Phenotypes, Genotypes, and Operators in Evolutionary Computation David B. Fogel Natural Selection, Inc. 1591 Calle De Cinco La Jolla, CA 92037 [email protected] ABSTRACT Evolutionary computation can be conducted at various levels of abstraction (e.g., genes, individuals, species). Recent claims have been made that simulated evolution can be made more biologically accurate by applying specific genetic operators that mimic low-level transformations to DNA. This paper argues instead that the appropriateness of particular variation operators depends on the level of abstraction of the simulation. Further, including specific random variation operators simply because they have a similar form as genetic operators that occur in nature does not, in general, lead to greater fidelity in simulation. 1. Introduction sult may be opposite: the explanatory power of such a model of evolution may decrease, not increase, with the Evolution is characterized by distinct levels of hierar- inclusion of successively more complicated genetic ef- chy (e.g., species, individuals, chromosomes, genes), and fects. quite naturally so is evolutionary computation. Typically, the elements in a simulated evolving population are con- 2. The Genotype, The Phenotype, and sidered to be analogous to species in evolutionary pro- Lewontin's Mappings gramming [9], individuals in evolution strategies [20], and chromosomes and genes in genetic algorithms [7, p. Living organisms can be viewed as a duality of their 2]. Recently, Collins [5] argued that evolutionary com- genotype (the underlying genetic coding) and their phe- putation can be made more “biologically accurate” by notype (the manner of response contained in the behav- including specific operators which mimic low-level ior, physiology, and morphology of the organism). changes that occur to DNA. Such a broad assertion ig- Lewontin [15] illustrated this distinction by specifying nores the level of abstraction of the simulation. The ap- two state spaces: a populational genotypic (informa- propriateness of particular operators in a simulation de- tional) space G and a populational phenotypic (behav- rives from the level of abstraction and not necessarily ioral) space P. Four functions map elements in G and P from any specific resemblance to a mechanism found in to each other (Figure 1). Atmar [1] modified these func- natural evolution. tions to be: In general, models of natural phenomena can be con- structed in two ways: bottom-up or top-down [10, pp. × → f1: I G P 255-258]. A bottom-up simulation emphasizes segrega- → f2: P P tion of individual components and their local interac- → f3: P G tions, reducing the total system into successively smaller f : G → G. subsystems that are then analyzed in a piecemeal fash- 4 ion. In contrast, top-down analyses emphasize extrinsi- ∈ The function f1, epigenesis, maps the element g1 G cally imposed forces and their associated physics as they into the phenotypic space P as a particular collection of pertain to the modeled system in its entirety. The appro- phenotypes p1 whose development is modified by its priateness of different operators in an evolutionary com- ∈ environment, an indexed set of symbols (i1, ..., ik) I, putation follows from the adopted perspective, either where I is the set of all such environmental sequences. bottom-up, emphasizing specific genetic mechanisms, The function f2, selection, maps phenotypes p1 into p2. or top-down, emphasizing phenotypic (behavioral) re- As natural selection operates only on the phenotypic lationships. A careful examination of the nature of geno- expressions of the genotype [17, p. 131; 12, p. 431], the types and phenotypes, and the mappings between them, underlying coding g is not involved in the function f . reveals that the simple inclusion of genetic mechanisms 1 2 The function f3, genotypic survival, describes the effects as they occur in nature does not by consequence lead to of selection and migration processes on G. Function f , more biologically accurate simulation. Indeed, the re- 4 ∈ mutation, maps the representative codings g2 G to the ∈ point g'1 G. This function represents the “rules” of mutation and recombination, and encompasses all ge- netic changes. With the creation of the new population of genotypes g'1, one generation is complete. Evolution- ary adaptation occurs over successive iterations of these mappings. The mapping between the genotype and the pheno- type is characterized by pleiotropy and polygeny (Fig- ure 2). Pleiotropy is the effect that a single gene may simultaneously affect several phenotypic traits. Polyg- eny is the effect that a single phenotypic characteristic of an individual may be determined by the simultaneous interaction of many genes. Pleiotropy and polygeny pro- hibit any useful reductionist simplification of the geno- type-phenotype mapping; the mapping is not linearly separable. Single genetic changes can result in multiple phenotypic changes and individual phenotypic traits can- not be assigned to a single genetic cause (with the ex- ception of gene defects, see [2]). 3. Levels of Hierarchy in Evolutionary Computation All efforts in evolutionary computation involve popu- Figure 1. The evolution of a population within a single lation-based search with random variation and selection. generation. Evolution can be viewed as occurring as a succession of four mapping functions (epigenesis, selec- But the multiple levels of hierarchy in evolution and the tion, genotypic survival, and mutation) relating the dual nature of individuals in terms of their genotype and genoptypic information state space and the phenotypic phenotype suggest various methods for simulating evo- behavioral state space. lution. In particular, attention may be focused on the genes of individuals, on the behavior of individuals, or on the behavior of reproducing populations (species). Each of these perspectives are represented by the ef- forts in genetic algorithms, evolution strategies, and evo- lutionary programming, respectively. Within the framework of genetic algorithms, atten- tion is given to the genotypes of individuals, and the elements (data structures) in an evolving population are considered analogous to chromosomes and genes [7, p. 2]. Each structure, which in this case is purely geno- typic, is decoded into a phenotypic expression which is in turn evaluated in light of a fitness function. Selection acts to replicate the structures in the population in pro- portion to their relative fitness. New structures are cre- ated by applying operators that mimic the forms of trans- formations that occur to natural chromosomes and genes (e.g., crossover, inversion, mutation). Within evolution strategies, attention is instead given to the phenotypes of individuals, and the elements in an evolving population are abstracted as vectors of behav- ioral traits [20]. Each vector of traits is evaluated in light of a fitness function. Selection acts to eliminate those Figure 2. The effects of pleiotropy (single gene—mul- behaviors having a performance that is below a pre- tiple phenotypic expressions) and polygeny (single phe- scribed threshold. New vectors are created by applying notypic character—multiple genes) (after [16, p. 265]). operators which mimic the functional change in pheno- typic traits in individuals. For example, for continuous- valued traits, the behavioral change may follow a zero- (a) GENETIC ALGORITHM mean Gaussian random variable. Further, the variabil- ity of the change may be determined in part by genetic • Decode Genotypes; structures (i.e., strategy parameters) which accompany Evaluate Fitness of each vector of phenotypic traits. These genetic struc- Phenotypes • Amplify Genotypes in tures may undergo varieties of recombination and mu- Proportion to Fitness g1 g’1 tation operators. • Use Crossover, Muta- Within evolutionary programming, attention is given tion, and Other Genetic to the phenotypes of entire reproductive populations Operators to Generate (species), and the elements in an evolving population New Genotypes. can again be abstracted as vectors of behavioral traits [10]. In a similar manner as with individual traits, each vector of behavioral traits is evaluated in light of a fit- (b) EVOLUTION STRATEGIES ness function and selection acts to eliminate those be- • Assess Fitness of haviors having a performance that is below a prescribed Phenotypes threshold. New vectors are created by applying opera- • Use Selection to Elimi- tors that mimic the functional change in phenotypic traits nate Poor Phenotypes in populations. Again, the variability of the change may p1 • Use Self-Adaptation, p’1 be determined in part by genetic strategy parameters. Including Recombina- But these genetic structures are only subject to individual tion, (Genetics) to random variation and do not undergo varieties of re- Generate New Distribu- combination operators because there is no recombina- tion of Phenotypes tion between species (using the biological species con- cept of Mayr [18, p. 318]. (c) EVOLUTIONARY PROGRAMMING 4. Phenotypic and Genotypic Operators • Assess Fitness of Phenotypes The level of abstraction in a simulation of evolution • Use Selection to dictates the appropriateness of phenotypic or genotypic (Stochastically) Elimi- P’ operators. In genetic-based simulations (e.g., genetic al- P1 nate Poor Phenotypes 1 gorithms, [7, 14]), operators which mimic the form of • Use Self-Adaptation (Genetics) to Generate genetic transformation in natural evolution are applied New Distribution of to abstracted