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SE 016 990 Adaptive Schemes Called Sequential Reproductive Computers

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SE 016 990 Adaptive Schemes Called Sequential Reproductive Computers DOCUMENT RESUME ED 099 184 SE 016 990 AUTHOR Martin, Nancy TITLE Convergence Properties of a Class of Probabilistic Adaptive Schemes Called Sequential Reproductive Plans. Psychology and Education Series, Technical Report No. 210. INSTITUTION Stanford Univ., Calif. Inst. for Mathematical Studies in Social Science. SPONS AGENCY National Science Foundation, Washington, D.C. PUB DATE 31 Jul 73 NOTE 63p. EDRS PRICE EF-$0.75 HC-$3.15 PLUS POSTAGE DESCRIPTORS Computers; *Computer Science Education; Educational Research; *Genetics; Linear Programing; *Mathematical Models; Reproduction (Biology); Science Education; Social Sciences ABSTRACT Presented is a technical report concerning the use of a mathematical model describing certain aspects of the duplication and selection processes in natural genetic adaptation. This reproductive plan/model occurs in artificial genetics (the use of ideas from genetics to develop general problem solving techniques for computers). The reproductive plan is a sequential stochastic process involving n-tuples (corresponding to chromosomes in genetics) which may be simple numeric constants or complex structures such as computer algorithms. The plan also involves a sequence of probability distributions defined over n-tuples,, The report consists of five chapters: introduction; reproductive plans; deterministic problem bases; a chapter divided into sections on the search for an arena, the linear additive model, the linear models and pure problem bases; and conclusions. An appendix illustrating the theorem involved and a list of references conclude the report. (PEB) ! ..,ot Jr HI AL TN t A ALLF Alit .e.S1,Ttatt OF BEST COPYAVAILABLE t % -ttr . % % ; - % ftt ts` !CONVERGENCE PROPERTIES OF A CLASS OF PROBABILISTIC ADAPTIVESCHEMES CALLED SEQUENTIAL REPRODUCTIVE PLANS BY NANCY MARTIN TECHNICAL REPORT NO. 210 JULY 31, 1973 PSYCHOLOGY AND EDUCATION SERIES INSTITUTE FOR MATHEMATICAL STUDIES IN THE SOCIAL SCIENCES STANFORD UNIVERSITY STANFORD, CALIFORNIA TLCHN(CAL REPORTS PSYCIIOtOCY SERIES BS1 COPYAVNUWLE INSTITUTE FOR MAThEMATiCAl. STUDIES 1W THf50C14. SCIENCES iPlac, of pcatoa ,hou.n In enthis U pIIIshed thIs tsMsIpti*hssis.twiIs dlifqvvit ft tftle of ?*daIaf Rpit, ...omno.1.44, uiT hnalRpIno. (25.) so R. C Atkrnson and R. C. Calf,.. Mait*na1icaI Iwntiq they.J4IUW)P2.1963. (In 8. 0. Wofis (U.), Sc**1floPRg0aII. Bail. Disks. tai., 1995. Pp. 254.275) SI P. Stis,. E. CItiurs, and P. Welt. *spftcet*jn of tIseaitIeat tsnto thry mi IIniIstc iysI 1. vu4 L.... sM 1* Ruulan*vds. Diaitsksr 28,1902. 52 B, C. AUtism, ft. CalIsi, 0, S, W. Js and B. SPIsisikse. A MIt it ill,. ms*Ii f$ftsifie.iiumdM cMt*ia. Jany 29,1903. U. Piyc(,d., (964, 67, 52.55 53 B.dkwi,OsissilMukir..114515(11 $ssittii . -61al (cr5stUsu5. Aisfi 8,190$. 54 J. L. v,.114 P. C. AUtaii. ChoIce bs$svfer andswwd sfrt*n.1W 24,1963, Cjawnat Pmyd., $964, j 55 B. B. RskMseii. AIM 0.1.1511051aoich to isçstcof ,1iIu1iisi Of isaiirstsint51111.11111. .Mi10,1963. S0 B. Mwa, B. We and P. Patisi. The role of Urwl$han Inthe W1n of thsthepspl'$e revanw(ans Of Buasfa. 57 P. Susiss. t45, Jiss $7, 1963. ci o$lndUtIon In Is1Pn109 a list ci sIi1* ft*.1s.July 22,1963. Un MaynwdW, Sholly, It and Clisil..ysi (Us), fn.*s and OpUsiJt. Niw Yet: Wiley. (964. Pp.1(0126) 58 B. C. *ibin.siandE. J. Crothura. Thserotical nMeta$l.-nan.1swn1nsnd I11wI51 fir5iUln5. Jtdy 24,190$. 59 B. C. Callus. Long-tstmbsftavI of ativa rblIIst1c roInfv.ua.* sa1wdes. Ditcb.r 1.1903. 60 P. C. Athinsiss and E. .1. Crcthu.s. Tests ofa Islttand 151si11an, silsia hr siWd.assuslat. lwnMs. of Ostikur 35,1965.1* ..s soc1at lianIis unduls Itsvfr.g diffurmnt acul1(tIOn andmtantloi,aidcua,.1, math. Psyctiul., (904,1, 2*5415) 61 W. J. (kOhl v,4J. GIbbon. The i.wat-çaana dlstilbutls.i and esieftanItsis. Nar 20, 94 Li, 62 V. P. Nuruan. InomimMal Iewffln9 øn 'ando.. trlali. Piyhol., (963.3, 1.45) Decemau, 9,1963. Li. Psycliof., 1904, , 336459) 63 P.Stein.Tbs &msnt of aulM.11Ica$ concepts In cN1*ss. Fskrowy 25764.i the behevbbsl fcsi45U Of sithimaticci cpts. Mcn*s's essurci iDsv&nt, 1965.30,60.96) 64 P. Supss. UathsumticaJ canospI torsitlon InchtSsi.*rll 10,1964. CAsir. P5yctndII1J, 1966,3! (39.190) 05 P.C. Calfis, B.C. Atkinson, and 7. Shoftcn,J. Jc51si45l,hrs,9a1WNq. 1*mtZS,1964, Whe,andJ. P. $chi45 (Ek.), C)tuns1 51 the Nurvs.a Pvcursss Raswch. AW45m, Tils NstM.Ian45t EIuvler PJ1IAMp Pp.333.349) C., 1969. 66 1.. KiUr, V. Cels, C. .1. 6.m, and W. K, Eatas. Paksd ussoclats lmasts wit,. dlftsrUa$ a.i..August 20,1964. (Bd . InfisistISn vofuis ii Dial a51canus In paftWss atsfmawfng. CPsi. .. (965,79,1-31) 07 V. P. Nanann. AubabI1tst1c usd51 fur fIvs.csspondM.Diau (4,1964. 68 W. K. Kiwi and N, A. Isytur. Visual dstsctlon fr. ralattsi to display its* and r,*mnoy of alilcofaI.uss.Jansany 23,1969, Psutsid 7+65. (PuiuiLJon and Ptychopluyslcs, 1966,!-' 69 P. Suppsi si14J. Donlo. FajatIons ci su -saq,lIn thoøy hrcmImsosi4lsiurtossius. F*y 9,1995. Ii, ft. 4,202.225) 19f7, 70 P. C. Atklnun mci P. A. KInchia. A lswnhn mo45I Ii. forced-dislcsdilutionaspurimsits. F*vay ID, (96$. 1. .1. Psj51., (965, Ia, (84.306) E. J. 'I oces. ProurAatlan irs fur Mal ('roe, diffursit catsur1us. V.1sti10,1905. 72 P Susi,s, 0. Gun, and U. ScIdaq4y. Sos. undiha Furrusponas latency Inbid4uociuse, 'miii,. IVy 5,1969. U. $966, , 99-1281 13 V. V. L.vins. the snurallulion functian In lii skIilty lmwthq sxpsiasM. Jus. 3,1905. 74 0. Nans.n and 1'. S. Ro45ur,. An exploration of payChOIIn$ISt1C unitsIn Initial .uadIn,. July 8,1965. 75 B. C. Arnold. A ce,,ciat.d twn-schsue for a coiljnw. ofrsipcnus. Jtdy 30,1963. 70 C. IzaandW. K. Estis. RsInf.veamt.4sgtsa5usncss akid.auictets (mviii,, Aiiust(, (965. 1Pycf. 77 S. L Blefs't. Pattsrnd 1966,18, 819.9997 winaticlwnlngwithRhosuaisosisys. Ssptas. 1, 1965. (P,yplte$. , 1960, 19, 311.324) 78 .1. 1.. PhillIps and P. C. AtkInson. Thee cidis$ay.tceansfs.t-Can.suu..1y. Aumt59, 1965. 79 P. C. Atkinson and R. U. SNifrIn. MaU.muudcai sodils Forumury and lm1A9. Sust.ndsr 20,1965. 80 P. Supsis. The issycboloilcal fcun(a ii thweeilcs, October 25(963. (C0los I &t*.1uadi F(stosIil EditIonsdiCunDe (kilonsi di Is Raclurche Sste.*196*. Pwls (96?. Pp.2(3.242) (If P.1'Toipttsi-assstid Instrtisn In use scheofa. pantialitiss,ribims..o.pects. Otosker 29, (965. 82 B. A. KIncliha, J. Townsind, J. Yallolt, k, an4 P. C. Atkinson.Influence of ccvrsiatedvlsvaf sims on audUy sipil diuictian. Novsndsr 2, 1905. (PerceptIon mci rchophydcs. 1990,1, 87-73) a, P. Suposs, U. Jernors, undO. (keen. ArlthnsUe dulls it'd essisuona cosps*er-based talilype. Nmse.uOs, 3,1965. (AftthniUc APrU (966. 3O33O9. 84 P. Suppes and L.. Hync.. Concept 'mwnlnq with nsa-verbalisositricli itimsit. Nasii6er (5, I%P. 85 P. Holland. A vwlatlan on ttw ..$nkeus cM-squu luSt. LI. !!yChOl.. (967. 17.4l3). 86 P. Supess. Accoleiltid proça.. In etsesntury-sthoot sittismaucs (Is secind sw Novea6sr 22, 1963. tatSehecli, (960, !.. 29401 87 P. Loenast' mid F. Stafeid. Lupte is idlslcqicalpsi. Noeer 29, (965. 88 L lImb,, W. .1. Thus,ia,, J. B. Tweedy, and B. C. Atkinson. The sffecta ofreInforcement (WIll on the acu1sSultci psses51 puiisss. Dus (0, (965. Ci, (967, 73, 268-2Th . 89 J, F. 'iifNu, N Sims sflstiannencoøMnt scap In lumsn r*ndh1Ity Isviun. D.osu8ei (5,1965. 90 P. 3upes and 0. Grsin. Sans Co.nttis usdits fur F bstçadi psrfo'sinc. diM en misple alt1on facts. Janusy 14,1900. (Ii J. U. Scindiis (Pd.), VLhssI1ICS WasMnptan, P. C. NCTU, 1967. Pp. 35-4*. 9$ P. Suppec. tnf&nsmtiin pscsssln mid chilci beheclur. Jam.anj 31,1966. 92 0. Gi'ans and ft. C. Atklisen. Vodiud F. iptiahilep the Imw,fi, ss. Filuwey U, (966. (Pgychof. DiHetti, (966, , 309-320) 9, P. C. AtkInson ad 0. Musmn. CemsuW-4u1111d InsUit1on in initIal ,sadh,t Stanfordurjsct. IWsO (7, (966. (R.i4M R*s#srcb Qst, (966, 2, 3-25) 94 P. Bapi,. Ptskchitlstic P*Iuncs aid the smicapt ci 11151 suldinoc. Musk 23,1906. (IiJ. HWM* aid P. Sispps, (Eds.), at M . A.sbedius Nmth.ilolfand Pl6llsMni Ce,, 1966. Ps. 49-65. 95 p.ii,pss. The aihimsils usthod I.. M-acheu1wetIwsustics. *jstl (2(966. (71. R* AsIoçp P*la. SWvkseusC,. The Cs% .ac of tI1u& Silences, WasfilAiton, 0. C. OPus mid C.., 1966. Pp.49.76. (Cositineedontiwids bepk csv) CONVERGENCE PROPERTIES OF A CLASS OF PROBABILISTIC ADAPTIVE SCHEMES CALLED SIQUENTIAL REPRODUCTIVE PLANS by Nancy Martin TECHNICAL REPORT NO. 210 July 31, 1973 PSYCHOLOGY AND EDUCATION SERIES Reproduction in Whole or in Part Is Permitted for Any Purpose of the United States Government This research was supported by National Science Foundation Grant No. GJ-443X to the Institute for Mathematical Studies in the Social Sciences, Stanford University, and by National Science Foundation Grant No. GJ-29989X to the Logic of Computers Group, University of Michigan. INSTITUTE FOR MATHEMATICAL STUDIES IN THE SOCIAL SCIENCES STANFORD UNIVERSITY STANFORD, CALIFORNIA ABSTRACT CONVERGENCE PROPERTIES OF A CLASS OFPROBABILISTIC ADAPTIVE SCHEMES CALLED SliQtfl REPRODUCTIVE PLANS by Nancy Martin Chairman: John H. Holland A reproductive plan isa mathematical model describing certain aspects of the duplication and selectionprocesses in natural genetic adaptation. These models occur in artificialgenetics, which is the use of ideas from genetics to developgeneral problem solving techniques for computers. A reproductive plan isa sequential stochastic process involving n-tuples which correspond to chromosomesi enetics. The individual elements of the n-tuples, which correspondtv enes, may be simple numeric constants or may be suchcomplex strixtures as computer algo- rithms. The plan also involves a sequence of probabilitydistributions defined over the n-tuples.
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