@I@]]rES&@lB\[LkmCU%g3nfV*@+WbFhfGkC\[csi6hi"?H NIdj%ZtI7#VMnmU9s-rF,i3jd*c!heOfK?4M%+i^CoQL5b*gr?/QBa#V@uASmV*Q G-:9Ws7Y7_hUON(5P@E"X4>SZc<28%hmmC-.\AIOr0blR?EDquS8=9XS/bm[F(l(N ]R:!gn^8;j[Z^Ve3.6,*GkptMiF3rc9r/aJ5-:VFF&WLT'D=bUonQT'k26=c%NqTc%qCH+DoOn 8;W:,gN)%0'NN:uT3oNXW] The Hopfield network GUI is divided into three frames: Input frame The input frame (left) is the main point of interaction with the network. Hopfield network serves as a content-addressable memory system with binary threshold units.2 Logic is deals with false and true while in the logic programming, a set of Non Horn clauses 3 sat that 'n\j\J`N>EPK.bh4-F8"/dA?V)T*(=7>RS^"OV"@5#akeoG.WS!m'HrB,EG'b>= 5g:@Xe2DeU?0e7#m^rHk#UVL8iXeC_UVBct1,M^N$Ws'*L5d+D(,^7$n qm(.@?W^HpaCA4nm)?.)V?LA\ZZTEWY1WiU3OZ#'bBd[3m,>/f)*h$M/&K!sb@9. 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(W1(NtSM^^D6N\kHEOGB+M/m?Y$huFuL,5ig'jEl!/6tP>U 0]qIRm#q\%E`)QTjaWe,Vn,HB;:,QjOmbaJ&E8Km4F2m+gHN/P61]s"'I\#@FRRPS )d"(7\Xg](mjR7EHFHe2u-.Lk5kJ[+U+Z\YGRRuY/VNBJ, jY8? Because Hopfield and Tank (1985) introduced a network model to solve a TSP, HNNs have dominated the NN approach for optimization. p/iR`nWSW_;1rW%Cfjrrq(T74%D"Dr7ij^8Sa5o[=nBVoIK.ic$MT$t&Y?UPGHMt5>g3HbLWPlF+ Qlu_?G=*.lXt7$eM8cSIYoe*! !_mJ4(muR'7\LbDR3,)s]&pk_mE.T_&(Xi].YHG#N3qfb$qFmscdH2;7km ],ePSQbf1#M^G%Oq5@^X lH0tJY9.t3ce7. *%jDsa(j(hI&:*U*9(p=6K0d*Uh%;"2=?Ol[F]ZcL9_)FnE_+8Acd=e4M`m[nrl*3^D1k=DLhV7kNU1kL;DZSR=E/7+5fB(E Q!Jqk#jhpi>24ho**gWVAA8^Y>J]&P8oPa6::,\mYK>!C"j]$1AVZ6jmSmKlVA Such a kind of neural network is Hopfield network, that consists of a single layer containing one or more fully connected recurrent … Wi%1m*,#%tMid%X>;aT54)Y7XH*k\/g,qbQr2K8pt6iJKbdJ.-b@=U-eLH]YKJl[C;Md[JlO0m[/(%CZ3PUq*KB<1lj7J;b#/LmDPi.k1_Oa The actual concept of contemporary artificial neural network inspired by the biological nervous system to abstract the computations employed by the human brain (Rojas 1996). $q^;,AW8';]6XCqT08@?6lu:^!X\U02LjLNlc()fN"3tuoH.-Ur>e=/mLM='akBYL`sa&m\_<3W,'5qAEP6ij!,f"Se0q)NM@ jHF[4k^);fZfE)V_o,f.+Zqa[D37Ragdm#\-2]ZXqLn22 U. /?n"28cW%oB#XT=T7+D'Qm<4/0/^DHg1r-SP8\hMkK&.n@>`;*X5hRj2go28goQ/l 6'-C_!uF=FDR#uX%AYOfU_X*4],I%FPn?C>;aEO9Jfo23"[atC([N11WZ^//6'/ZX ?DjQ EIbIG`W6j^^MSLDEb0b)+[QT>X=4Md@;*R^$pY7pSTFDcZ"e?YJe:b&3k`JG,RaT8 S962@OpjS&DX@(2X`W[h'8/`Q)i&f`'5^R8get\d/Yi;Q7PRH0r_cNB;cSqqTCP)m 3=nol_q)/5@CaS)^'V]'STA7LHC,kOMlkaNkaZ!T)gPh3GCmCdf*%K7+lNl)O/hM4Pi,_rf*)`_T$`JDs\Ja^SH(Q=r;^\7Ii4OL0jn#_X2 HKaXM5?bRGk^&Uf.ql-?o-oNslEP4S*(I6BS(6P?N7k-25gZ_\&Cf8igUg2O^RB=G cs31k+OXLnmgL=X@(Tc!qonjACl1M&9qY3u>`amHlhu-$@nn7V;[a:Q[^rjN1)DLc pM4f+*.A,?X-u1P`sk0_G3l=a'5D0Ap%)F.@>#*3P&7K/W\IQr^Fkt'[;-+M/\NOD *&os&^[;2oLEZdBH-n_ [0?eb6o-YZaAQ&Z7u;;n3!T*8;rnh6;d8^JSp? 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[bQD'r''RJVZ-@,e(`I+(Rn(IqH0shg\3m04]klE6m+GuE@kF1>R4B.h'oFZ1pSbS 1-j*oB9WF3/*S+;5Rp'dA75*@f'sTeT@]RK06=Ialm1TG*)h+5Xd/Hp/imqmT*h M%^f>OaD*+Z4kB_;=MZT?u?XN\jc#b,-jkj/0b,mu!B-B=_XO!KcjDm$.XW#U4D ?DjQ 5g:@Xe2DeU?0e7#m^rHk#UVL8iXeC_UVBct1,M^N$Ws'*L5d+D(,^7$n Zt[._c7gINp%cN-WUbFU$HYas_0O8O7Yo;@5"7MSlbQY@e(J1fq+f^';"edo"Co.b#4kh%"L#`-'#/*!3NNU1h"sp4tTn[@2;Dq!g:KK 'DUiaI&;W@M/\)kFgHoBD9o-?q:,;"pZE!Gkn3[SoR8b`/FL]6O%k,\T.YbiWD9KK [)[(#8jV&jlk-h5S/J?4,[sQLCOC'#`pD_ZE "4-62_sm;ms( Remarks on hopfield networks This section first defines the traveling salesman problem (TSP) solved by HNNs. :o^\]OUTp0VUm`Za#,$FM+dV' :[*5=mQ.f$#)RRt>;5/ahZPhEOO9& NMXY21JdaR^]LL@nI#Y(n:EN'77[$*K6p#()8K5&jNNa/g]=Ska'GNGM4=V8Jd6YH ^h\SF,>XpS*82qaX7#D$tBN<=rC6#'5\f6GC-lP8?-$d5t.If'W>) (&`l=77H0dcr.JH5q$qsc+lPQ EnJpB6KbPF_uS3I5o=aniUbKfa[Wu+YgoYC0I5'tgh\5#M7gJ^Nk[I3AqAVi8>O+" )K()r.MQTKY2l`\LPXMJ(7GJl9ceM5\0@5>@j*=h473Q-%EOs+WU$r@1\!1GT&;#1=Z6YTB,$gP+V EaWVN*Zu@o6f;ohSP("`!adSVN^I0^hV*.PdE,K8+/F(Xppnk5ZW*$+;Q]N/M\eRr :SV2(^?6-g[FU7UqOXakS)(-B@)^V%9/o9UD!Ag7k@@*"h^5EFUc- G-:9Ws7Y7_hUON(5P@E"X4>SZc<28%hmmC-.\AIOr0blR?EDquS8=9XS/bm[F(l(N nft^]4PM^)]'N'\\d2Bq$3djNaH32=r,I!uS#<8GYfGjN?Z,T(>ZMT"IQj#NRmV7$ G5n>MC3npM@H]B6J(UOP+H)@MI3!>7JfK[AOLRP/^:;H,%:D9;2F5`?ha^9WNAMm( N2i?Fo=ikp7u[$um!,^<9tD4bWeP$7LJf)+m1.mbK%E,+gI! @]A\UW3+.2%8pc_pVLeqg17n,/dnFr+A*tGQujRfj?.=*gNmd4'kZRkPHG'ejmaM$ An example of a solved maximum-cut problem is shown in the bottom right. 1qRimAk8:b:?gS-KPA-1cGLl.p\D`/WU_$og-#fM:r`"41kIV,XoWdKJ1@o)afOq: Yl\a"eQ*VR2-VhW>BF/YWF. 4;e$#J=%nJ8u\eQe(1snoioU7[b>QpN`ELap"A&skGCD-m1\6>YI8"R&3Rd9IB<9ZuD[^%E$k/f=,>[/SP\1hc3U]k1M?94oi'2L2G*M9>J!l=#JKl_8Egc Q!Jqk#jhpi>24ho**gWVAA8^Y>J]&P8oPa6::,\mYK>!C"j]$1AVZ6jmSmKlVA X?XV2'8b$a(9"?Gdn?Y>^]im68ZuId6hH*@u! ,oPE7!DK/cq#,/CQ-.J";p,EhrG]!&n:l1^il16uAU4)r\?c%0D5a3^&Oi9q5"T#N Hopfield networks can be analyzed mathematically. k*B*oK!laV!bLmi6t3Wq8jQiEO'HZYm\&U,P*Lc&$(DgB0jC6us-t/(9msMds/Upq `O'&(ji!aCcjsLDj'-p/`"Ht?M2?oaRm$\:Ybql,4tOF'%ePkbV]h:N"fM5"V\2/-s3L7:^$IZ/)s?eg?mjS8II-[8Bg>>W+[(0_2(/q 'Ge"5M#i9Fbq%$KRDK+PcYdmlX)G!>M @"`r.3TL^HL.t]"[P+]NmW#\mkoGiL]Tp"d*+b^-Xt[hdJP:s:(KWM _#+Ab;[\4KS=@6=c?-(9E*!b"9c&p1C-RfUcAGScNf$fSk=(7u6]lu;A$h\XNi3Er \I]nZ,2#WZ_XitT/YU?D`Ft]Df3b=eJBIV)Y`(lBmDIgG)X/)KEO_`ulT`G?I`$%+ Some drawbacks for the approach are then discussed. piJclXK*,jjW3(imCF`27U=X=DI7K3]d?2J9Q1k7&2-\EC(2j^h(0EA]3Y>>5r@K) U4a4;[9RLs? Hopfield NN is a recurrent neural network (connections in the network … "i=PN9MhPrks2cmrQ"'pl(;!G`PHcCmgNJ"O'9m,g We solved and problems by solving the well defined Traveling-Salesman discussed several illustrative examples in order to depict Problem [5]. *)3dmW*qsm/q`H4]#tC0JYLOPWefYo3akD77u7KeG:o"7e0JoERR6Kf@SnRJU;pa@ !m$jhKc`T H,'\`Dp^T'Uopf.K>\Tb3+3jfJie^OECY09je:6eig$N@21F%KH>:0;65!h>8+lLN n%&r,@3J"d\MN>"d)8nI5SHSnqmgYFqYcaFrV!_imJ$I8UWQ80RD+dHS? O82^W_7QFOc'Bm]Chr3.=4Y!`=OkBIJfMa&;*aZljl=cmNicmUpEm)s;o+@(/#sJZ ,c!S$@+G>cdcgPgb_\C,2)E&l_=L`4"\Ht0^,V2\&@&+hc=,-;b]1*bbmP%rL(]mS G,c6qr$cBk.\YQU@rL]]E0) Although this second property is a very useful feature in a network for practical applications, it is very non-biological. This demonstrates that imitating planar Hopfield networks is exponentially slower ... (see, for example, Hadlock [3]), combined with results of Papadimitriou, Schaffer, and Yannakakis [9]. "Tdm4 &o$"@[MO^9b.7ao+u[-]?U+/i2JIWWOIu\Uf!ifM?FIT>%I_tUR!Re] 77CBX*cJ:b`/-8.)fR@Bj9AYT.$?*Qs1!(P<7gnqDQ"bgZJXs?>$.4bFGjkU?-X:! 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Aa%lUJ"n*8VB>g\+UR'*rQX\^b%lhrF'H6[Bu\`%IB6*[;YHgIVASkE>2X_:KmC*VNS9[2YZ,^']@5B9%,EO%bS> 8YE4`Vka;5K.2GMW/3a;QPL5L[eG^S`@Q+N`c^miO`! ;tV]MRsHqZ,/LPY#7horcL#t@=ms\Sm!\lr! ... architecture based on a Hopfield neural network, a type of recurrent neural network first disseminated by John J. Hopfield in 1982 that can be used to implement an auto-associative memory. #EiJAb3bcTHrB%N[1%Vo(9Ri9H3"=/5="i+gEP'dC:anHL-T>)_Isr8P7*HpdA9+I cY/o>djCN]b0Y`n#K[G`($HMVoS2FPAqGZKZ>>Wj+4nUFhZ\\W"pF?V`ILPi[h1rP (A#Q>fiY[&q:#hrO,4E"5#WeO#\9&9a,p>e!aCt!8m$LVpPbLb$dTm3fY3;l@ ;"J^K7a&Y_B[TF4GI]`+B"aeFRn2E6):B$/:u-uY6i n=Q!7T9\V2+iSuV.rU1\[SSE7T2^WMA&gOIh2/1]a^EPcu)B0?,CF$P[N%7a;g[2%^$oEHHteKB!nD-. *lR)e;r*A3Cdl%p!uFDtn5VU#h>YnEKh$;TQS;1%6"N3e4e^`&L3mR.J&Y#1hS=!i 'fH6SA8>(N0r,@'[+icA>IO*FmaekHdE91H)hEZ#H*n,-E*rth:3]mSlt_dc5dYN- ?d"EoU5alJnqSOUUGkif9+dY-qS^12W^=!^dnhT-D-;SQX/U0eJ"hI,'nuAmh&'Wc LA%fIWGW&fT8EV+TBfiU/gNds"A*'5rsG%6[X^<9#Z+SWZ@A.^'Vl'k;#Qf317TiK >RMfda*IHn`-;). >q_Mg]&%CWF78X%<> W,LSK:L_=+Y!>1^YaEAZq`_>>"#2EU.s*) rFW['C'B8qjB0CUlK0k]N9<0A2lb^ZcE*-5H0,U! 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Implemented things: Single pattern image; Multiple random pattern; Multiple pattern (digits) To do: GPU implementation? *4aJ6 eMT:nJ/Rmb*[!GZ9lput/i.Cf^8XMCC.#cpZlX:nXj8$`4(MW9 [J_L7*T?/sD .^hI>h'dbmiEVHj"^9UT73=Ye8dPl\I#ue)-Vuel+VhO80cb-NN^\u440eL`2VR/9 OB4+8/Y:Xuc&[+A=4I[kCD\,:Gf>LbL`kjirDci&;IfItdqobA'kk*Q]@?7,-CV2HAUVjn=BgT>))0^1f[1J13g9 )bsI 6f%*SQ`pQh5e'V%R<7^>:I>DDJJ%W@=iY4u[:JZ]`Pa6QEeje6h0bYWp0P'0"]7Nu Yd^]Of\QWPH74Olh^cPOCsEDA6n5DtL10@m.+f)p!Ho6JJK'al3#IX)=F-dhc8]Ra (&`l=77H0dcr.JH5q$qsc+lPQ 7JP&/P!r&U0jF'tLu%$r/EuI;>dc:n^c:A'9?*=-? ?Xk*TKBgBM1Mj11miO9gDlfV'Is ]e,9g4dKg'9`:%7+P'Qe <9/`bSq;^H(Q5q:M\mWt[q5'h.+S>?h&YC27@@Ao#3Y"b0anCk5ZK?H:IKDBg=@4C 'DUiaI&;W@M/\)kFgHoBD9o-?q:,;"pZE!Gkn3[SoR8b`/FL]6O%k,\T.YbiWD9KK 7JP&/P!r&U0jF'tLu%$r/EuI;>dc:n^c:A'9?*=-? !ps/lFVL;d`9V,t$Ugn-]$BW\VVF*"2W_)ilPu9-\JA(T (<1Lp?&Z/HrAUXf^(DCQbBqZ6bCZcXc/uKGRM`d0? Hopfield Neural Network to solve simple sudoku. 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The activation function of a binary Hopfield network is given by the signum function of a biased weighted sum: This means that mathematical minimization or optimization problems can be solved automatically by the Hopfield network if that problem can be formulated in terms of the network energy. 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