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# An empirical study of the efficiency of learning ... - ResearchGate

An empirical study of the efficiency of learning ... - ResearchGate

## 100). Thus for instance,

100). Thus for instance, in row two of Table 2, R(z) is 2, indicating that 2 runs of population 4, lasting 721 generations (including the initial population), is required to give a success probability of 0.99. GENETIC ALGORITHM RESULTS EVEN PARITY FUNCTIONS Breeding rate = 100.0%, Crossover rate = 50%, Crossover type = uniform, Mutation rate = 0.25%, acceptance probability = 0.7 (tournament size 2), Unless stated to the contrary gate set = {and, nand, or, nor}. In Tables 4 and 5 all gates were used because with geometries of 10 x 10 and 3 x 3 it was not possible to produce a sufficiently high numbers of 100% functional solutions with the gate set consisting of {and, nand, or, nor}. N denotes the number of generations. Table 2: 3 bit even parity (geometry = 16 x 16) N Pop. size, M R(z) I(M, N, z) 6,000 2 1 (100) 6,002 4,000 4 2 (100) 5,768 3,000 6 1 (100) 4,326 1,000 10 2 (100 ) 5,620 1,000 20 1 (100 ) 6,420 1,000 30 1 (100 ) 7,230 I(M,N,z) 7.5E+03 7.0E+03 6.5E+03 6.0E+03 5.5E+03 5.0E+03 4.5E+03 4.0E+03 2 4 6 10 20 30 pop. size Figure 2: Variation of I(M,N, z) with population size for 3- bit even parity (16 x 16) Table 3: 4 bit even parity (geometry = 16 x 16) N Pop. size, M R(z) I(M, N, z) 10,000 4 1 (100) 18,404 6,000 10 1 (100 ) 26,410 3,000 20 1 (100 ) 40,820 I(M,N,z) 4.5E+04 3.5E+04 2.5E+04 1.5E+04 4 10 20 pop. size Figure 3: Variation of I(M,N, z) with population size for 4- bit even parity (16 x 16) Table 4: 4 bit even parity (geometry = 10 x 10, gate set = {all}) N Pop. size, M R(z) I(M, N, z) 6,000 4 1 (100) 15,364 6,000 10 4 (100) 24,040 6,000 20 2 (100) 19,240 6,000 30 3 (100 ) 21,690 6,000 40 2 (100 ) 28,880 6,000 50 1 (100 ) 30,050 I (M, N, z) 3.5E+04 3.0E+04 2.5E+04 2.0E+04 1.5E+04 1.0E+04 5.0E+03 0.0E+00 4 10 20 30 40 50 pop. size Figure 4: Variation of I(M,N, z) with population size for 4- bit even parity (10 x 10, all gates) Table 5: 4 bit even parity (geometry = 3 x 3, gate set ={all}) N Pop. size, M R(z) I(M, N, z) 25,000 10 152 (3) 761,520 25,000 20 2 (90) 110,220 25,000 30 1 (100) 75,150 10,000 40 2 (100 ) 48,240 5,000 50 1 (99 ) 40,400

I (M, N, z) 8.0E+05 6.0E+05 4.0E+05 2.0E+05 0.0E+00 10 20 30 40 50 pop. size Figure 5: Variation of I(M,N, z) with population size for 4- bit even parity (3 x 3, all gates) Table 6: 5-bit even parity (geometry = 16x16, * = 30x30) N Pop. size, M R(z) I(M, N, z) 15,000 4* 1 (99) 49,204 10,000 10 2 (97 ) 152,020 15,000 20 2 (100 ) 264,020 10,000 30 2 (98 ) 348,060 I(M,N,z) 4.0E+05 3.0E+05 2.0E+05 1.0E+05 0.0E+00 4 10 20 30 pop. size Figure 6: Variation of I(M,N, z) with population size for 5- bit even parity ( 16 x 16, population size 4 used a 30 x 30 geometry) Table 7: 2-bit multiplier (geometry=4x4, gate set={all}) Pop Breeding rate 100% Breeding rate 0% size R(z) I(M,N,z) R(z) I(M,N,z) M6 3 (93) 900,018 2 (95) 816,012 8 2 (93) 1,344,016 2 (95) 1,312,016 10 4 (88) 1,760,040 3 (88) 2.040,030 20 3 (97) 1,560,060 2 (95) 2,080,040 30 2 (99) 1,800,060 3 (97) 2,340,090 40 3 (96) 2,880,120 3 (99) 2,400,120 50 2 (97) 2,600,100 2 (100) 3,200,100 Table 8: 2-bit multiplier (geometry = 7x7, gate set = {all}) Pop Breeding rate = 100% Breeding rate = 0% size, R(z) I(M,N,z) R(z) I(M,N,z) M2 1 (99) 188,002 2 (98) 248,004 3 2 (99) 264,006 1 (100) 246,003 4 1 (100) 168,004 1 (100) 192,004 6 1 (100) 288,006 1 (100) 300,006 8 1 (100) 256,008 2 (100) 288,016 10 1 (100) 320,010 2 (100) 360,020 20 1 (100) 400,020 1 (100) 640,020 30 2 (100) 480,060 1 (100) 780,030 40 2 (100) 640,080 2 (100) 1,120,080 50 1 (100) 800,050 1 (100) 800,050 Table 9: 2-bit multiplier (geometry = 10 x 10, gate set = {all}) Pop Breeding rate = 100% Breeding rate = 0% size, R(z) I(M,N,z) R(z) I(M,N,z) M2 1 (100) 124,002 2 (94) 164,004 3 1 (100) 156,003 2 (98) 192,006 4 1 (100) 152,004 1 (99) 160,004 6 1 (100) 216,012 2 (99) 264,012 8 1 (100) 176,008 - - 10 1 (100) 240,010 1 (100) 300,010 20 1 (100) 320,020 1 (100) 340,020 30 1 (100) 360,030 1 (100) 540,030 40 1 (100) 480,040 2 (100) 640,080 50 2 (100) 600,100 2 (100) 800,100 I (M, N, z) 3.0E+06 2.5E+06 2.0E+06 1.5E+06 1.0E+06 5.0E+05 0.0E+00 geometry 2 3 4 6 8 10 20 30 40 50 pop. size 4 x 4 7 x 7 10 x 10 Figure 7: Variation of I(M,N,z) with population size, and different geometries (breeding rate=100%)

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