JAIST Repository: 動的な関数における複雑さの発展
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(2) Evolution of Complexity in Dynamical Functions Jun Namikawa School of Knowledge Science, Japan Advanced Institute of Science and Technology March 2001 Keywords: Dynamical Functions, Homomorphism, Complexity of Functions, Redundancy, Functional Entropy. Abstract A new approach to study the evolution of complexity is presented as a formal system describing a dynamic change of functions. We study, analytically and numerically, how complexity of functions changes by transformation. A dynamical function fn : Sn → S (Sn ⊆ S) is defined by fn+1 = g ◦ fn ◦ gn∗. (1). ∗ ∗ where g : S → S and an indexed family {gi∗ }∞ i=0 with gn : Sn+1 → Sn are g|Sn ◦ gn = in+1 . Here ,in+1 : Sn+1 → S is inclusion mapping. For any function g, fn and Sn ⊆ S, if x ∈ gn∗ (Sn+1 ) then g ◦ fn (x) = fn+1 ◦ g(x). Thus the Eq. (1) is homomorphic mapping in gn∗ (Sn+1 ) ⊆ Sn . We propose measures of complexity of functions, “redundancy” and “functional entropy”. Redundancy evaluates the number of inputs giving the same output in a function. Functional entropy measures the degree of randomness of outputs from a function when inputs are given randomly. Given any function describing a dynamical system, these measures identify whether it has periodicity. Thus, these measures represent complexity of functions. If there is an isomorphism between f and f , then both functions have the same value for each measure. We find that these measures have the following characters:. 1. If g is an injection, max redundancy decreases. 2. If gn∗ is an surjection, average redundancy increases. 3. If g is an injection and gn∗ is an surjection, redundancy is invariable.. c 2001 by Jun Namikawa Copyright . 1.
(3) We study how redundancy and functional entropy change in terms of numerical simulations of Eq. (1), by choosing logistic map ax(1 − x) and tent map a(0.5 − |x − 0.5|) as function g, and x2 , x + 0.3 (mod1), and π2 arctan(2x − 1) + 0.5 as the initial function f0 . Here, the domain and range of g and fn are chosen to be [0, 1]. As results of simulations, we confirm that 1) redundancy changes largely for logistic map and not so large for tent map; 2) functional entropy is associated with lyapunov exponent.. 2.
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