain the minimized
version of F (called Fmin). Figure 10.a depicts the minimization process. Figure 10.b shows
the implementation of Fmin using standard AND-OR-NOT gates.
Combinational Logic Circuits
Famous combinational circuits that are widely adopted in digital systems include Encoders,
Decoders, Multiplexers, Adders, some Programmable Logic Devices (PLDs), etc. The
basic operation of Multiplexers, Half-adders, and simple PLDs (SPLDs) is described in the
following lines.
A multiplexer (MUX) selects one of n input lines and provides it on a single output. The
select lines, denoted S, identify or address one of the several inputs. Figure 11.a shows the
block diagram of a 2-to-1 multiplexer. The two inputs can be selected by one select line,
S. If the selector S = 0, input line d0 will be the output O, otherwise, d1 will be produced at
the output. An MUX implementation of the majority function F(A, B, C) is shown in Figure
11.b.
A half-adder inputs two binary digits to be added and produces two binary digits
representing the sum and carry. The equations, implementation and symbol of a half-adder
are shown in Figure 12.
BC
A
00
01
11
10
0
0
0
1
0
1
0
1
1
1
A
C
B
Fmin = AC + BC + AB
(a)
A
C
Fmin(A, B, C)
B
C
A
B
(b)
Figure 10. (a) Minimization of a 3 variables majority function. (b) Implementation of a minimized
three variables majority function.
MUX
S
d0
d1
O
(a)
MUX
S0
d0
d1
O
d2
d3
d4
d5
d6
d7
S1S2
0
0
0
0
1
1
1
1
A B C
F
(b)
Figure 11. (a) Minimization of a 3 variables majority function. (b) Implementation of a minimized
three variables maj`ority function.
Half Adder
A0
B0S = A0 Å B0
C = A0 . B0
Figure 12. The equations, implementation and symbol of a half-adder. The used symbol for a XOR
operation is ‘Å’.
Simple PLDs (SPLDs) are usually built from combinational logic blocks with pre-routed
wiring. In implementing a function on a PLD, the designer will only decide of which wires
and blocks to use; this step is usually referred to as programming the device. Programmable
Logic Array (PLA) and the Programmable Array Logic (PAL) are commonly used SPLD.
A PLA has a set of programmable AND gates, which link to a set of programmable OR
gates to produce an output (See Figure 13.a). A PAL has a set of programmable AND gates,
which link to a set of fixed OR gates to produce an output (See Figure 13.b). The AND-OR
layout of a PLA/PAL allows for implementing logic functions that are in an SOP form. A
PLA implementation of the majority function f(A, B, C) is shown in Figure 13.c.
Standard Multiple
AND Gate Symbol AND Array
Symbol
Standard Multiple
OR Gate Symbol OR Array
Symbol
AB C
(a)
AB C
(b)
AB C
x x x
x x x
x x x
x x x
x
x
x
x
xFuse not blown, the literal is
included in the term
A’BC
AB’C
ABC’
ABC
F
(c)
Figure 13. (a) A 3-input 2-output PLA with its AND Arrays and OR Arrays. An AND array is
equivalent to a standard multiple-input AND gate, and an OR array is equivalent to a standard
multiple-input OR gate. (b) A 3-input 2-output PAL. (c) A PLA implementation of the majority
function F(A, B, C)
Sequential Logic
In practice, most digital systems contain combinational circuits along with memory; these
systems are known as sequential circuits. In sequential circuits, the present outputs depend
on the present inputs and the previous states stored in the memory elements. Sequential
circuits are of two types synchronous and asynchronous. In a synchronous sequential
circuit, a clock signal is used at discrete instants of time to synchronize desired operations.
A clock is a device that generates a train of pulses as shown in Figure 14. Asynchronous
sequential circuits don’t require synchronizing clock pulses; however, the completion of
an operation signals the start of the next operation in sequence.
Rising Edge
Falling Edge
Time
Figure 14. Clock pulses.
In synchronous sequential circuits, the memory elements are called flip-flops and can store
only one bit. Arrays of flip-flops are usually used to accommodate for bit-width
requirements of binary data. A typical synchronous sequential circuit contains a
combinational part, sequential elements, and feedback signals coming from the output of
the sequential elements.
Flip-flops
Flip-flops are volatile elements, where the stored bit is stored if power is available. Flip-
flops are designed using basic storage circuits called latches. The most common latch is
the SR (Set to 1 - Reset to 0) latch. An SR latch could be formed with two cross-coupled
NAND gates as shown in Figure 15. The responses to various inputs to the SR latch are
setting Q to 1 for an SR input of 01 (S is active low, i.e. S is active when it is equal to 0),
resetting Q to 0 for an SR input of 10 (R here is also active low), memorizing the current
state for an SR input of 11. The SR input of 00 is considered invalid.
A flip-flop is a latch with a clock input. A flip-flop that changes state either at the positive
(rising) edge or the negative (falling) edge of the clock is called an edge-triggered flip-flop
(See Figure 14). The three famous edge-triggered flip-flops are the RS, JK, and D flip-
flops.
An RS flip-flop is a clocked SR latch with two more NAND gates (See Figure 15.b). The
symbol and the basic operation of an RS flip-flop are illustrated in Figure 16.a. The
operation of an RS flip-flop is different from that of an SR latch and responds differently
to different values of S and R. The JK and D flip-flops are derived from the SR flip-flop.
However, the JK and D flip-flops are more widely used (2). The JK flip-flop is identical to
the SR flip-flop with a single difference, where it has no invalid state (See Figure 16.b).
The D flip-flop has only one input formed with an SR flip-flop and an inverter (See Figure
16.c); thus, it only could set or reset. The D flip-flop is also known as transparent flip-flop,
where output will have the same value of the input after one clock cycle.
S
RQ’
Q
(a)
S
RQ’
Q
CLK
(b)
Figure 15. (a) An SR latch. (b) an RS flip-flop.
S
R
Next State Q(t + 1)
0
0
Q(t)
Unchanged
0
1
0
Reset
1
0
1
Set
1
1
-
Invalid
Q
Q
SET
CLR
S
R
CLK
(a)
J
K
Next State Q(t + 1)
0
0
Q(t)
Unchanged
0
1
0
Reset
1
0
1
Set
1
1
Q’(t)
Complement
Q
Q
SET
CLR
S
R
CLK
J
K
(b)
D
Next State Q(t + 1)
0
0
Reset
1
1
Set
Q
Q
SET
CLR
S
R
CLK
D
K
(c)
Figure 16. (a) The symbol and the basic operation of (a) RS flip-flop (b) JK flip-flop (c) D flip-flop.
Sequential Logic Design
The basic sequential logic design steps are generally identical to those for combinational
circuits; these are Specification, Formulation, Optimization, and the Implementation of the
optimized equations using a suitable hardware (IC) technology. The differences between
sequential and combinational design steps appear in the details of each step.
The specification step in sequential logic design usually describes the different states the
sequential circuit goes through. A typical example for a sequential circuit is a counter that
undergoes 8 different states, for instance, 0, 1, 2, 3, up till 7. A classic way to describe the
state transitions of sequential circuits is a state diagram. In a state diagram, a circle
represents a state, and an arrow represents a transition. The proposed example assumes no
inputs to control the transitions among states. Figure 17.a shows the state diagram of the
specified counter. The number of states determines the minimum number of flip-flops to
be used in the circuit. In the case of the 8-states counter, the number of flip-flops should be
3; in accordance with the formula 8 equals 23. At this stage, the states could be coded in
binary. For instance, the stage representing count 0 is coded to binary 000; the stage of
count 1 is coded to binary 001, etc.
The state diagram is next to be described in a truth table style, usually known as state table,
from which the formulation step could be carried forward. For each flip-flop an input
equation is derived (See Figure 17.b). The equations are then minimized using K-maps
(See Figure 17.c). The minimized input equations are then implemented using a suitable
hardware (IC) technology. The minimized equations are then to be implemented (See
Figure 17.d).
0
(000) 1
(001)
2
(010)
3
(011)
4
(100)
6
(110)
5
(101)
7
(111)
(a)
Present State
Next State
QA(t)
QB(t)
QC(t)
QA(t + 1)
QB(t + 1)
QC(t + 1)
0
0
0
0
0
1
0
0
1
0
1
0
0
1
0
0
1
1
0
1
1
1
0
0
1
0
0
1
0
1
1
0
1
1
1
0
1
1
0
1
1
1
1
1
1
0
0
0
(b)
BC
A
00
01
11
10
0
0
0
1
0
1
1
1
0
1
A
C
B
QA(t+1) = AB’ + A’BC + AC’
BC
A
00
01
11
10
0
0
1
0
1
1
0
1
0
1
A
C
B
QB(t + 1) = B’C + BC’
BC
A
00
01
11
10
0
1
0
0
1
1
1
0
0
1
A
C
B
QB(t + 1) = C’
(c)
Q
Q
SET
CLR
S
R
CLK
D
K
Q
Q
SET
CLR
S
R
CLK
D
K
Q
Q
SET
CLR
S
R
CLK
D
K
QA(t+1)
QB(t+1)
QC(t+1)
QA(t)
CLK
QB(t) QC(t)
(d)
Figure 17. (a) The state diagram of the specified counter. (b) The state table. (c) Minimization of
input equations. (d) Implementation of the counter.
Modern Logic Design
The task of manually designing hardware tends to be extremely tedious, and sometimes
impossible, with the increasing complexity of modern digital circuits. Fortunately, the
demand on large digital systems has been accompanied with a fast advancement in IC
technologies. Indeed, IC technology has been growing faster than the ability of designers
to produce hardware designs. Hence, there has been a growing interest in developing
techniques and tools that facilitate the process of logic design.
Two different approaches emerged from the debate over ways to automate hardware logic
design. On one hand, the capture-and-simulate proponents believe that human designers
have good design experience that cannot be automated. They also believe that a designer
can build a design in a bottom-up style from elementary components such as transistors
and gates. Since the designer is concerned with the deepest details of the design, optimized
and cheap designs could be produced. On the other hand, the describe-and-synthesize
advocates believe that synthesizing algorithms can out-perform human designers. They
also believe that a top-down fashion would be better suited for designing complex systems.
In describe-and-synthesize methodology, the designers firstly describe the design. Then,
computer aided design (CAD) tools can generate the physical and electrical structure. This
approach describes the intended designs using special languages called hardware
description languages (HDLs). Some HDLs are very similar to traditional programming
languages like C, Pascal, etc. (3). Verilog (4) and VHDL (Very High Speed Integrated
Circuit Hardware Description Language) (5) are by far the most commonly used HDLs in
industry.
Hardware synthesis is a general term used to refer to the processes involved in
automatically generating a hardware design from its specification. High-level Synthesis
(HLS) could be defined as the translation from a behavioral description of the intended
hardware circuit into a structural description. The behavioral description represents an
algorithm, equation, etc., while a structural description represents the hardware
components that implement the behavioral description.
The chained synthesis tasks at each level of the design process include system synthesis,
register-transfer synthesis, logic synthesis, and circuit synthesis. System synthesis starts
with a set of processes communicating though either shared variables or message passing.
Each component can be described using a register-transfer language (RTL). RTL
descriptions model a hardware design as circuit blocks and interconnecting wires. Each of
these circuit blocks could be described using Boolean expressions. Logic synthesis
translates Boolean expressions into a list of logic gates and their interconnections (netlist).
Based on the produced netlist, circuit synthesis generates a transistor schematic from a set
of input-output current, voltage and frequency characteristics or equations.
The Logic synthesis step automatically converts a logic-level behavior, consisting of logic
equations and/or a Finite State Machines (FSMs), into a structural implementation (3).
Finding an optimal solution for complex logic minimization problems is very hard. As a
consequence, most logic synthesis tools use heuristics. A heuristic is a technique whose
result can hopefully come close to the optimal solution. The impact of complexity and of
the use of heuristics on logic synthesis is significant. Logic synthesis tools differ
tremendously according to the heuristics they use. Some computationally-intensive
heuristics requires long run times and thus powerful workstations producing high-quality
solutions. However, other logic synthesis tools use fast heuristics that are typically found
on personal computers (PCs) producing solutions with less quality. Tools with expensive
heuristics usually allow a user to control the level of optimization to be applied.
Continuous efforts have been made paving the way for modern logic design. These efforts
included the development of many new techniques and tools. An approach to logic
minimization using a new sum operation called multiple valued EXOR is proposed in (6)
based on neural computing.
In (7), Tomita et al. discuss the problem of locating logic design errors, and proposes an
algorithm to solve it. Based on the results of logic verification, the authors introduce an
input pattern for locating design errors. An algorithm for locating single design errors with
the input patterns has been developed.
Efforts for creating tools with higher levels of abstraction in design lead to the production
of many powerful modern hardware design tools. Ian Page and Wayne Luk developed a
compiler that transformed a subset of Occam into a netlist (8). Nearly ten years later we
have seen the development of Handel-C, the first commercially available high-level
language for targeting programmable logic devices (such as field programmable gate arrays
- FPGAs) (9).
A prototype HDL called Lava is developed by Satnam Singh at Xilinx and Mary Sheeran
and Koen Claessen at Chalmers University in Sweden (10). Lava allows circuit tiles to be
composed using powerful high-order combinators. This language is embedded in the
Haskell lazy functional programming language. Xilinx implementation of Lava is designed
to support the rapid representation, implementation and analysis of high performance
FPGA circuits.
Logic design has an essential impact on the development of modern digital systems. In
addition, logic design techniques are a primary key in various modern areas, such as,
embedded systems design, reconfigurable systems (11), hardware/software co-design, etc.
Bibliography.
[1] F. Vahid et al., Embedded System Design: A Unified Hardware/Software Introduction,
New York: John Wiley & Sons, 2002.
[2] T. Floyd , Digital Fundamentals with PLD Programming, New Jersey: Prentice Hall,
2006.
[3] S. Hachtel, Logic Synthesis and Verification Algorithms, Norwell: Kluwer, 1996.
[4] IEEE, Verilog HDL language reference manual, IEEE Standard 1364, 1995.
[5] IEEE, Standard VHDL reference manual, IEEE Standard 1076, 1993.
[6] A. Hozumi, N. Kamiura, Y. Hata, K. Yamato, Multiple-valued logic design using
multiple-valued EXOR, Proc. Multiple-Valued Logic, 290 - 294, 1995.
[7] M. Tomita, H. Jiang, T. Yamamoto, Y. Hayashi, An algorithm for locating logic
design errors, Proc. Computer-Aided Design, 468 – 471, 1990.
[8] I. Page and W. Luk, Compiling Occam into field-programmable gate arrays, Proc.
Workshop on Field Programmable Logic and Applications: 271–283, 1991.
[9] S. Brown and J. Rose, Architecture of FPGAs and CPLDs: A Tutorial, IEEE Design
and Test of Computers, 2: 42–57, 1996.
[10] K. Claessen. Embedded Languages for Describing and Verifying Hardware. PhD
Thesis, Chalmers University of Technology and Göteborg University, 2001.
[11] E. Mirsky and A. DeHon, MATRIX: A reconfigurable computing architecture with
configurable instruction distribution and deployable resources, Proc. IEEE Workshop on
FPGAs for Custom Computing Machines, 157 – 166, 1996.
Reading List
T. Floyd , Digital Fundamentals with PLD Programming, New Jersey: Prentice Hall,
2006.
M. Mano et al., Logic and Computer Design Fundamentals, New Jersey: Prentice Hall,
2004.
Cross-references
Programmable Logic Devices, See PLDs.
Synthesis, See High-level Synthesis
Synthesis, See Logic Synthesis
