$q ([dplqdwlrq Ri 7xqdeoh 5dqgrp 6hdufk /dqgvfdshv

This paper carefully considers random landscapes related to Kauffman’s NK model. In particular, it considers a superset of this model (the NKP model) recently suggested in the GA-analytic literature. Landscapes are exhaustively examined for both the distribution of local optima relative to the global optima, and for characteristics that would effect juxtapositional (building block based) search. The later is accomplished through a Walsh-based analysis. The results indicate that K and P have distinct effects on peak distribution, K controlling peak placement, and P affecting relative peak height. The results indicate that P has little effect on a landscape’s expected juxtapositional complexity. Moreover, the results suggest that landscapes developed by the NKP procedure are unlikely to have substantial juxtapositional complexity. A Walsh-based procedure that embodies the flavor of the NKP procedure, but can allow for greater expected juxtapositional complexity, is suggested. ,QWURGXFWLRQ ,Q H[DPLQLQJ HYROXWLRQDU\ FRPSXWDWLRQ (& V\VWHPV LW LV XVHIXO WR FRQVLGHU WKH QDWXUH RI WKH VHDUFK SUREOHPV WR ZKLFK WKH\ DUH DSSOLHG $ JUHDW GHDO RI WKH JHQHWLF DOJRULWKP *$ OLWHUDWXUH LV GLUHFWHG DW W KLV H[DPLQDWLRQ &DUHIXO FRQVLGHUDWLRQ R I VHDUFK SUREOHPV RU VHDUFK ODQGVFDSHV DV WKH\ DUH RIWHQ FDOOHG DOVR K DV XVHV LQ GHVFULSWLYH VFLHQFH )RU LQVWDQFH FRQVLGHU .DXIIPDQ©V H[DPLQDWLRQ R I UDQGRP ODQGVFDSHV ZLWK WXQDEOH OHYHOV RI HSLVWDVLV DV DQ H[SODQDWLRQ R I FRPSOH[LW\ LQ ELRORJLFDO V\VWHPV .DXIIPDQ 6LQFH HSLVWDVLV LV D SKHQRPHQRQ WKDW LV HDVLO\ UHODWHG WR WKH *$ VHDUFK SURFHVV .DXIIPDQ©V 1. ODQGVFDSHV DUH UHFHLYLQJ LQFUHDVLQJ XVH DV WHVW SUREOHPV IRU *$V 7KLV SDSHU H[DPLQHV 1. ODQGVFDSHV DQG WKHLU H[WHQVLRQV DV (& WHVW SUREOHPV 0RUHRYHU LW IRUPV UHODWLRQVKLSV EHWZHHQ .DXIIPDQ EDVHG DQG :DOVK EDVHG GHFHSWLRQ EDVHG ODQGVFDSH DQDO\VLV DQG GHVLJQ )LQDOO\ LW VXJJHVWV QHZ PHWKRGV IRU WHVW ODQGVFDSH GHVLJQ DQG IRU XQGHUVWDQGLQJ RI UHDO ZRUOG VHDUFK ODQGVFDSHV $GYDQWDJHV RI 1. /DQGVFDSHV 7KHUH DUH WZR PDLQ DGYDQWDJHV WR XVLQJ 1. ODQGVFDSHV DV (& WHVW SUREOHPV x The landscapes have two tunable parameters, (N and K) that can be used to control characteristics (e.g., landscape size and epistasis, respectively) of landscapes. x For given values of N and K, one can generate a large number of landscapes, at random. 7KHVH DGYDQWDJHV DOORZ RQH WR HDVLO\ JHQHUDWH D ODUJH QXPEHU RI (& WHVW SUREOHPV DQG H[DPLQH (& SHUIRUPDQFH YHUVXV WKH SDUDPHWHUV 1 DQG . +RZHYHU WKH PHDQLQJIXOQHVV RI VXFK H[DPLQDWLRQV GHSHQGV RQ WKH UHODWLRQVKLS E HWZHHQ WKHVH SDUDPHWHUV DQG WKH GLIILFXOW\ RI WKH ODQGVFDSHV IRU VHDUFK LQ JHQHUDO DQG (& VHDUFK LQ SDUWLFXODU .DXIIPDQ©V ZRUN SURYLGHV V RPH LQGLFDWLRQ R I WKH UHODWLRQVKLS EHWZHHQ WKH SDUDPHWHUV DQG FHUWDLQ DVSHFWV RI ODQGVFDSH F RPSOH[LW\ 7KLV FRPSOH[LW\ PD\ RU PD\ QRW LQGLFDWH VHDUFK GLIILFXOW\ 7KH F RPSOHWH UHODWLRQVKLS E HWZHHQ WKH 1. SDUDPHWHUV DQG WKH GLIILFXOW\ RI WKH VHDUFK ODQGVFDSHV WKH\ LQGXFH LV QRW HQWLUHO\ FOHDU IRU VHYHUDO UHDVRQV )LUVW .DXIIPDQ©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©V ZRUN DORQJ ZLWK HDUOLHU ZRUN E\ $OWHQEHUJ VXJJHVWV D EURDGHU FODVV RI UDQGRP ODQGVFDSHV WKDW FRYHU WKH F RPSOHWH VSDFH RI VL]H 1 SUREOHPV ZLWK HSLVWDVLV . WKURXJK WKH LQWURGXFWLRQ R I DQRWKHU SDUDPHWHU 3 :H ZLOO FDOO WKHVH 1.3 ODQGVFDSHV 7KLV SDSHU SURYLGHV DQ H[WHQVLYH DQDO\VLV RI 1.3 ODQGVFDSHV 7KH 1. ODQGVFDSHV DUH D VXEVHW RI WKH 1.3 ODQGVFDSHV DQG WKHUHIRUH WKLV LPSOLHV D PRUH H[WHQVLYH DQDO\VLV RI 1. ODQGVFDSHV DV ZHOO 7KH SDSHU DWWHPSWV WR DQVZHU VHYHUDO TXHVWLRQV x Does the P parameter change the complexity characteristics of the NK landscapes examined by Kauffman? If so, how? x Does an exhaustive examination of several NKP landscapes yield complexity characteristics that agree with those reported by Kauffman? x How does the difficulty of NKP landscapes for juxtapositional operators vary with N, K, and P (e.g., how are N, K, and P related to deception and other GAanalytic measures of difficulty)? 7KH SUDFWLFDO LPSOLFDWLRQV RI WKHVH H[DPLQDWLRQV DUH WZR IROG x They provide greater understanding of the expected difficulty of EC test problems generated via the NKP (and therefore, the NK) landscape procedure. x They suggest possible alternative methods for generating random landscapes with tunable difficulty for use as EC test problems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©V 1. /DQGVFDSHV ,Q WKLV GLVFXVVLRQ ZH ZLOO DVVXPH DOO JHQHV DUH ELQDU\ IRU FRQYHQLHQFH +RZHYHU WKH UHVXOWV DUH H[WHQVLEOH WR SUREOHPV ZLWK ODUJHU JHQH DOSKDEHWV 6SHFLI\LQJ DQ 1. ODQGVFDSH UHTXLUHV WKH IROORZLQJ SDUDPHWHUV 1 WKH WRWDO QXPEHU RI ELWV JHQHV . WKH DPRXQW RI HSLVWDVLV (DFK ELW GHSHQGV RQ . RWKHU ELWV WR GHWHUPLQH WKDW LWV ILWQHVV FRQWULEXWLRQ :H ZLOO FDOO WKH . E LWV LQYROYHG LQ HDFK FRQWULEXWLRQ D VXEIXQFWLRQ EL 1 SRVVLEO\ UDQGRP ELWPDVNV L ž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̧̧1· ̈̈©§ d d .1 3 0RUHRYHU WKLV PHDQV WKDW RQH PXVW VSHFLI\ 3 ELWPDVNV UDWKHU WKDQ RQ O\ 1 7KLV PHDQV WKDW WDEOH ; PXVW EH H[WHQGHG WR EH 3 E\ . 2WKHUZLVH WKH FRQVWUXFWLRQ DQG GHFRGLQJ SURFHGXUH LV WKH VDPH DV WKDW IRU 1. ODQGVFDSHV :DOVK EDVHG (SLVWDWLF /DQGVFDSHV $ VHW RI IXQFWLRQV VLPLODU WR WKH 1.3 ODQGVFDSHV FDQ EH FRQVWUXFWHG XVLQJ :DOVK FRHIILFLHQWV 7KLV LV FRQYHQLHQW VLQFH :DOVK FRHIILFLHQWV KDYH RIWHQ EHHQ XVHG LQ *$ DQDO\VLV 0RUHRYHU LW ZLOO KHOS WR LOOXVWUDWH +HFNHQGRUQ DQG :KLWOH\©V FRQFOXVLRQV +RZHYHU RQH PXVW PDNH QRWH RI ZKDW LV PHDQW E\ HSLVWDVLV LQ :DOVK EDVHG ODQGVFDSHV 6RPH WHUPLQRORJ\ ZLOO EH XVHIXO KHUH &RQVLGHU D VHDUFK VSDFH GHILQHG RYHU OHQJWK / ELQDU\ VWULQJV ,Q WHUPV RI WKH :DOVK FRHIILFLHQWV GHILQH WKH ILWQHVV RI DQ LQGLYLGXDO DV 0 10 20 30 0 50 100 150 HSLA (Red) 0 10 20 30 0 50 100 150 200 HSLA (Blue) 0 10 20 30 0 50 100 150 SSLA (Red) 0 10 20 30 0 50 100 150 200 SSLA (Blue) 0 10 20 30 0 50 100 150 200 B2BH (Red) 0 10 20 30 0 50 100 150 B2BH (Blue) 0 10 20 30 0 50 100 150 200 DEF (Red) 0 10 20 30 0 50 100 150 200 DEF (Blue) 0 10 20 30 0 50 100 150 200 OFF (Red) 0 10 20 30 0 50 100 150 200 OFF (Blue) ZKHUH [ LV WKH ELW VWULQJ UHSUHVHQWLQJ WKH LQGLYLGXDO [L LV WKH LWK ELW LQ WKDW VWULQJ ZM LV WKH :DOVK FRHIILFLHQW FRUUHVSRQGLQJ WR WKH SDUWLWLRQ ELWPDVN QXPEHUHG M LV WKH ELWPDVN ZKLFK LV WKH 1 ELW ELQDU\ LQWHJHU UHSUHVHQWDWLRQ RI M -L LV WKH LWK ELW LQ WKH ELWPDVN DQG < LV WKH IXQFWLRQ GHVFULEH LQ 7DEOH [L -L < [ L L