The Building-Block Hypothesis

to be sure, not every possible subset of the solution-space can be described as a schema. Simple counting shows that a length-A chromosome can have 2 possible configurations, and therefore 2 possible subsets, but only 3 different schemas. Nonetheless, it is a central axiom of the building-block hypothesis that it is precisely the set of schemas that are effectively being processed by GAs. 

The schema population can be estimated using a simple mean-field-like argument, Let S represent a schema in a size-A population V t) at time t, and Z V,t) instances of the schema at time t. Let f s) be the fitness of the string s, fs be the average fitness of instances of S at time t, and / = Y2ifi be the average fitness of the population. Then the expected number of instances of S at time t + l, Z V, t -f 1), is equal to 

This basic difference equation - known as the Schema Theorem [holl92] - expresses the fact that the sample representation of schemas whose average fitness remains above average relative to the whole population increases exponentially over time. As it stands, however, this equation addresses only the reproduction operator, and ignores effects of both crossover and mutation. 

A lower bound on the overall effect of crossover, which can both create and destroy instances of a given schema, can be estimated by calculating the probability, Pc S), that crossover leaves a schema S unaltered. Let be the probability that the crossover operation will be applied to a string. Since a schema S will be destroyed by crossover if the operation is applied anywhere within its defining length, the probability that S will be destroyed is equal to Pc x 6 S)/ K — 1), where 6 S) is the defining length of S. Hence, the probability of survival ps = 1 — PcS S)/ K — 1), and equation 11.9 takes the updated form  

Finally, in order to also take into account the mutation operator, we note that the probability that a schema S survives under mutation is given by pu S) = (1 — Pm) where pm is the single-bit mutation probability and 0( S) is the number of fixed-bits (i.e. the order) or S. With this we can now express the Schema Theorem that (partially) respects the operations of reproduction, crossover and mutation  

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Forrest, M. and Mitchell, M. Relative building-block fitness and the building-block hypothesis. In Foundations of Genetic Algorithms 2, Ed. Whitley, D., Morgan Kaufmann, San Mateo, CA, 1993, pp. 109-126. 

Schema and the Building Block Hypothesis. Once a population of good local substates has been established, then crossover... 

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Numerical evaluation of the numerator of Eq. (7) reduces to a time-invariant two-component parallel system reliability analysis. It is clear that the first part of Eq. (5) represents the building block for the solution of both time-invariant and time-variant reliability problems (Der Kiureghian 1996). Using Eq. (7), Poisson approximation to the failure probability, Pf oisson(T), is obtained as (under the hypothesis that... 

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Already in 1964 Zweig and Gell-Mann had postulated the existence of quarks as building blocks of hadrons, ihus establishing the premises necessary for a dynamical interpretation of the SU(3) already well-established symmetry.82 At the time of the Solvay conference, however, the quark hypothesis was considered only as a convenient model. 

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