Inducing Sequentiality Using
Grammatical Genetic Codes
Kei Ohnishi, Kumara Sastry,
Ying-ping Chen, and David E. Goldberg
Illinois Genetic Algorithms Laboratory (IlliGAL)
University of Illinois at Urbana-Champaign

Contents
Background and Objective
Uniformly-scaled problems
Sequentiality
Grammatical Genetic Codes
Experiments
Summary and Conclusion

Backgrounds
niformly-scaled problems
proportion of a correct BB
ontribution to a fitness value
1
no priority
0
…
BB1 BB2 BB3 BBm generation

Objective
Uniformly-scaled problems
Sequentiality
proportion of a correct BB
1
BB1 BB2 BB3 BBm
…
rammatical Genetic Code
artificially brings priority among
phenotypic genes 0
generation

Contents
Background and Objective
Uniformly-scaled problems
Sequentiality
Grammatical Genetic Codes
Experimental Results
Summary and Conclusion

Grammatical Genetic Codes
General Characteristics
( 1 ) Genotypic genes are decoded in a certain order.
( 2 ) Grammar determines how to decode genotypic genes.
( 3 ) Interpretation of genotypic genes depends on genotypic ones
decoded earlier on.
Example ) If g1 is changed, interpretation of other genes is also changed.
decoding order ( left to right )
g3
g1 g2 g3
g2
h1
enotype
decoding
p1 p1
p3 p3
henotype

Our expectation
Due to grammar, interpretation of genotypic genes
depends on genotypic ones decoded earlier on.
Grammar induces sequential interactions among
phenotypic genes.
The sequential interactions among phenotypic genes
could induce prioritized phenotypic convergence
for search problems including uniformly-scaled problems.

GAuGE Code
Genetic Algorithms using Grammatical Evolutio
decoding order ( left to right )
p1 v1 p2 v2 pl vl
genotype
pi % ( l – i + 1) = phenotypic position
vi % 2 = phenotypic value
01234 l-2 l-1
phenotype 1
l-3 l-2
12 34
0
1
1
11110 00

Complex and Cellular Grammatical Codes
gl
g1 g2
genotype
Complex
operation to
complex numbers
p1 v1 p2 v2 pl vl Cellular
cellular automaton
pi % ( l – i + 1) = phenotypic position
vi % 2 = phenotypic value
01234 l-2 l-1
henotype 1

Contents
Background and Objective
Uniformly-scaled problems
Sequentiality
Grammatical Genetic Codes
Experimental Results
Summary and Conclusion

Experiments
Test Problems
OneMax Problem with l bits ( OneMax-l )
4-bit Trap Deceptive Function with Tightly
Linked m BBs ( (m,4)-Trap-T )
4-bit Trap Deceptive Function with Loosely
Linked m BBs ( (m,4)-Trap-L )

Experimental plan
( 1 ) Can the grammatical codes change uniformly-scaled problems
to non-uniformly-scaled ones?
If yes
( 2 ) Can GAs using the grammatical codes induce the sequentiality
for uniformly-scaled problems?
If yes
( 3 ) How is the scalability of the GAs using the grammatical codes?

( 1 ) Can the grammatical codes change uniformly-scaled
problems to non-uniformly-scaled ones?
Genotypic Space
A randomly generated genotype
Genotypes obtained by changing
only a value at the i-th position of
H ( #, % ) : Hamming distance between phenotypes of # and %
hi = ( Σ H ( , ) ) / ( the number of )
Hi = ( Σ hi ) / ( the number of )
F ( # ) : fitness value of #
fi = ( Σ | F ( ) – F ( ) | ) / ( the number of )
Fi = ( Σ fi ) / ( the number of )

Results
GAuGE Complex Cellular
Hi
locus
i
eMax-32
Fi
4)-Trap-T
locus
i
4)-Trap-L

Experimental plan
( 1 ) Can the grammatical codes change uniformly-scaled problems
to non-uniformly-scaled ones?
YES
( 2 ) Can GAs using the grammatical codes induce the sequentiality
for uniformly-scaled problems?
If yes
( 3 ) How is the scalability of the GAs using the grammatical codes?

( 2 ) Can GAs using the grammatical codes induce
the sequentiality for uniformly-scaled problems?
Convergence of phenotypic values
Obtaining generations at which
proportion of a correct BB
proportion of correct BBs is over 0.9
generation
gi
g3
g2
g1
g1 g2 g3 gi
generation 1 2 3 i

Results on convergence of phenotypic values
(8,4)-Trap-T
OneMax-80
(8,4)-Trap-L
Standard
GAuGE
Complex
Cellular

( 2 ) Can GAs using the grammatical codes induce
the sequentiality for uniformly-scaled problems?
Convergence of phenotypic positions
i l
1 2 1 2 l
… pl
p1 p2
phenotype
1. Observing proportion that the i-th set of genotypic
genes are mapped onto each phenotypic locus
2. Selecting the highest proportion from among them
Ph ( i )
Ph ( i )
Population (genotypes)
1 Ph ( 1 )
Ph ( l )
0

Results on convergence of phenotypic positions
GAuGE Complex Cellular
eMax-32
4)-Trap-T
4)-Trap-L

Experimental plan
( 1 ) Can the grammatical codes change uniformly-scaled problems
to non-uniformly-scaled ones?
YES
( 2 ) Can GAs using the grammatical codes induce the sequentiality
for uniformly-scaled problems?
YES
( 3 ) How is the scalability of the GAs using the grammatical codes?

( 3 ) How is the scalability of the GAs using
the grammatical codes?
GA settings Obtaining reliable population size
one-point crossover
• Minimal population size with which
no mutation
• GA can find the optimum over 95 times
minimal generation gap model
• out of 100 independent runs
genetic code
•
log ( reliable population size )
1. Standard genetic code
2. GAuGE code
3. Complex genetic code
4. Cellular genetic code

Results
(m,4)-Trap-T
OneMax-l
(m,4)-Trap-L
Standard
GAuGE
Complex
Cellular

Contents
Background and Objective
Uniformly-scaled problems
Sequentiality
Grammatical Genetic Codes
Experimental Results
Summary and Conclusion

Summary
The grammatical genetic codes get uniformly-scaled
problems to be non-uniformly-scaled problems.
The grammatical genetic codes help GAs induce
sequentiality together with the genetic operator.
GAs using the grammatical genetic codes scale-up
exponentially with problem size.

Conclusion
GAs using the grammatical genetic codes scale-up
exponentially with problem size.
Fixation of the genotypic genes from left to right is
not strong enough for a recombination operator
to mix genotypes effectively.
Selectormutative GAs
Selectorecombinative GAs

Relations to Grammatical Evolution
Interactions among components Inherent interactions among
of a program induced by grammar components of a program
well-balanced
Diversity due to binary-coded
genotypic genes
Recombination Mutation