steps) = 1 − ( 1 − ( 1 − 1 n )t )⌊n/2⌋ t := (n − 1) ln n Prob (· · · ) = 1 − ( 1 − ( 1 − 1 n )(n−1) ln n )⌊n/2⌋ ≥ 1 − ( 1 − (1 e )ln n )⌊n/2⌋ ≥ 1 − ( 1 − 1 n )⌊n/2⌋ ≥ 1 − (1 e )1/2 = 1 − e−1/2 > 0.39 372 [...] together. . . Prob ( T(1+1) EA,OneMax > (n − 1) ln n | A ) ≥ 1 − e−1/2 E ( T(1+1) EA,OneMax | A ) ≥ (n − 1) ln n · Prob ( T(1+1) EA,OneMax > t | A ) ≥ ( 1 − e−1/2 ) (n − 1) ln n E ( T(1+1) EA,OneMax ) ≥ Prob [...] (II) ∀i ∈ {0, 1, . . . , n − 1} : Li := { x ∈ {0, 1}n \ ( i−1⋃ j=0 Lj ) | BinVal(x) < i∑ j=0 2n−1−j } obvious si ≥ 1 n ( 1 − 1 n )n−1 ≥ 1 en Theorem E ( T(1+1) EA,BinVal ) ≤ en2 365 Introduction Fitness-Based …

(be−1c)3 = (b−1a−1bd−1e−1)2 = (eb−1)5 = (eb)5 = (ed−1e−1b−1d)2 = b−1d−1e−1b−1d−1ebd−1bd−1e−2 = (dbede−1)3 = (d−1bd−1e−2)3 = ((ed−1e−1b−1d−1e2db−1d)2eb−1eb−1)2 = (abebea−1e−1b−1)2 = ebced−1e−1b−1d−1e2db−1dceb [...] (b−1a−1bd−1e−1)2 = (eb−1)5 = (eb)5 = (ed−1e−1b−1d)2 = b−1d−1e−1b−1d−1ebd−1bd−1e−2 = (dbede−1)3 = (d−1bd−1e−2)3 = (cebceb−1eb−1)2 = (abebea−1e−1b−1)2 = ebced−1e−1b−1d−1e2db−1dceb = ede−1d−2e2db−1dcb−1e−1b [...] = (be−1c)3 = (b−1a−1bd−1e−1)2 = (eb−1)5 = (eb)5 = (ed−1e−1b−1d)2 = b−1d−1e−1b−1d−1ebd−1bd−1e−2 = (dbede−1)3 = (d−1bd−1e−2)3 = (cebceb−1eb−1)2 = (abebea−1e−1b−1)2 = ebced−1e−1b−1d−1e2db−1dceb = 1〉. Nach …

· p+1 +p−1 p+1 · · · p+i−1+p−i−1 p+i−1 sowie der Voraussetzung folgt E0 · (1− p−i p+i + p−i ) ≤ 1 p+0 + 1 p+1 + p−1 + · · ·+ 1 p+i−1+ p−i−1 + 1 p+i + p−i · p+1 + p−1 p+1 · · · p+i−1+ p−i−1 p+i−1 + p+i [...] p+i Ei+1 p+i + p−i . Schlussendlich ist (1− p−i p+i +p−i ) · p+i +p−i p+i = 1 und es gilt E0 ≤ 1 p+0 + 1 p+1 + p−1 + · · ·+ 1 p+i−1+ p−i−1 + 1 p+i + p−i · p+1 + p−1 p+1 · · · p+i−1+ p−i−1 p+i−1 · p+i [...] Pr[Tf ,(1?̄λ)-EA > 1+λ | x1 = x(1)] = Pr[Tf ,(1?λ)-EA > 1+λ | x1 = x(1)]≤ min{tmax f ,(1?λ)-EA/(1+λ), 1}. Wir betrachten nun die i-te Phase, i ≥ 0, des (1?̄λ)-EA. Diese umfasst 7d(tmax f ,(1?λ)-EA−1)/λe=: …

min{1, 1−A(a) + B(b)} R a=3 a=4 a=5 a=6 b=1 1 1/9 1/9 1 b=2 1 1 1 1 b=3 1 1/9 1/9 1 Generalized Modus tollens (GMT) using max-prod composition: B′ ◦R−1 = A′ = (0 0 1) ◦  1 1/9 1/9 1 1 1 1 1 1 1/9 1/9 [...] + (1− b)(1− a) = 1− 1 + b + 1− b− a + ab = 1− a + ab = Imp(a, b) Lukaciewicz: Imp(c(b), c(a)) = min{1, 1−(1−b)+(1−a)} = min{1, 1−1+b+1−a} = min{1, 1−a+b} = Imp(a, b) Gödel: Imp(c(b), c(a)) = { 1 (1− b) [...] 1/9 1  = (1 1/9 1/9 1) max-prod: (P •Q)(x, z) = maxy∈Y {P (x, y) ·Q(y, z)}, P,Q fuzzy relations (B′ •R−1)(1, a) = maxb{B′(b) ·R−1(b, a)} e.g (B′ •R−1)(1, 4) = maxb{0 · 1/9, 0 · 1, 1 · 1/9} = 1/9 Exercise …

benutzen können. x1 x2 y -1.5 -2.0 +1 -1.0 0.0 +1 -0.5 1.0 +1 0.0 2.0 +1 0.5 1.0 +1 1.0 2.0 +1 1.5 3.75 +1 -1.0 -2.0 -1 -0.5 -1.0 -1 0.0 -3.0 -1 0.5 -0.5 -1 1.0 -2.0 -1 1.5 1.5 -1 -4 -3 -2 -1 0 1 2 3 4 -4 -3 [...] Abbildung 1): D = { (1, 3,−1), (2, 5,−1), (3, 6,−1), (4, 4,−1), (5, 2,−1), (6, 3,−1), (8, 4,−1), (9, 4,−1), (3, 9, +1), (4, 8, +1), (6, 8, +1), (7, 9, +1), (8, 7, +1), (9, 8, +1), (10, 6, +1), (11, 7, +1) } Dabei [...] Welche Funktion ermöglicht eine lineare Trennung der Daten? (a) Φ1(x1, x2) = (x2 1, x2) (b) Φ2(x1, x2) = (x3 1 − 2x1, x2) (c) Φ3(x1, x2) = (x3 1, x2) Hinweis: Die graphische Darstellung ist als Hilfe für Sie …

−4 −1 −3 −3 −2 11  ⊕ ( 1 0 0 1 ) h1p3R2 =  2 −1 0 1 −1 −1 2 −1 2 0 −1 1 3 −2 0 0 3 0 0 0 0 1 −1 0 2 0 −2 −1 −1 1 0 0 4 1 −1 −1 3 0 −2 1 8 0 2 −2 0 −1 −1 0 9 ⊕ ( 1 1 1 1 2 1 1 1 [...] 4,0) 1p3R1 1 1/155.520 (5,0,1) 1p3R3 1 1/311.040 (1,2,1) 1p3R2 1 1/7.776 (2,0,2) 1p3R4 1 1/2.592 Ordnung 5: ZG-Struktur orthogonale Darstellung max k max ω (0,2,0) 1p5R2 1 1/600 (3,0,1) 1p5R1 1 1/1.200 [...]  1 −1 1 0 0 −1 3 −1 0 −1 1 −1 2 0 0 0 0 0 3 0 0 −1 0 0 2 ⊕  1 ζ3 + 1 0 0 0 −ζ3 2 0 0 0 0 0 2 2ζ3 + 1 −ζ3 0 0 −2ζ3 − 1 3 −ζ3 − 2 0 0 ζ3 + 1 ζ3 − 1 2  h1p3R2 =  1 1 1 1 0 0 1 1 7 −2 …

z 20 1.49 155 18 1.43 143 42 1.84 329 20 1.46 156 Saarland 18 1.51 146 16 1.45 130 42 1.96 317 24 1.72 173 Sachsen 20 1.24 246 19 1.21 224 42 1.55 486 19 1.24 227 Sachsen-Anhalt 15 1.16 180 18 1.27 200 [...] 23 1.64 188 13 1.34 100 1 0.45 11 Sachsen 72 1.40 859 21 1.28 254 5 0.68 70 1 0.39 16 Sachsen-Anhalt 58 1.60 651 26 1.44 292 15 1.14 182 1 0.32 15 Schleswig-Holstein 72 1.71 686 19 1.47 187 8 1.02 84 1 0 [...] berg 22 1.45 212 19 1.39 187 41 1.75 374 18 1.40 163 Bayern 31 1.59 303 19 1.37 188 30 1.59 289 19 1.39 173 Berlin 22 1.74 211 16 1.34 178 41 2.05 380 20 1.72 176 Brandenburg 19 1.27 215 22 1.35 251 42 …

1≤κ≤n−k−1 (2κ− 1)! (κ− 1)! ∏ 1≤κ≤k 1≤λ≤n−k ( −1 2 + λ− κ ) ∏ 1≤κ≤k 1≤λ≤n−k ( −3 2 + λ + κ ) ·(n− 1)!−1 · ∏ 1≤κ≤n−1 (2κ− 1)!−1 = ± ∏ 1≤κ≤k (2κ− 2)! · ∏ 1≤κ≤n−k−1 (2κ− 1)! · (n− k − 1)! · ∏ 1≤κ≤k 1≤λ≤n−k ( −1 2 [...] (2κ− 1)! (κ− 1)! ∏ 1≤κ≤k 1≤λ≤n−k ( −1 2 + λ− κ ) ∏ 1≤κ≤k 1≤λ≤n−k ( −3 2 + λ + κ ) ·(n− 1)!−1 · ∏ 1≤κ≤n−1 (2κ− 1)!−1 = ± ∏ 1≤κ≤k (2κ− 2)! · ∏ 1≤κ≤n−k−1 (2κ− 1)! · (n− k − 1)! · ∏ 1≤κ≤k 1≤λ≤n−k ( −1 2 + λ− [...] 2λ−1 + (1 + yκ) 2λ−1 ) 1≤κ,λ≤n = 2l l∏ ν=1 (2ν − 1) · det   λ∈{1,...,l}︷ ︸︸ ︷ x2λ−2 κ + y2λ−2 κ (1− xκ) 2l+1 + (1 + xκ) 2l+1 + (1− yκ) 2l+1 + (1 + yκ) 2l+1 (1− xκ) 2λ−1 + (1 + xκ) 2λ−1 + (1− yκ) …

Sachsen 2,858 1,212 42.4 163 5.7 1,425 49.9 40 1.4 18 0.6 Sachsen- Anhalt 1,518 144 9.5 92 6.1 1,261 83.1 . . . . Schleswig- Holstein 1,870 389 20.8 52 2.8 1,385 74.1 21 1.1 23 1.2 Thüringen 1,567 231 14 [...] 6.5 57 3.1 97 5.3 4,518 122 2.7 1,478 32.7 1,578 34.9 794 17.6 353 7.8 123 2.7 70 1.5 2,987 34 1.1 1,014 33.9 1,385 46.4 459 15.4 81 2.7 10 0.3 4 0.1 Berlin 2,832 929 415 44.7 66 7.1 131 14.1 78 8.4 43 [...] 30.9 1,100 11.7 545 5.8 354 3.8 1,396 14.8 28,837 1,484 5.1 3,817 13.2 9,363 32.5 7,813 27.1 3,723 12.9 1,559 5.4 1,078 3.7 17,835 521 2.9 1,814 10.2 7,185 40.3 6,657 37.3 1,350 7.6 247 1.4 61 0.3 1 Lei …