Digital Combinational Circuit Optimization Using Chaos-Genetic Algorithm
AmirMohsen Toliyat Abolhassani
Department of Electrical and Computer Engineering
Islamic Azad University Mashhad Branch
Mashhad, Iran
[email protected]
Abstract – Genetic algorithms (GA) have been widely applied
to digital circuit optimization. With the increase of the
complexity and the larger problem scale of digital circuits, GAs
are most frequently faced with the problems of premature
convergence, slow iterations to reach the global optimal solution
and getting stuck at a local optimum. A novel chaos genetic
algorithm (CGA) based on the chaos optimization algorithm
(COA) and genetic algorithm (GA), which makes use of internal
randomness of chaos iterations, is proposed to overcome the
problem of premature local optimum and also to increase the
convergence speed of genetic algorithm. CGA integrates
powerful global searching capability of the GA with powerful
local searching capability of the COA.
Two measures are adopted in order to improve the
performance of the GA. The first one is the adoption of chaos
optimization of the initialization to improve species quality and
to maintain the population diversity. The second is the utilization
of annealing chaotic mutation operation to replace standard
mutation operator in order to avoid the search being trapped in
local optimum. The Particle Swarm Optimization (PSO)
algorithm[11-13] and multi-objective genetic algorithm
(MOGA), n-cardinally alphabet and a two stage fitness function
genetic algorithm (NGA), which often used as benchmarks for
contemporary optimization algorithms and evolutionary
computation, are first employed to evaluate the performance of
the GA and CGA. The test results indicate that CGA can
improve convergence speed and solution accuracy. Furthermore,
the developed algorithm is applied to several test circuits and it
is benchmarked with different optimization algorithms. The
results show that CGA has the best and its convergent speed not
only is faster than dynamic programming largely, but also
overpasses the standard GA.
Index Terms—Chaos Algorithm, Genetic Algorithm, Digital
Circuit Optimization, evolutionary hardware design
I. INTRODUCTION
Recently, many researchers have been focused on
evolutionary optimization of digital combinational circuits. It
is clear that by optimizing the number of components per
system, the reliability of the system will improve, the parasitic
effect is reduced, power consumption is decreased and more
important the hardware cost goes down.
The conventional digital circuit design is a very
complicated task which requires much knowledge in domainspecific rules. The design procedures involve the process such
Mahdi Yaghoobi
Department of Electrical and Computer Engineering
Islamic Azad University Mashhad Branch
Mashhad, Iran
[email protected]
as choosing the suitable gate types to match the logical
specification, minimizing and optimizing the boolean
representations with respect to the user defined constraints
[1].
In addition to conventional method, Evolutionary
Algorithms (EA) has been widely used in complex
optimization problems. Evolutionary algorithms have been
successfully applied to digital combinational circuits too [2].
Particle Swarm Optimization (PSO) algorithm has been
successfully used as a nonlinear optimization technique [1113]. The main idea behind PSO is to simulate the movement
of a flock of birds seeking food.
Genetic algorithms (GA) have been widely applied to system
optimization. The application of GA in combination logic
design optimization was proposed by Louis [3]. In this work,
GA was combined with knowledge based systems and uses
masked crossover operator to solve the combination logic
circuit. This method can solve the functional output for the
combination logic but it does not emphasize on the
optimization of the gate usage. With the increase of the
complexity and the larger problem scale of the logic circuits,
GAs are most frequently faced with the problems of premature
convergence, slow iterations to reach the global optimal solution
and getting stuck at a local optimum. The chaos is a general
phenomenon in nonlinear system and has some special
characteristics such as ergodicity, regularity, randomicity, and
acquiring all kinds of states in a self-rule in a certain range. It is
highly sensitive to change of initial condition that an small
change to initial condition can lead to a big change of the
behavior of the system. Based on the two advantages of the
chaos, a chaos optimization algorithm (COA) was proposed that
can solve complex function optimization and have a high
efficiency of calculation [7]. The basic idea of the algorithm is to
transform the variable of problems from the solution space to
chaos space and then perform search to find out the solution by
virtue of the randomicity, orderliness and ergodicity of the chaos
variable. Although the chaos optimization method has many
advantages such as sensitive to the initial value, easy to skip out
of the locally minimum value, speeding up search because of
reducing the search space by carrier wave, it makes no use of the
experiential information previously acquired. As a result, search
effect of chaos optimization has its own limitation.
In order to overcome the shortcomings of both chaotic
optimization method and GA, one method is to integrate the
COA with GA to fully utilize their respective searching
capabilities. Lü et al. [8] applied a chaotic approach to maintain
the population diversity of genetic algorithm in network training.
In [18] chaos search genetic algorithm and meta-heuristics
method was combined for short-term load forecasting.
However, most of these algorithms just try to use the feature
of randomicity of chaos sequences to generate individuals and
do not effectively combine the spatial search advantage of these
two methods. Taking account of the search efficiency of the GA
and the application of chaos sequences can preferably simulate
chaotic evolutionary process of biology. This paper presents a
CGA based on utilizing characteristic of both COA and GA. The
proposed CGA adopts chaos optimization in initialization to
improve species quality and maintain the population diversity.
Annealing chaotic mutation operation is utilized to replace
standard mutation operator in GA in order to avoid the search
being trapped in local optimum.
The main characteristic of this new method is that the
mechanism of the GA is not changed but the search space and
the coefficient of adjustment are reduced continually. This can
facilitate the evolution of the next generation in order to produce
better optimization individuals, which can improve the
performance of the GA and overcome the disadvantages of the
GA. The correlative examination indicates that the CGA has fast
convergent velocity and powerful search capabilities while at the
same time maintaining the population diversity of the
conventional GA to generate satisfactory results. The
effectiveness of the algorithm has been tested by several test
circuits and benchmarked with different optimization
methods.
II.PROPOSED OPTIMIZATION ALGORITHM
II.1. Chaotic Sequence
Chaos is a universal non-linear phenomenon in natural
world and is the highly unstable motion of deterministic
systems in finite phase space. Roughly speaking, a nonlinear
system is said to be chaotic if it exhibits sensitive dependence
on initial conditions and has an infinite number of different
periodic responses. This sensitive dependence on initial
conditions is generally exhibited by systems containing
multiple elements with nonlinear interactions, particularly
when the system is forced and dissipative.
The chaotic system was initially proposed by Lorenz [10]
and by Henon (1976). Chaos states disorder and irregularities
within a system. In order to enforce non-chaotic behavior, it is
necessary to design a control of chaos. Two possibilities exist
in order to accomplish a system that does not converge to an
attractor or diverge to an edge. The first solution is to detect
chaotic system whenever it is about to arise and design a
feedback system in order to bypass the chaotic region. In
order to find the global minima, the population does not need
to converge, but it is required to stay robust. Robustness is
critical in order to map the solution space. Even when the
objective function has converged, the local or individual
solution may not be converged. Therefore the proposed
approach is to keep the solutions diverse throughout the
evolution, by generating a distance between the solutions
spread instead of the objective function of the solution. In
order to do this, intelligence has to be incorporated within the
solutions. The overriding approach is to incorporate
population dynamics within the solutions in order to organize
a feasible propagation approach.
Chaos theory stipulates that the emergence of chaotic
behavior is invariably linked to initial conditions of the
system. When observing all EA’s, it becomes clear that little
attention is paid to the initial conditions like population. The
overriding approach is to have a population created using
random generation, which the search heuristic will guide
towards the global minima.
In this approach, random generation of initial conditions is
emphasized. Propagation will only occur, if an acceptable
starting point is achieved. If the initial population is very far
away from the global minima, then a large number of
generations would be required in order to find the correct
route.
In this work the creation of the initial population is
becoming very important. Using chaos theory as a guide, the
initial conditions has to be such which can be mapped and its
structure identified.
The chaotic sequence can usually be produced by the
following well-known one dimensional logistic map defined
by May (1976):
x k 1   x k 1  x k  , xk x k  0,1 , k=0,1,2,….
(1)
where μ is a control parameter, and x is a variable. It is clear
that equation (1) is a deterministic dynamic system. The
variable x is also called as chaotic variable.
The value of the control parameter μ determines whether x
stabilizes at a constant size, oscillates between a limited
sequences of sizes, or whether x behaves chaotically in an
unpredictable pattern. Usually, [0, 4] has been defined as
domain area of control parameter μ.
The basic characteristic of chaos could be presented by Eq.
(1). Small variations in the initial value of x will cause large
differences in its long-term behavior of the optimization
algorithm.
The Eq. (1) can be distinguished by four behaviors in
regard with the value of μ. First, when the value of μ is
smaller than 1.0, the chaotic variable xk+1 converges to a
stable point 0.0. Then, if the value of μ is between 1.0 and
3.0, for all the initial values x0, xk+1 would converge to a
certain value between 0.0 and 0.63665. And, the bifurcation
occurs from μ≧3.0. The system will enter the chaos domain,
if μ reaches a critical point of 3.5699. Finally, when μ =4.0
the values of xk+1 would take any real numbers between 0.0
and 1.0 and no redundant value will be generated. The
optimization process of the chaotic variables could be defined
through the Eq. 1 when μ=4 and the system exhibits chaotic
behavior [7]. The equation can be written as:
xk 1,i  4 xk ,i 1  xk ,i  , x k , i  0,1
k=0, 1, 2
(2)
, i=1, 2,…..n
where xk,i is the ith chaotic variable and k denotes the iteration
number. The initial x 0,i ∈ (0,1) and that x 0,i ∉ {0.25, 0.5,
0.75}. In Chaos optimization method, optimization variables
are varied to chaos variables. In next step the ergodic area of
chaotic motion is amplified to the variation ranges of every
variable, as the chaos system was selected has a certain
ergodic area of 0–1. Finally, the chaos search method is used
to optimize the problem.
II.2. Chaotic-Genetic Algorithm
As it was mentioned earlier, when applying GAs to solve
large-scale and complex real-world problems, premature
convergence is one of the most frequently encountered
difficulties. The GA process may also take large number of
iterations to reach the global optimal solution and the
optimization may get stuck at a local optimum. In order to
prevent these issues, it is necessary to find an effective
approach to improve GA to increase speed of convergence
and effectiveness of GA. The chaos as it was explained in
previous section, is a general phenomenon in nonlinear
system that has some special characteristics such as
ergodicity, regularity, randomicity, and acquiring all kinds of
states in a self-rule in a certain range. Based on the two
advantages of the chaos, a chaos optimization algorithm
(COA) was proposed that can solve complex function
optimization and have a high efficiency of calculation [7].
The basic idea of the algorithm is to transform the variables
of problems from the solution space to chaos space and then
perform search to find out the solution by applying of the
randomicity, orderliness and ergodicity of the chaos variables.
The chaos optimization method is very sensitive to the initial
value and can easily skip from localized minimum value.
Therefore, the search space is reduced and the optimization
process will be faster. However, it does not use of the
experimental information that previously acquired. Hence,
search effect of chaos optimization has some limitation.
Genetic algorithms (GA) are designed by randomized
search and optimization techniques. The principles of
evolution and natural genetics are built in functions that are
accompanied by large amount of implicit parallel features.
GA contains a fixed-size population of potential solutions
over the search space. The population could be created by an
objective or fitness function or based on the domain
knowledge of GA.
These potential solutions are named individuals or
chromosomes. GA consists not only of binary strings
chromosomes, but other encoding values are also possible. In
[9] a real-coded GA was proposed and the individual vector
was coded as the same as the solution vector. The evolution
usually starts from a population of randomly generated
chromosomes and continued by selection, crossover, and
mutation in next iterations.
In each GA approach, a new chromosome based on the
following four steps is created:
(1) Evaluation: each chromosome of the population will be
evaluated and assigned a value derived from fitness function.
(2) Selection: chromosomes with higher fitness value will
be more likely to be selected for next generation. In this
study, a competitive strategy is used in selection to improve
its performance.
(3) Crossover: in crossover process two chromosomes as
parents are randomly chosen.
(4) Mutation: The mutation process is a probability-based
procedure in which a heuristic operation was employed to
find shortest path from a random point. Then, a correction
action is taken to keep chromosomes meeting the legal
requirements, it is necessary.
The above four steps are iterated in GA optimization
method until a satisfactory solution is found or the
terminating criterion is met.
To improve the performance of GA search, chromosomes
should be kept scattered in the entire searching space. After
adopting the nature of the chaotic process, a new GA search
method is performed. Chaos- Genetic Algorithms (CGA) that
integrate GA with chaotic variables.
CGA method holds both advantages of GA and the chaotic
variables. CGA method can keep the chromosomes
distributed ergodically in the defined space and prevent
premature generations. However, CGA also takes the inherent
advantage of GA in solution convergence to overcome the
randomness of the chaotic process and hence it increases the
probability of finding the global optimal solution. In proposed
CGA, two measures are adopted to improve the GA’s
performance. First is the adoption of chaos optimization of
the initialization to improve species quality, maintain the
population diversity and finally realize the global
optimization.
Another is the utilization of annealing chaotic mutation
operation to replace standard mutation operator in order to
avoid the search being trapped in local optimum.
(1) Generating initial population by chaotic optimization:
In GA method, the convergence problem is relevant to
initial population. The initial populations generated by
random approach might be far from optimal solution. Hence
the algorithmic efficiency can be very low and more number
of iterations is needed to find the global optimum. The
diverse global search by chaotic ergodictity usually will
acquire better species than randomized search. This will
improve the quality of chromosomes in initial population and
the efficiency of optimization algorithm. The m initial values
xk (0≤x k,i≤1,i=1, 2, 3,...m) are embedded in Eq. 2 where x 0,i ∉
{0.25, 0.5, 0.75}. Then it will generate m chaotic variable xk,i
(xk,i , i=1, 2,...m). The m chaotic variables are mapped into
variable space of optimization. The fitness value of every
feasible solution is calculated and n chromosomes with
highest fitness value form the initial population are selected.
(2) Chaotic mutation operation: Mutation is an effective
operator to increase the population diversity. It is also an
efficient method to escape from local optimum solution and to
prevent the premature convergence. The mutation process can
generate the chromosomes with higher fitness value and guide
evolution of the whole population. The GA process is useful
in generating chromosomes which have the high average
fitness value. Nonetheless GA process can not generate the
optimum chromosomes with higher fitness value. Therefore,
large population should be generated in mutation process to
capture the optimum solution. Having large population will
increase the probability of the process getting stuck in local
optimum or take too much time to converge. To prevent these
issues, chaotic mutation operation can be adopted.
The Mutation processes directly adopts chaotic variable to
carry through ergodic search of solution space and the process
of search is continued based on chaos movement.
stable therefore the mutation operation becomes slow and the
crossover operation becomes more important. Hence
integrating crossover operation with selection operation can
perform accurate search in local solution space. The flowchart
of chaos genetic algorithm for optimization of digital circuit
is shown in figure 1.
Evolution parameters setting
Chaos optimization generates initial population
Calculate fitness value
Evaluate with TruthTable
No
Satisfy
Yes
The main process is as following [4]:
Yes
The nth generation population (yn1 , yn2 ,..., ynm) of current
solution space (a,b) are mapped to chaotic variable interval
[0,1] and formed chaotic variable space :

*
*
*
Y *n  y n1, y n 2 ,.... y nm
 y a
 ,
y *ni   ni
 ba 



(3)
,
Output
No
n=1, 2, …, Gmax
The ith chaotic variable xk,i is regenerated and is added to
mapped chromosome y *ni , and the chaotic mutation
chromosomes are mapped into interval [0, 1]:
*
Z *ni  y ni   x k , i
(4)
where  is the annealing operation.
n 1
n
Termination
Condition
Crossover
i=1, 2, …, m
where Gmax is the maximum evolutional generation of the
population.
  1
Trash
k
(5)
where k is an integer.
Finally, the chaotic mutation chromosome obtained in
interval [0, 1] is mapped to the solution space (a, b) by
definite probability. As it is clear from equation (4) and (5)
the annealing mutation operation simulates process of species
evolution of nature. Because of higher evolutionary attempt,
the population diversity is higher. However, with increase of
evolutionary generation, the population gradually becomes
Chaos mutation operation
Figure 1 : Chaos genetic algorithm flow chart for digital
circuit optimization
III. EXPERIMENTAL CASE STUDIES
To show the effectiveness of the proposed Chaos-genetic
algorithm, several experimental case studies have been
performed. The problem tested in [14-18 ] have been
examines and compared. These case studies could be treated
as representative problems for logic circuit optimization
studies. The chaos-genetic optimization algorithm has been
simulated using Matlab. The truth tables of 7 demonstrative
examples have been shown in table 1-7. Table 8 depicts the
experimental results of comparison of the optimized circuit in
correspond to Example 1 through Example 7. Human design
approach, NGA, MOGA, PSO and CGA have been
compared. As it is clear from the Table 8, the number of the
optimized gates for the specified examples, for PSO and GA
and CGA are the same.
TABLE 1: TRUTH TABLE FOR THE 3 INPUT/1 OUTPUT
i1
0
0
0
0
1
1
1
1
i2
0
0
1
1
0
0
1
1
i3
0
0
0
1
0
1
0
1
O
0
0
0
1
0
1
1
0
TABLE 4: TRUTH TABLE FOR THE 5 INPUT/1 OUTPUT
TABLE 2: TRUTH TABLE FOR THE 4 INPUT/1 OUTPUT
i1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
i2
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
i3
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
i4
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
O
1
1
0
1
0
0
1
1
1
0
1
0
0
1
0
0
TABLE3: TRUTH TABLE FOR THE 4 INPUT/1 OUTPUT
i1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
i2
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
i3
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
i4
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
been significantly reduced. In result the time needed for
optimization has been reduced as well.
O
1
0
0
0
1
1
1
1
1
1
1
0
0
1
0
1
Table 9 through 13 compare the number of iterations and
population size for PSO, MGA, and CGA approach for truth
table 3 through 7. Number of iteration in conjunction with
population size can be treated as operation time. Since the
proposed chaos-genetic method use chaos mutation search
method, it is expected to need less time to find the optimized
solution. As it is understood from table 9 through 13, the
number of iteration in conjunction with population size has
i1 i2
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 1
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
i3
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
i4
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
i5
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
O
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
TABLE 5: TRUTH TABLE FOR 4 INPUT/2 OUTPUT
i1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
i2
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
i3
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
i4
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
O1
1
1
1
0
1
1
0
0
1
0
0
0
0
0
0
0
O2
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
1
TABLE 6: TRUTH TABLE FOR 4 INPUT/3 OUTPUT
i1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
i2
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
i3
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
I4
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
O1
0
0
0
0
0
0
0
1
0
0
1
1
0
1
1
1
O2
0
0
1
1
0
1
1
0
1
1
0
0
1
0
0
1
O3
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
Table 10: Comparison for Table 4
Technique Population size Iteration
MGA
330
6,000
PSO
50
39,600
CGA
100
1500
Table 11: Comparison for Table 5
Technique Population size Iteration
MGA
330
610
PSO
50
4,000
CGA
100
350
TABLE 7: TRUTH TABLE FOR 4 INPUT/4 OUTPUT
i1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
i2
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
i3
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
i4
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
O1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
O2
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
0
O3
0
0
0
0
0
0
1
1
0
1
0
1
0
1
1
0
O4
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
1
TABLE 8: NUMBER OF GATES USING DIFFERENT
APPROACHES
Problem
In/Out
Example1
(3/1)
Example2
(4/1)
Example3
(4/1)
Example4
(5/1)
Example5
(4/2)
Example6
(4/3)
Example7
(4/4)
Table 9: Comparison for Table 3
Technique Population size Iteration
MGA
490
1,500
PSO
50
14,700
CGA
100
700
Human NGA MOGA PSO GA CGA
Table 12: Comparison for Table 6
Technique Population size Iteration
MGA
490
1,500
PSO
50
14,700
CGA
100
900
Table 13: Comparison for Table 7
Technique Population size Iteration
MGA
650
500
PSO
50
19,500
CGA
100
230
IV. CONCLUSIONS
Genetic Algorithms GA’s have been widely used in
optimization of the logic circuit. However, the two problem
of GA method is premature convergence and slow iterations
to reach the global optimization. A chaos-genetic algorithm to
optimized digital circuit has been proposed in this paper. The
algorithm was tested on several reprehensive examples. The
results of experimental testing show that Chaos-genetic
algorithm can perform significantly better than the existing
method. In particular the number of iterations and consumed
time to find the optimized solution has been reduced.
5
4
4
4
4
4
11
10
8
7
7
7
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