859:
881:
The reason that the Schema
Theorem cannot explain the power of genetic algorithms is that it holds for all problem instances, and cannot distinguish between problems in which genetic algorithms perform poorly, and problems for which genetic algorithms perform well.
148:
is the average fitness of all strings matching the schema. The fitness of a string is a measure of the value of the encoded problem solution, as computed by a problem-specific evaluation function. Using the established methods and
278:
623:
830:
is clearly pessimistic: depending on the mating partner, recombination may not disrupt the scheme even when a cross point is selected between the first and the last fixed position of
866:
The schema theorem holds under the assumption of a genetic algorithm that maintains an infinitely large population, but does not always carry over to (finite) practice: due to
27:. The Schema Theorem says that short, low-order schemata with above-average fitness increase exponentially in frequency in successive generations. The theorem was proposed by
900:
35:. However, this interpretation of its implications has been criticized in several publications reviewed in, where the Schema Theorem is shown to be a special case of the
758:
533:
138:
316:
729:
702:
436:
655:
385:
104:
157:, the schema theorem states that short, low-order schemata with above-average fitness increase exponentially in successive generations. Expressed as an equation:
848:
828:
808:
784:
675:
500:
480:
460:
409:
356:
336:
766:
rather than an equality. The answer is in fact simple: the
Theorem neglects the small, yet non-zero, probability that a string belonging to the schema
995:
163:
870:
in the initial population, genetic algorithms may converge on schemata that have no selective advantage. This happens in particular in
950:
910:
541:
990:
43:
70:
describes the set of all strings of length 6 with 1's at positions 1, 3 and 6 and a 0 at position 4. The * is a
871:
786:
will be created "from scratch" by mutation of a single string (or recombination of two strings) that did
24:
31:
in the 1970s. It was initially widely taken to be the foundation for explanations of the power of
154:
71:
28:
734:
505:
114:
946:
906:
55:
32:
286:
150:
707:
680:
414:
631:
361:
108:
80:
50:
of strings with similarities at certain string positions. Schemata are a special case of
858:
867:
833:
813:
793:
769:
660:
485:
465:
445:
394:
341:
321:
36:
984:
875:
51:
926:
74:
symbol, which means that positions 2 and 5 can have a value of either 1 or 0. The
966:
940:
902:
An analysis of reproduction and crossover in a binary-coded genetic algorithm
731:
is the probability of crossover. So a schema with a shorter defining length
140:
is the distance between the first and last specific positions. The order of
273:{\displaystyle \operatorname {E} (m(H,t+1))\geq {m(H,t)f(H) \over a_{t}}.}
106:
is defined as the number of fixed positions in the template, while the
482:
is the probability that crossover or mutation will destroy the schema
970:, The MIT Press; Reprint edition 1992 (originally published in 1975).
47:
39:
with the schema indicator function as the macroscopic measurement.
23:, is an inequality that results from coarse-graining an equation for
942:
Genetic algorithms with sharing for multimodal function optimization
857:
945:. 2nd Int'l Conf. on Genetic Algorithms and their applications.
905:. 2nd Int'l Conf. on Genetic Algorithms and their applications.
874:, where a function can have multiple peaks: the population may
762:
An often misunderstood point is why the Schema
Theorem is an
810:
in the previous generation. Moreover, the expression for
618:{\displaystyle p={\delta (H) \over l-1}p_{c}+o(H)p_{m}}
836:
816:
796:
772:
737:
710:
683:
663:
634:
544:
508:
488:
468:
448:
417:
397:
364:
344:
324:
289:
166:
117:
83:
842:
822:
802:
778:
752:
723:
696:
669:
649:
617:
527:
494:
474:
454:
430:
403:
379:
350:
330:
310:
272:
132:
98:
878:to prefer one of the peaks, ignoring the others.
899:Bridges, Clayton L.; Goldberg, David E. (1987).
66:Consider binary strings of length 6. The schema
862:Plot of a multimodal function in two variables.
975:Hidden Order: How Adaptation Builds Complexity
929:. Foundations of genetic algorithms, 3, 23-49.
318:is the number of strings belonging to schema
8:
967:Adaptation in Natural and Artificial Systems
939:David E., Goldberg; Richardson, Jon (1987).
835:
815:
795:
771:
736:
715:
709:
688:
682:
662:
633:
609:
584:
551:
543:
513:
507:
487:
467:
447:
422:
416:
396:
363:
343:
323:
288:
244:
206:
165:
116:
82:
21:fundamental theorem of genetic algorithms
891:
144:is 4 and its defining length is 5. The
927:The Schema Theorem and Price’s Theorem
7:
704:is the probability of mutation and
167:
14:
996:Theorems in discrete mathematics
462:. The probability of disruption
46:is a template that identifies a
760:is less likely to be disrupted.
747:
741:
644:
638:
602:
596:
563:
557:
442:average fitness at generation
374:
368:
305:
293:
264:
252:
236:
230:
224:
212:
200:
197:
179:
173:
127:
121:
93:
87:
1:
657:is the order of the schema,
502:. Under the assumption that
677:is the length of the code,
1012:
753:{\displaystyle \delta (H)}
535:, it can be expressed as:
528:{\displaystyle p_{m}\ll 1}
391:average fitness of schema
133:{\displaystyle \delta (H)}
17:Holland's schema theorem
872:multimodal optimization
925:Altenberg, L. (1995).
863:
844:
824:
804:
780:
754:
725:
698:
671:
651:
619:
529:
496:
476:
456:
432:
405:
381:
352:
332:
312:
311:{\displaystyle m(H,t)}
274:
134:
100:
861:
845:
825:
805:
781:
755:
726:
724:{\displaystyle p_{c}}
699:
697:{\displaystyle p_{m}}
672:
652:
620:
530:
497:
477:
457:
433:
431:{\displaystyle a_{t}}
406:
382:
353:
333:
313:
275:
135:
101:
25:evolutionary dynamics
977:, Helix Books; 1996.
834:
814:
794:
770:
735:
708:
681:
661:
650:{\displaystyle o(H)}
632:
542:
506:
486:
466:
446:
415:
395:
380:{\displaystyle f(H)}
362:
342:
322:
287:
164:
115:
99:{\displaystyle o(H)}
81:
146:fitness of a schema
54:, and hence form a
991:Genetic algorithms
864:
840:
820:
800:
776:
750:
721:
694:
667:
647:
615:
525:
492:
472:
452:
428:
401:
377:
348:
328:
308:
270:
155:genetic algorithms
130:
96:
33:genetic algorithms
19:, also called the
843:{\displaystyle H}
823:{\displaystyle p}
803:{\displaystyle H}
779:{\displaystyle H}
670:{\displaystyle l}
578:
495:{\displaystyle H}
475:{\displaystyle p}
455:{\displaystyle t}
404:{\displaystyle H}
351:{\displaystyle t}
331:{\displaystyle H}
250:
151:genetic operators
76:order of a schema
56:topological space
1003:
957:
956:
936:
930:
923:
917:
916:
896:
849:
847:
846:
841:
829:
827:
826:
821:
809:
807:
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801:
785:
783:
782:
777:
759:
757:
756:
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730:
728:
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719:
703:
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695:
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692:
676:
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673:
668:
656:
654:
653:
648:
624:
622:
621:
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614:
613:
589:
588:
579:
577:
566:
552:
534:
532:
531:
526:
518:
517:
501:
499:
498:
493:
481:
479:
478:
473:
461:
459:
458:
453:
437:
435:
434:
429:
427:
426:
410:
408:
407:
402:
386:
384:
383:
378:
357:
355:
354:
349:
337:
335:
334:
329:
317:
315:
314:
309:
279:
277:
276:
271:
251:
249:
248:
239:
207:
143:
139:
137:
136:
131:
105:
103:
102:
97:
69:
1011:
1010:
1006:
1005:
1004:
1002:
1001:
1000:
981:
980:
961:
960:
953:
938:
937:
933:
924:
920:
913:
898:
897:
893:
888:
856:
832:
831:
812:
811:
792:
791:
768:
767:
761:
733:
732:
711:
706:
705:
684:
679:
678:
659:
658:
630:
629:
605:
580:
567:
553:
540:
539:
509:
504:
503:
484:
483:
464:
463:
444:
443:
418:
413:
412:
393:
392:
360:
359:
340:
339:
320:
319:
285:
284:
240:
208:
162:
161:
141:
113:
112:
109:defining length
79:
78:
67:
64:
12:
11:
5:
1009:
1007:
999:
998:
993:
983:
982:
979:
978:
971:
959:
958:
951:
931:
918:
911:
890:
889:
887:
884:
868:sampling error
855:
852:
839:
819:
799:
775:
749:
746:
743:
740:
718:
714:
691:
687:
666:
646:
643:
640:
637:
626:
625:
612:
608:
604:
601:
598:
595:
592:
587:
583:
576:
573:
570:
565:
562:
559:
556:
550:
547:
524:
521:
516:
512:
491:
471:
451:
425:
421:
400:
376:
373:
370:
367:
347:
338:at generation
327:
307:
304:
301:
298:
295:
292:
281:
280:
269:
266:
263:
260:
257:
254:
247:
243:
238:
235:
232:
229:
226:
223:
220:
217:
214:
211:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
175:
172:
169:
129:
126:
123:
120:
95:
92:
89:
86:
63:
60:
37:Price equation
13:
10:
9:
6:
4:
3:
2:
1008:
997:
994:
992:
989:
988:
986:
976:
972:
969:
968:
963:
962:
954:
952:9781134989737
948:
944:
943:
935:
932:
928:
922:
919:
914:
912:9781134989737
908:
904:
903:
895:
892:
885:
883:
879:
877:
873:
869:
860:
853:
851:
837:
817:
797:
789:
773:
765:
744:
738:
716:
712:
689:
685:
664:
641:
635:
610:
606:
599:
593:
590:
585:
581:
574:
571:
568:
560:
554:
548:
545:
538:
537:
536:
522:
519:
514:
510:
489:
469:
449:
441:
423:
419:
398:
390:
371:
365:
345:
325:
302:
299:
296:
290:
267:
261:
258:
255:
245:
241:
233:
227:
221:
218:
215:
209:
203:
194:
191:
188:
185:
182:
176:
170:
160:
159:
158:
156:
152:
147:
124:
118:
111:
110:
90:
84:
77:
73:
61:
59:
57:
53:
52:cylinder sets
49:
45:
40:
38:
34:
30:
26:
22:
18:
974:
973:J. Holland,
965:
964:J. Holland,
941:
934:
921:
901:
894:
880:
865:
787:
763:
627:
439:
388:
282:
145:
107:
75:
65:
41:
29:John Holland
20:
16:
15:
62:Description
985:Categories
886:References
854:Limitation
790:belong to
764:inequality
739:δ
572:−
555:δ
520:≪
259:−
204:≥
171:
119:δ
440:observed
389:observed
72:wildcard
438:is the
387:is the
949:
909:
628:where
142:1*10*1
68:1*10*1
48:subset
44:schema
876:drift
283:Here
947:ISBN
907:ISBN
411:and
788:not
153:of
987::
850:.
358:,
58:.
42:A
955:.
915:.
838:H
818:p
798:H
774:H
748:)
745:H
742:(
717:c
713:p
690:m
686:p
665:l
645:)
642:H
639:(
636:o
611:m
607:p
603:)
600:H
597:(
594:o
591:+
586:c
582:p
575:1
569:l
564:)
561:H
558:(
549:=
546:p
523:1
515:m
511:p
490:H
470:p
450:t
424:t
420:a
399:H
375:)
372:H
369:(
366:f
346:t
326:H
306:)
303:t
300:,
297:H
294:(
291:m
268:.
265:]
262:p
256:1
253:[
246:t
242:a
237:)
234:H
231:(
228:f
225:)
222:t
219:,
216:H
213:(
210:m
201:)
198:)
195:1
192:+
189:t
186:,
183:H
180:(
177:m
174:(
168:E
128:)
125:H
122:(
94:)
91:H
88:(
85:o
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