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Inducing Sequentiality Using Grammatical Genetic Codes

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This paper studies the inducement of sequentiality in genetic algorithms (GAs) for uniformly-scaled problems. Sequentiality is a phenomenon in which sub-solutions converge sequentially in time in ...

This paper studies the inducement of sequentiality in genetic algorithms (GAs) for uniformly-scaled problems. Sequentiality is a phenomenon in which sub-solutions converge sequentially in time in contrast to uniform convergence observed for uniformly-scaled problems. This study uses three different grammatical genetic codes to induce sequentiality. Genotypic genes in the grammatical codes are interpreted as phenotypes according to the grammar, and the grammar induces sequential interactions among phenotypic genes. The experimental results show that the grammatical codes can indeed induce sequentiality, but the GAs using them need exponential population sizes for a reliable search.

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    Inducing Sequentiality Using Grammatical Genetic Codes Inducing Sequentiality Using Grammatical Genetic Codes Presentation Transcript

    • 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