Chapter 4: Theoretical Foundations of Genetic Algorithms come up with the best encoding for the GA. In fact, coming up with the best encoding is almost tantamount to solving the problem itself! An example of this was seen in the discussion on evolving cellular automata in chapter 2 above. The original lexicographic ordering of bits was arbitrary, and it probably impeded the GA from finding better solutions quickly—to find high−fitness rules, many bits spread throughout the string had to be coadapted. If these bits were close together on the string, so that they were less likely to be separated under crossover, the performance of the GA would presumably be improved. But we had no idea how best to order the bits ahead of time for this problem. This is known in the GA literature as the "linkage problem"—one wants to have functionally related loci be more likely to stay together on the string under crossover, but it is not clear how this is to be done without knowing ahead of time which loci are important in useful schemas. Faced with this problem, and having notions of evolution and adaptation already primed in the mind, many users have a revelation: "As long as I'm using a GA to solve the problem, why not have it adapt the encoding at the same time!" A second reason for adapting the encoding is that a fixed−length representation limits the complexity of the candidate solutions. For example, in the Prisoner's Dilemma example, Axelrod fixed the memory of the evolving strategies to three games, requiring a chromosome of length 64 plus a few extra bits to encode initial conditions. But it would be interesting to know what types of strategies could evolve if the memory size were allowed to increase or decrease (requiring variable−length chromosomes). As was mentioned earlier, such an experiment was done by Lindgren (1992), in which "gene doubling" and "deletion" operators allowed the chromosome length—and thus the potential memory size—to increase and decrease over time, permitting more "open−ended" evolution. Likewise, tree encodings such as those used in genetic programming automatically allow for adaptation of the encoding, since under crossover and mutation the trees can grow or shrink. Meyer and Packard's encoding of condition sets also allowed for individuals of varying lengths, since crossovers between individuals of different lengths could cause the number of conditions in a set to increase or decrease. Other work along these lines has been done by Schaefer (1987), Harp and Samad (1991), Harvey (1992), Schraudolph and Belew (1992), and Altenberg (1994). Below I describe in detail three (of the many) approaches to adapting the encoding for a GA.
Inversion Holland (1975) included proposals for adapting the encodings in his original proposal for GAs (also see Goldberg (1989a). Holland, acutely aware that correct linkage is essential for single−point crossover to work well, proposed an "inversion" operator specifically to deal with the linkage problem in fixed−length strings. Inversion is a reordering operator inspired by a similar operator in real genetics. Unlike simple GAs, in real genetics the function of a gene is often independent of its position in the chromosome (though often genes in a local area work together in a regulatory network), so inverting part of the chromosome will retain much or all of the "semantics" of the original chromosome.
To use inversion in GAs, we have to find some way for the functional interpretation of an allele to be the same no matter where it appears in the string. For example, in the chromosome encoding a cellular automaton (see section 2.1), the leftmost bit under lexicographic ordering is the output bit for the neighborhood of all zeros. We would want that bit to represent that same neighborhood even if its position were changed in the string under an inversion. Holland proposed that each allele be given an index indicating its "real" position, to be used when evaluating a chromosome's fitness. For example, the string 00010101 would be encoded as
with the first member of each pair giving the "real" position of the given allele. This is the same string as, say,
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