Global+Optimization+Algorithms+Theory+and+Application_Part5

Global+Optimization+Algorithms+Theory+and+Application_Part5...

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1.5 Formae and Search Space/Operator Design 81 1 for the input 0, and false otherwise. Assume that the formulas were decoded from a binary search space G = B n to the space of trees that represent mathematical expression by a genotype-phenotype mapping. A genotypical property then would be if a certain sequence of bits occurs in the genotype p . g and a phenotypical property is the number of nodes in the phenotype p . x , for instance. If we try to solve a graph-coloring problem, for example, a property φ 3 ∈{ black , white , gray } could denote the color of a specific vertex q as illustrated in Figure 1.29 . A f 3 = A f 3 = black q G 1 q G 3 q G 2 q G 4 q G 6 q G 7 q G 8 q G 5 A f 3 = gray Í Pop X Figure 1.29: An graph coloring-based example for properties and formae. In general, we can imagine the properties φ i to be some sort of functions that map the individuals to property values. φ 1 and φ 2 would then both map the space of mathematical functions to the set B = { true , false } whereas φ 3 maps the space of all possible colorings for the given graph to the set { white , gray , black } . On the basis of the properties φ i we can define equivalence relations 74 φ i : p 1 φ i p 2 φ i ( p 1 ) = φ i ( p 2 ) p 1 ,p 2 G × X (1.48) Obviously, for each two solution candidates and x 1 and x 2 , either x 1 φ i x 2 or x 1 negationslash∼ φ i x 2 holds. These relations divide the search space into equivalence classes A φ i = v . Definition 1.57 (Forma). An equivalence class A φ i = v that contains all the individuals sharing the same characteristic v in terms of the property φ i is called a forma [1691] or predicate [2122]. A φ i = v = {∀ p G × X : φ i ( p ) = v } (1.49) p 1 ,p 2 A φ i = v p 1 φ i p 2 (1.50) The number of formae induced by a property, i. e., the number of its different character- istics, is called its precision [1691]. The precision of φ 1 and φ 2 is 2, for φ 3 it is 3. We can define another property φ 4 f (0) denoting the value a mathematical function has for the input 0. This property would have an uncountable infinite large precision. Two formae A φ i = v and A φ j = w are said to be compatible , written as A φ i = v ⊲⊳ A φ j = w , if there can exist at least one individual which is an instance of both. 74 See the definition of equivalence classes in Section 27.7.3 on page 464 .
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82 1 Introduction f (x)=x+1 1 f (x)=x +1.1 2 2 f (x)= 3 x+2 f (x)=2( 4 x+1) f (x)=(sin x)(x+1) 6 f (x)=(cos x)(x+1) 7 f (x)=(tan x)+1 8 f (x)=tan x 5 f 1 f 4 f 6 f 7 A f 1 = true f 2 f 3 f 8 f 5 A f 1 = false f 1 f 2 f 7 f 8 A f 2 = true f 3 f 4 f 6 f 5 A f 2 = false A f 4 = 1 f 1 f 2 f 7 f 8 A f 4 = 0 f 6 f 5 f 3 f 4 A f 4 = 2 ~ f 1 ~ f 2 ~ f 4 A f 4 = 1.1 Í Pop X Figure 1.30: Example for formae in symbolic regression. A φ i = v ⊲⊳ A φ j = w A φ i = v A φ j = w negationslash = (1.51) A φ i = v ⊲⊳ A φ j = w ⇔∃ p G × X : p A φ i = v p A φ j = w (1.52) A φ i = v ⊲⊳ A φ i = w w = v (1.53) Of course, two different formae of the same property φ i , i. e., two different charac- teristics of φ i, are always incompatible. In our initial symbolic regression example hence A φ 1 = true negationslash
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