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1.0015 -1.2007 TD
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1.0015 -1.2007 TD
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-1.0015 -1.2007 TD
[(wi)-8.2(th)-453.8(similar)-460.3(structura)-9.7(l)-453.1(prope)-7.6(rties.)-458.5(Indeed,)-460.1(rando)-7.6(m)-453.8(networ)-9.3(ks)-452.3(can)-461.1(be)-452.9(artic)-9.2(ially)-457.8(select)-11(ed)-452.9(for)-455.8(robus)-7(tness,)]TJ
T*
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[(prope)-7.6(rties)-8.6(.)-415.8(Since)-419.1(the)-422.3(space)-420.1(of)-418.9(model)-421.6(states)-420.1(is)-420.9(expone)-8.1(ntially)-422.2(large,)-422.2(mapping)-423.1(its)-418.7(attractors)-424.7(and)-419.5(basins)-421.2(of)]TJ
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[(we)-303.2(u)-6.5(s)0(e)-300.4(rando)-7.6(mized)-303.9(mode)-8.4(ls)-301.4(that)-300.6(maint)-9(ain)-302.7(the)-302.7(topology)-307.5(and)-300(logical)-307.4(com)-7.6(plexity)-305.6(of)-299.4(the)-302.7(real)-307.3(models)-8.1(.)-296.3(W)0(e)-307.5(nd)]TJ
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[(13)-7.2(4)-30433.8(BRU)-9.2(NER)-339.2(AND)-338.4(SHAR)-8(AN)]TJ
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[(.)-245.1(W)0(e)-239.2(s)0(t)17.3(u)0(d)15.5(y)-245.5(th)18.5(e)-247.2(i)0(m)20.6(p)0(a)13.8(c)0(t)-233.7(o)0(f)-242.5(s)0(u)20(c)0(h)-236.6(c)0(h)13.8(a)0(n)19.4(g)0(e)19.4(s)-246.7(on)-240.5(t)13.5(h)0(e)-242.3(s)0(e)18.3(t)-242.5(of)-236.8(mo)12(de)19.4(l)-248.2(a)0(t)22.4(t)0(r)16.5(a)0(c)17.7(t)0(o)18.5(r)0(s)-238(b)0(y)-240.5(q)0(u)21.2(a)0(n)13.8(t)13.5(if)16.5(yi)18.5(ng)]TJ
-10.2544 -1.2007 TD
[(fo)19.3(r)-327.1(e)0(a)17.7(c)0(h)-316.3(c)0(h)19.4(a)0(n)19.4(g)0(e)-316.3(t)0(h)18.5(e)-326.9(pe)19.4(rc)17.5(en)13.8(t)-322.2(o)0(f)-316.5(m)0(o)17.7(d)0(e)19.4(l)-327.9(st)23(at)16.7(es)12.6(,)-319.1(l)0(e)16.7(a)0(d)13.8(i)13.5(ng)-314.5(to)-317.3(a)-326.9(d)0(i)18.5(f)0(f)17.3(e)0(r)17.5(e)0(n)19.4(t)-322.2(at)16.7(tr)16.5(ac)17.7(to)18.5(r)-327.1(i)13.5(n)-325.1(t)0(h)18.5(e)-326.9(pe)19.4(rt)16.5(ur)19.3(be)19.4(d)-325.1(m)0(o)17.7(d)0(e)13.8(l)0(/)21.4(s)0(t)17.3(a)0(t)16.7(e)0(.)]TJ
1.0015 -1.2007 TD
-.0186 Tc
[(Since)-300(t).6(he)-300(space)-300(of)-300.2(model)-301(s)2.2(tates)-299.4(i).6(s)-299.4(e)-4.1(xponentially)-298.2(large,)-297.8(ma)]TJ
22.6427 0 TD
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[(pping)-298.4(its)-299.6(attractors)-299.6(a)1.4(nd)-298.4(basins)-299.6(of)-300.4(attraction)-298.4(i).4(s)-299.6(v)-2.5(ery)]TJ
-23.6442 -1.2007 TD
-.018 Tc
[(costly.)-428.1(T)4.2(o)-428.5(t)1.2(ack)4(l)1.2(e)-430.3(t)1.2(his)-429.7(c)2.2(o)4(m).5(putatio)4(nal)-431.2(p)4(rob)4(l)1.2(em,)-428.1(w).1(e)-430.3(c)2.2(om)]TJ
21.9997 0 TD
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[(b)4.6(i)1.8(ne)-424(no)4.6(vel)-430.6(I)2.6(LP)-427.1(fo)4.6(rmu)4.6(l)1.8(atio)4.6(ns)-423.4(fo)4.6(r)-429.9(m)1.1(o)4.6(d)-1.1(el)-430.6(learning)-422.2(an)4.6(d)]TJ
-21.9997 -1.195 TD
-.0191 Tc
[(attractor)-243.8(c)1.1(omputations)-248.7(with)-247.5(stochastic)-249.3(simulations)-248.7(for)-243.8(e)]TJ
21.2542 0 TD
-.0187 Tc
[(stimating)-241.4(t).5(he)-248.9(size)-248.9(of)-249.1(each)-247.1(basin)-247.1(o)3.3(f)-249.1(a)1.5(ttraction.)-246.7(To)-241.4(describe)]TJ
-21.2542 -1.2007 TD
-.0181 Tc
[(our)-265.5(f)1.9(ormulation)-263.6(f)1.9(or)-265.5(attractor)-265.5(represen)]TJ
14.4085 0 TD
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[(tatio)-4.4(n,)-271.5(we)-273.6(rs)-5.6(t)-268.9(de)-6.2(scr)-6.4(i)-1.5(be)-273.6(a)-267.9(s).1(e)-6.2(t)-268.9(of)-273.8(su)-4.4(bp)-4.4(ro)]TJ
14.9605 0 TD
-.0189 Tc
[(grams)-271.2(t).3(hat)-267.1(f)1.1(orm)-267.9(its)-265.6(building)-264.4(b)-2.6(locks.)]TJ
/F4 1 Tf
10.959 0 0 10.959 107.6031 571.351 Tm
0 Tc
[(3.1.)-496.9(State)-334.5(comparison)8.4(s)]TJ
/F1 1 Tf
9.9626 0 0 9.9626 117.5811 553.4361 Tm
[(The)-328.2(repr)-8.8(esentatio)-10.3(n)-330.8(o)0(f)-327.9(attra)-9.2(ctor)-328.2(sta)-8(tes)-332.4(requires)-336.2(efcient)-336.8(mea)-9.4(ns)-327.1(to)-334.3(compare)-334.9(them)-335.5(to)-328.7(one)-334.1(anoth)-8.4(er)-329.7(to)-328.7(mak)-7.6(e)]TJ
-1.0015 -1.2007 TD
[(sure)-298.2(that)-294.9(they)-297.8(repr)-8.8(esent)-295.9(a)-298.5(cyclic)-300(path)-297.8(of)-293.7(disti)-7.6(nct)-297.1(states)-300.6(and)-294.3(that)-300.6(no)-297.4(two)-293.5(attra)-9.2(ctors)-296.1(shar)-8(e)-292.8(the)-297.1(same)-301.6(state.)-299(One)]TJ
T*
[(way)-400.7(to)-402.6(tackle)-402.4(this)-402.4(problem)-402.3(as)-402.8(well)-401.2(as)-402.8(remove)-406.3(degrees)-406.5(of)-396.2(freedom)-409(from)-397.5(solu)-7(tions)-403.2(is)-398.1(to)-396.9(assign)-404.2(a)-400.9(uniqu)-6.6(e)]TJ
T*
[(numeri)-8.9(c)-247.2(key)-254.5(to)-249(each)-258.8(possibl)-7.6(e)-247.2(networ)-9.3(k)-251.2(state.)-253.5(Th)-7(e)-247.2(resu)-8(lting)-251.1(order)-254.2(can)-256.2(then)-252.3(be)-253.7(used)-256.4(to)-249(imple)-9(ment)-250.1(equal)-10.1(ity)-252.5(and)]TJ
T*
[(inequal)-8.7(ity)-332.2(const)-9.6(raints.)]TJ
1.0015 -1.2007 TD
[(A)-323.4(natural)-329.4(key)-328.4(for)-324.9(a)-321.2(sta)-8(te)-324.7(is)-324.2(the)-325.5(binary)-329.8(number)-330.5(it)-320(repr)-8.8(esents,)-326.6(but)-329.4(the)-325.5(represe)-10.2(ntation)-328.4(size)-329.2(may)-326.3(exceed)-330.4(the)]TJ
-1.0015 -1.2007 TD
[(numb)-6.6(er)-369.5(of)-367.7(bits)-374(in)-368.5(a)-372.4(compute)-9.5(r)-366.9(word.)-373.5(Instead)-9.1(,)-370.3(w)0(e)-371.4(impleme)-10.1(nt)-368.5(this)-374(representa)-11.3(tion)-372.8(by)-365.7(assum)-11.5(ing)-369.2(an)-367.5(arbi)-9.6(trary)]TJ
0 -1.195 TD
[(upper)-258(bound)-259.9(of)]TJ
/F4 1 Tf
6.3905 0 TD
(M)Tj
/F1 1 Tf
1.0869 0 TD
[(bits)-260.2(per)-256.4(word)-259.3(and)-260.2(splitting)-260.9(the)-257.2(numb)-6.6(er)-255.7(into)]TJ
/F10 1 Tf
17.4415 0 TD
()Tj
/F6 1 Tf
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(+)Tj
/F1 1 Tf
.717 0 TD
[(1)-256.8(pieces,)-264.6(where)]TJ
/F10 1 Tf
6.4872 0 TD
()Tj
/F8 1 Tf
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(:)Tj
/F6 1 Tf
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.165 Tc
(=)Tj
ET
458.305 472.025 8.674 .39685 re
f
BT
/F7 1 Tf
6.9738 0 0 6.9738 458.3054 474.4629 Tm
0 Tc
(j)Tj
/F4 1 Tf
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(V)Tj
/F7 1 Tf
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(j)Tj
/F4 1 Tf
-.7804 -1.1625 TD
(M)Tj
/F6 1 Tf
9.9626 0 0 9.9626 466.9794 469.7574 Tm
.167 Tc
(-)Tj
/F1 1 Tf
1.269 0 TD
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[(1.)-257.2(For)-259.6(any)-254.5(two)]TJ
-37.3414 -1.2007 TD
(states)Tj
/F4 1 Tf
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(a)Tj
/F1 1 Tf
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()Tj
/F4 1 Tf
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(b)Tj
/F7 1 Tf
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.2772 Tc
(2f)Tj
/F1 1 Tf
1.434 0 TD
0 Tc
[(0)-498.6(1)]TJ
/F7 1 Tf
1.838 0 TD
(g)Tj
6.9738 0 0 6.9738 191.1685 462.5574 Tm
(j)Tj
/F4 1 Tf
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(V)Tj
/F7 1 Tf
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(j)Tj
/F1 1 Tf
9.9626 0 0 9.9626 202.9606 457.7952 Tm
[(represen)-10.9(ted)-268.6(by)]TJ
/F4 1 Tf
6.1629 0 TD
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(0)Tj
9.9626 0 0 9.9626 272.8629 457.7952 Tm
()Tj
/F8 1 Tf
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.1682 Tc
(...)Tj
/F4 1 Tf
1.3145 0 TD
0 Tc
(x)Tj
/F10 1 Tf
6.9738 0 0 6.9738 297.0708 456.2645 Tm
()Tj
/F1 1 Tf
9.9626 0 0 9.9626 303.0235 457.7952 Tm
(and)Tj
/F4 1 Tf
1.7185 0 TD
(y)Tj
/F1 1 Tf
6.9738 0 0 6.9738 324.6235 456.2645 Tm
(0)Tj
9.9626 0 0 9.9626 328.592 457.7952 Tm
()Tj
/F8 1 Tf
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.1682 Tc
(...)Tj
/F1 1 Tf
1.3145 0 TD
0 Tc
()Tj
/F4 1 Tf
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(y)Tj
/F10 1 Tf
6.9738 0 0 6.9738 357.7889 456.2645 Tm
()Tj
/F1 1 Tf
9.9626 0 0 9.9626 361.1338 457.7952 Tm
[(,)-267.8(respective)-12.1(ly,)-266.4(to)-266.1(im)-7.8(plement)-274(an)-265.1(indicat)-11.5(or)-265.3(for)]TJ
-25.4481 -1.2007 TD
[(an)-373.2(inequal)-8.7(ity)-372(such)-375.9(as)]TJ
/F4 1 Tf
9.1333 0 TD
(a)Tj
/F10 1 Tf
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(>)Tj
/F4 1 Tf
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(b)Tj
/F1 1 Tf
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[(,)-376(w)0(e)-371.4(use)-375.2(auxi)-7.6(liary)-377.3(binary)-375.3(variables)-379.4(\()]TJ
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15.097 0 TD
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6.9738 0 0 6.9738 376.044 444.3022 Tm
(0)Tj
9.9626 0 0 9.9626 380.0125 445.8329 Tm
()Tj
/F8 1 Tf
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[(..)5.7(.)]TJ
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(z)Tj
/F10 1 Tf
6.9738 0 0 6.9738 408.5857 444.3022 Tm
()Tj
/F1 1 Tf
9.9626 0 0 9.9626 411.9306 445.8329 Tm
[(\))-372.6(and,)-374.3(for)-376.1(conven)-8.2(ience,)-376.5(denot)-8.4(e)]TJ
/F4 1 Tf
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(z)Tj
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(-)Tj
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(=)Tj
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21.0607 0 TD
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-10.9543 -2.0372 TD
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(M)Tj
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6.9738 0 0 6.9738 297.581 412.1007 Tm
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()Tj
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6.9738 0 0 6.9738 317.7637 412.1007 Tm
(j)Tj
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(-)Tj
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6.9738 0 0 6.9738 333.5243 412.1007 Tm
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6.9738 0 0 6.9738 348.6613 412.1007 Tm
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9.9626 0 0 9.9626 261.6944 397.4739 Tm
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(M)Tj
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9.9626 0 0 9.9626 278.2487 397.4739 Tm
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(M)Tj
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9.9626 0 0 9.9626 381.203 397.4739 Tm
(:)Tj
/F1 1 Tf
14.5678 .8194 TD
(\(2\))Tj
-41.0289 -2.8168 TD
[(It)-330.6(holds)-337.8(that)]TJ
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[(1)-330.8(iff)-339.1(\(if)-333.4(and)-339.8(only)-335.9(if\))]TJ
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6.9738 0 0 6.9738 298.7149 376.1007 Tm
(0)Tj
9.9626 0 0 9.9626 306.0283 377.5747 Tm
[(and)-334.1(for)]TJ
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6.9738 0 0 6.9738 255.118 356.9385 Tm
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6.9738 0 0 6.9738 287.8298 356.9385 Tm
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6.9738 0 0 6.9738 308.0125 356.9385 Tm
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9.9626 0 0 9.9626 312.6613 358.4692 Tm
(_)Tj
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6.9738 0 0 6.9738 329.2157 356.9385 Tm
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6.9738 0 0 6.9738 344.9763 356.9385 Tm
(j)Tj
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9.9626 0 0 9.9626 349.6818 358.4692 Tm
(^)Tj
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6.9738 0 0 6.9738 362.2676 356.9385 Tm
(j)Tj
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9.9626 0 0 9.9626 376.1007 358.4692 Tm
(=)Tj
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(1\))Tj
/F10 1 Tf
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(:)Tj
/F1 1 Tf
-27.5024 -1.9405 TD
[(This)-336.9(implies)-344.5(that)]TJ
/F4 1 Tf
7.2668 0 TD
(z)Tj
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6.9738 0 0 6.9738 193.833 337.6629 Tm
()Tj
/F6 1 Tf
9.9626 0 0 9.9626 198.822 339.1369 Tm
(=)Tj
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[(1)-336.5(iff)]TJ
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2.1226 0 TD
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(>)Tj
/F4 1 Tf
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(b)Tj
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[(,)-336.1(and)-339.8(we)-343(can)-335.9(d)-6.5(enote)]TJ
/F4 1 Tf
8.6326 0 TD
(I)Tj
6.9738 0 0 6.9738 339.7038 337.6629 Tm
(a)Tj
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(b)Tj
/F6 1 Tf
9.9626 0 0 9.9626 356.5416 339.1369 Tm
(=)Tj
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(z)Tj
/F10 1 Tf
6.9738 0 0 6.9738 367.54 337.6629 Tm
()Tj
/F1 1 Tf
9.9626 0 0 9.9626 370.8849 339.1369 Tm
[(.)-336.1(Using)]TJ
/F4 1 Tf
3.3119 0 TD
(z)Tj
/F6 1 Tf
6.9738 0 0 6.9738 408.9259 337.6629 Tm
(-)Tj
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(1)Tj
/F6 1 Tf
9.9626 0 0 9.9626 419.5841 339.1369 Tm
(=)Tj
/F1 1 Tf
.717 0 TD
[(1)-336.5(capt)-9.4(ures)-338.1(a)-338.3(weak)-340.6(order)-9.6(ing)]TJ
-32.0321 -1.2007 TD
[(and)-334.1(allow)-9.4(s)-332(u)0(s)-338.5(t)0(o)-334.3(impleme)-10.1(nt)-334.3(the)-336.9(indicator)-9.3(s)]TJ
/F4 1 Tf
17.5781 0 TD
(I)Tj
6.9738 0 0 6.9738 286.0724 325.7007 Tm
(a)Tj
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(=)Tj
/F4 1 Tf
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(b)Tj
/F1 1 Tf
9.9626 0 0 9.9626 303.0235 327.1747 Tm
(and)Tj
/F4 1 Tf
1.7868 0 TD
(I)Tj
6.9738 0 0 6.9738 324.1133 325.7007 Tm
(a)Tj
/F7 1 Tf
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(6)Tj
T*
()Tj
/F4 1 Tf
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(b)Tj
/F1 1 Tf
9.9626 0 0 9.9626 340.3275 327.1747 Tm
[(as)-334.6(well)-8.6(.)]TJ
-22.3582 -1.2007 TD
[(We)-267.6(can)-261.9(use)-267(thes)-8.8(e)-264.3(indicator)-9.3(s)-263.7(t)0(o)-260.4(i)0(m)-7.8(plement)-268.3(a)-264.3(set)-264.1(membe)-8.7(rship)-271.5(indicator)]TJ
/F4 1 Tf
28.2593 0 TD
(I)Tj
6.9738 0 0 6.9738 402.4629 313.7385 Tm
(a)Tj
/F7 1 Tf
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(2)Tj
/F4 1 Tf
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(A)Tj
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(=)Tj
/F4 1 Tf
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(F)Tj
/F1 1 Tf
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(\()Tj
/F4 1 Tf
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(a)Tj
/F1 1 Tf
6.9738 0 0 6.9738 152.3338 702.0281 Tm
(0)Tj
9.9626 0 0 9.9626 156.3023 703.5588 Tm
[(\))-281.6(t)0(o)-283.1(refer)-286.5(to)-283.1(this)-288.6(subprogra)-8.1(m.)-283.5(We)-284.7(can)-284.7(also)-287.6(derive)-286(a)-281.4(formul)-9.1(ation)-284.3(for)-285.1(requiring)-291.3(the)-285.7(states)]TJ
-8.2513 -1.2007 TD
[(no)-7.2(t)-333.6(t)0(o)-334.3(form)-334.9(a)-332.6(p)-6.5(ath,)-331.6(whi)-9(ch)-333.4(we)-337.3(denote)-341(by)]TJ
ET
241.001 699.42 25.455 .39685 re
f
BT
/F4 1 Tf
9.9626 0 0 9.9626 241.0015 691.5967 Tm
(PATH)Tj
/F1 1 Tf
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6.9738 0 0 6.9738 274.7905 690.066 Tm
(0)Tj
9.9626 0 0 9.9626 278.8157 691.5967 Tm
()Tj
/F8 1 Tf
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(...)Tj
/F1 1 Tf
1.3145 0 TD
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()Tj
/F4 1 Tf
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(a)Tj
6.9738 0 0 6.9738 308.5228 690.066 Tm
(T)Tj
/F1 1 Tf
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(\).)Tj
/F4 1 Tf
10.959 0 0 10.959 74.0976 667.6722 Tm
[(3.3.)-502(Attractor)-330.1(learning)]TJ
/F1 1 Tf
9.9626 0 0 9.9626 84.0755 649.7573 Tm
[(We)-392.8(are)-389.1(now)-387.5(read)-8.6(y)-387.7(t)0(o)-385.6(present)-396.1(our)-385.5(formul)-9.1(ations)-388.7(for)-387.5(attra)-9.2(ctor)-390.8(learning)-392.8(under)-394.5(various)-388.6(const)-9.6(raints.)-391(Using)]TJ
-1.0015 -1.2007 TD
[(x)-6.4(ed)-276.5(values)-279.6(for)-273.7(the)-280(outputs)-279.5(of)]TJ
/F4 1 Tf
12.3997 0 TD
(F)Tj
/F1 1 Tf
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[(,)-273.5(w)0(e)-274.7(can)-279(use)-272.7(such)-279.2(formul)-9.1(ations)-280.5(to)-271.7(com)-7.6(pute)-275.1(the)-274.3(attra)-9.2(ctors)-279(of)-276.7(a)-270(g)-6.5(iven)-275.1(model.)]TJ
-13.02 -1.2007 TD
[(By)-353.4(substitut)-8.9(ing)-352.2(the)-348.3(outpu)-6.6(ts)-346.9(of)]TJ
/F4 1 Tf
12.6159 0 TD
(F)Tj
/F1 1 Tf
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[(with)-353.9(bina)-7.6(ry)-344.9(vari)-9.6(ables,)-353.2(we)-348.7(get)-354(a)-349.7(formulat)-9.5(ion)-352.2(whose)-352.2(solution)-357(represe)-10.1(nts)-347.7(a)]TJ
-13.5776 -1.2007 TD
[(po)-7.2(ssible)-355.7(mode)-8.4(l)-350.6(o)0(n)]TJ
/F4 1 Tf
7.8643 0 TD
(G)Tj
/F1 1 Tf
1.0755 0 TD
[(and)-356.9(its)-356.1(attract)-9.6(ors.)-353(Fo)-6.4(r)-349.9(brevit)-8.2(y,)-354(in)-357.1(the)-354(followi)-8.1(ng)-354.3(formul)-9.1(ations)-354.5(whe)-8(never)-356.5(we)-360.1(write)-358.5(an)]TJ
-8.9399 -1.2007 TD
[(indi)-8.6(cator)-262.5(as)-260.6(a)-264.3(constra)-9.1(int,)-258.6(we)-263.3(require)-271.7(its)-259.4(value)-266.2(to)-260.4(be)-265.1(1.)-257.2(All)-262.1(formul)-9.1(ations)-263.5(depend)-269.9(on)-257.6(specify)-9.1(ing)-261.1(in)-260.4(advanc)-9.9(e)-264.3(the)]TJ
T*
[(up)-7.2(per)-296.3(bounds)-301.7(on)-297.4(the)-297.1(number)-302.1(and)-294.3(leng)-7.6(th)-294.5(of)-299.4(the)-297.1(model)-8.9(s)-292.2(attra)-9.2(ctors.)-302.1(For)-293.7(clari)-9.2(ty,)-294.9(we)-297.5(assume)-304.3(at)-296.3(rst)-298.4(that)-300.6(there)]TJ
T*
[(are)-502.9(exactl)-9.8(y)]TJ
/F4 1 Tf
5.1386 0 TD
(K)Tj
/F1 1 Tf
1.1666 0 TD
[(attra)-9.2(ctors)-500.9(of)-498.6(leng)-7.6(th)]TJ
/F4 1 Tf
8.5927 0 TD
(T)Tj
/F1 1 Tf
1.0471 0 TD
[(each.)-509.6(We)-500.9(apply)-503.4(the)-501.9(conditional)-506.7(constra)-9.1(int)-497.2(mec)-9.4(hanism)-503.2(below)-501.7(to)]TJ
-15.9449 -1.2007 TD
[(gener)-9.4(alize)-339.1(the)-336.9(formul)-9.1(ations)-337.4(to)-334.3(the)-336.9(case)-339.7(that)]TJ
/F4 1 Tf
18.1016 0 TD
(K)Tj
/F1 1 Tf
1.0015 0 TD
(and)Tj
/F4 1 Tf
1.7868 0 TD
(T)Tj
/F1 1 Tf
.882 0 TD
[(are)-337.9(upper)-337.6(b)-6.5(ounds.)]TJ
-20.7705 -1.195 TD
(Let)Tj
/F7 1 Tf
1.6673 0 TD
(f)Tj
/F4 1 Tf
.5008 0 TD
(a)Tj
6.9738 0 0 6.9738 110.6645 552.5857 Tm
(k)Tj
/F1 1 Tf
.4634 0 TD
()Tj
/F4 1 Tf
.504 0 TD
(t)Tj
/F7 1 Tf
9.9626 0 0 9.9626 120.0189 554.1164 Tm
(g)Tj
/F1 1 Tf
6.9738 0 0 6.9738 125.0078 551.3384 Tm
(1)Tj
/F7 1 Tf
.504 0 TD
()Tj
/F4 1 Tf
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(k)Tj
/F7 1 Tf
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()Tj
/F4 1 Tf
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(K)Tj
/F1 1 Tf
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(0)Tj
/F7 1 Tf
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0 Tc
()Tj
/F4 1 Tf
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(t)Tj
/F7 1 Tf
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()Tj
/F4 1 Tf
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(T)Tj
/F1 1 Tf
9.9626 0 0 9.9626 175.5779 554.1164 Tm
[(denot)-8.4(e)-315.5(order)-9.6(ed)-316.3(set)-8(s)-320.6(o)0(f)-316.5(bina)-7.6(ry)-316.5(vect)-9.4(or)-316.5(variabl)-10.7(es)-317.5(repr)-8.8(esenting)-323.7(the)]TJ
/F4 1 Tf
25.8692 0 TD
(K)Tj
/F1 1 Tf
.9845 0 TD
[(attract)-9.6(ors.)-318.8(Th)-7(at)]TJ
-37.0398 -1.2007 TD
[(is,)-239.2(each)]TJ
/F4 1 Tf
3.2379 0 TD
(a)Tj
6.9738 0 0 6.9738 111.3448 540.6235 Tm
(k)Tj
/F1 1 Tf
.4634 0 TD
()Tj
/F4 1 Tf
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(t)Tj
/F1 1 Tf
9.9626 0 0 9.9626 123.0236 542.1542 Tm
[(is)-233.1(a)-235.9(v)-6.5(ector)-239.7(of)]TJ
/F4 1 Tf
5.4003 0 TD
(n)Tj
/F1 1 Tf
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[(binary)-238.7(variables)-242.8(denot)-8.4(ing)-232.7(the)-240.2(state)-241.7(of)-236.8(attractor)]TJ
/F4 1 Tf
18.6252 0 TD
(k)Tj
/F1 1 Tf
.6829 0 TD
[(at)-239.4(time)]TJ
/F4 1 Tf
2.9932 0 TD
(t)Tj
/F1 1 Tf
.2845 0 TD
[(.)-233.7(These)-241.7(variables)-242.8(can)-239.2(be)]TJ
-33.6368 -1.2007 TD
[(po)-7.2(pulated)-338.8(using)-337.8(the)-336.9(followi)-8.1(ng)-331.6(formul)-9.1(ation)-335.5(\(w)-7.4(ith)-332.2(emp)-7.6(ty)-334.3(objective\))-11.9(:)]TJ
19.3592 -1.9234 TD
(Max)Tj
/F8 1 Tf
2.1681 0 TD
()Tj
/F1 1 Tf
1.1097 0 TD
(s)Tj
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(t)Tj
/F10 1 Tf
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(:)Tj
/F7 1 Tf
-10.1861 -1.747 TD
(8)Tj
/F4 1 Tf
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(k)Tj
/F7 1 Tf
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()Tj
/F4 1 Tf
1.0471 0 TD
(K)Tj
/F8 1 Tf
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(:)Tj
/F4 1 Tf
3.5509 0 TD
(PATH)Tj
/F1 1 Tf
2.5551 0 TD
(\()Tj
/F4 1 Tf
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(a)Tj
6.9738 0 0 6.9738 310.0535 492.1511 Tm
(k)Tj
/F1 1 Tf
.4634 0 TD
.171 Tc
(0)Tj
9.9626 0 0 9.9626 320.7684 493.6251 Tm
0 Tc
()Tj
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(...)Tj
/F1 1 Tf
1.3145 0 TD
0 Tc
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(a)Tj
6.9738 0 0 6.9738 350.5322 492.1511 Tm
(k)Tj
/F1 1 Tf
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/F4 1 Tf
.504 0 TD
(T)Tj
/F1 1 Tf
9.9626 0 0 9.9626 362.2109 493.6251 Tm
(\))Tj
/F7 1 Tf
-15.5295 -1.5023 TD
(8)Tj
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()Tj
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(K)Tj
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/F4 1 Tf
8.5529 0 TD
(I)Tj
6.9738 0 0 6.9738 329.3857 477.1841 Tm
(a)Tj
4.9813 0 0 4.9813 332.9007 476.1636 Tm
(k)Tj
/F1 1 Tf
.4666 0 TD
.1678 Tc
(0)Tj
/F6 1 Tf
6.9738 0 0 6.9738 341.8582 477.1841 Tm
0 Tc
(=)Tj
/F4 1 Tf
.7235 0 TD
(a)Tj
4.9813 0 0 4.9813 350.4188 476.1636 Tm
(k)Tj
/F1 1 Tf
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()Tj
/F4 1 Tf
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(T)Tj
/F7 1 Tf
9.9626 0 0 9.9626 207.496 463.7479 Tm
(8)Tj
/F4 1 Tf
.552 0 TD
(k)Tj
/F7 1 Tf
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()Tj
/F4 1 Tf
1.0471 0 TD
(K)Tj
/F1 1 Tf
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()Tj
/F4 1 Tf
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(t)Tj
/F10 1 Tf
.5747 0 TD
(>)Tj
/F1 1 Tf
1.0528 0 TD
(0)Tj
/F8 1 Tf
.7796 0 TD
(:)Tj
/F4 1 Tf
6.5498 0 TD
(I)Tj
6.9738 0 0 6.9738 335.6786 462.2172 Tm
(a)Tj
4.9813 0 0 4.9813 339.1936 461.1967 Tm
(k)Tj
/F1 1 Tf
.4666 0 TD
()Tj
/F4 1 Tf
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(T)Tj
/F10 1 Tf
6.9738 0 0 6.9738 347.6975 462.2172 Tm
(>)Tj
/F4 1 Tf
.7723 0 TD
(a)Tj
4.9813 0 0 4.9813 356.5416 461.1967 Tm
(k)Tj
/F1 1 Tf
.478 0 TD
()Tj
/F4 1 Tf
.5008 0 TD
(t)Tj
/F7 1 Tf
9.9626 0 0 9.9626 207.496 448.781 Tm
(8)Tj
/F4 1 Tf
.552 0 TD
(k)Tj
/F10 1 Tf
.7455 0 TD
(<)Tj
/F4 1 Tf
1.0471 0 TD
(K)Tj
/F8 1 Tf
1.0072 0 TD
(:)Tj
/F4 1 Tf
8.5529 0 TD
(I)Tj
6.9738 0 0 6.9738 329.3857 447.3069 Tm
(a)Tj
4.9813 0 0 4.9813 332.9007 446.2298 Tm
(k)Tj
/F6 1 Tf
.6373 0 TD
(+)Tj
/F1 1 Tf
.717 0 TD
(1)Tj
/F4 1 Tf
1.0015 0 TD
(T)Tj
/F10 1 Tf
6.9738 0 0 6.9738 348.3212 447.3069 Tm
(>)Tj
/F4 1 Tf
.7723 0 TD
(a)Tj
4.9813 0 0 4.9813 357.1653 446.2298 Tm
(k)Tj
/F1 1 Tf
.4666 0 TD
()Tj
/F4 1 Tf
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(T)Tj
/F1 1 Tf
9.9626 0 0 9.9626 366.1794 448.781 Tm
[()-12385.4(\(4\))]TJ
-29.3177 -1.9974 TD
[(whe)-8(re)-449.2(degre)-9.4(es)-448.4(of)-447.4(fre)-8(edom)-452.2(in)-448.2(attract)-9.6(or)-447.4(represe)-10.1(ntation)-453.6(are)-451.7(elim)-8.2(inated)-451.8(by)-451.1(forcing)-452(the)-450.7(n)-6.4(al)-449.9(state)-452.3(of)-447.4(each)]TJ
0 -1.2007 TD
[(attra)-9.2(ctor)-271.3(to)-266.1(be)-270.8(the)-268.6(largest)-273.7(in)-266.1(terms)-271.1(of)-265.3(it)-7.1(s)-263.7(assoc)-9.8(iated)-269(nume)-8.4(ric)-264.9(valu)-7.6(e,)-264.7(and)-271.5(the)-268.6(nal)-268.6(sta)-8(tes)-269.8(of)-265.3(different)-278(attractors)]TJ
T*
[(to)-340(be)-333.4(ordered)-342.9(as)-334.6(well.)]TJ
1.0015 -1.2007 TD
[(To)-439.5(gener)-9.4(alize)-447.2(the)-439.3(program)-447.1(from)]TJ
/F4 1 Tf
14.0386 0 TD
(T)Tj
/F1 1 Tf
.6317 0 TD
()Tj
/F4 1 Tf
.8365 0 TD
(K)Tj
/F1 1 Tf
1.1666 0 TD
[(bein)-7.6(g)-438.9(exact)-444.4(numbers)-446.3(to)-436.8(upper)-445.7(bounds,)-444.3(we)-439.7(int)-7.8(roduce)-442.6(activit)-8.6(y)]TJ
-17.6748 -1.2007 TD
[(vari)-9.6(ables)]TJ
/F4 1 Tf
3.9379 0 TD
(w)Tj
6.9738 0 0 6.9738 119.9622 379.559 Tm
(k)Tj
/F1 1 Tf
.4715 0 TD
()Tj
/F4 1 Tf
.4959 0 TD
(t)Tj
/F1 1 Tf
9.9626 0 0 9.9626 132.3212 381.033 Tm
[(for)-307.8(each)-304.3(state)]TJ
/F4 1 Tf
5.7702 0 TD
(a)Tj
6.9738 0 0 6.9738 194.7968 379.559 Tm
(k)Tj
/F1 1 Tf
.4715 0 TD
()Tj
/F4 1 Tf
.4959 0 TD
(t)Tj
/F1 1 Tf
9.9626 0 0 9.9626 204.1511 381.033 Tm
[(,)-302(with)]TJ
/F4 1 Tf
2.6404 0 TD
(w)Tj
6.9738 0 0 6.9738 237.0897 379.559 Tm
(k)Tj
/F1 1 Tf
.4715 0 TD
()Tj
/F6 1 Tf
.6666 0 TD
(-)Tj
/F1 1 Tf
.7154 0 TD
(1)Tj
/F6 1 Tf
9.9626 0 0 9.9626 255.685 381.033 Tm
(=)Tj
/F1 1 Tf
.7227 0 TD
[(0)-302.4(for)-302.2(all)]TJ
/F4 1 Tf
3.585 0 TD
(k)Tj
/F1 1 Tf
.4496 0 TD
[(.)-302(These)-310(activity)-305.2(vari)-9.6(ables)-307.3(affect)-306.9(which)-308.2(constra)-9.1(ints)]TJ
-22.9841 -1.2007 TD
[(are)-417.6(in)-408.3(effect)-415(and)-413.8(what)-409.9(variabl)-10.7(es)-408.5(are)-411.9(zero)-8.6(ized.)-413.8(Their)-414.2(application)-417.4(to)-408.3(the)-410.9(progr)-7.8(am)-410.9(require)-9.9(s)-406(a)-412.3(conditioni)-9.1(ng)]TJ
T*
[(const)-9.6(ruct)-242.9(that)-238(enfor)-9.6(ces)-240.4(a)-235.9(const)-9.6(raint)-240.7(condi)-8.4(tioned)-239.5(on)-240.5(the)-240.2(valu)-7.6(e)-235.9(o)0(f)-242.5(the)-240.2(activit)-8.6(y)-234.1(v)-6.5(ariables.)-242.5(Su)-6.4(ch)-236.6(condi)-8.4(tioning)-241.3(can)]TJ
0 -1.195 TD
[(be)-424.4(construc)-9.9(ted)-422.3(by)-422.6(inequal)-8.7(ity)-423.2(constraint)-11.3(s)-417.4(that)-425.8(take)-424.8(effect)-426.4(iff)-424.4(the)-422.2(condition)-426.7(is)-420.9(met.)-426.1(For)-424.6(example,)-427.1(we)-422.7(can)]TJ
0 -1.2007 TD
[(formul)-9.1(ate)-617.5(the)-615.7(constra)-9.1(int)]TJ
/F4 1 Tf
10.9486 0 TD
(a)Tj
/F7 1 Tf
.7796 0 TD
(!)Tj
/F4 1 Tf
1.6161 0 TD
(b)Tj
/F7 1 Tf
1.1097 0 TD
()Tj
/F4 1 Tf
1.3828 0 TD
(c)Tj
/F1 1 Tf
.4496 0 TD
[(,)-615(which)-615.5(require)-9.9(s)]TJ
/F4 1 Tf
7.7733 0 TD
(b)Tj
/F7 1 Tf
.7796 0 TD
()Tj
/F4 1 Tf
1.3828 0 TD
(c)Tj
/F1 1 Tf
1.0641 0 TD
(iff)Tj
/F4 1 Tf
1.5592 0 TD
(a)Tj
/F6 1 Tf
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(=)Tj
/F1 1 Tf
.7227 0 TD
[(1,)-615.7(by)-610.4(the)-615.7(modi)-9.3(ed)-612.2(constra)-9.1(int)]TJ
/F4 1 Tf
-30.2339 -1.2007 TD
(b)Tj
/F7 1 Tf
.7796 0 TD
()Tj
/F4 1 Tf
1.0527 0 TD
(c)Tj
/F6 1 Tf
.6146 0 TD
(-)Tj
/F1 1 Tf
.717 0 TD
(\(1)Tj
/F6 1 Tf
1.0015 0 TD
(-)Tj
/F4 1 Tf
.7227 0 TD
(a)Tj
/F1 1 Tf
.5008 0 TD
[(\).)-333.2(The)-339.6(general)-9.7(ized)-339.4(formulation)-344.6(is)-329.8(as)-340.2(follows:)]TJ
14.0272 -1.9234 TD
(Max)Tj
/F8 1 Tf
2.1112 0 TD
()Tj
/F1 1 Tf
1.0528 0 TD
(s)Tj
/F10 1 Tf
.387 0 TD
(:)Tj
/F1 1 Tf
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(t)Tj
/F10 1 Tf
.2845 0 TD
(:)Tj
/F7 1 Tf
-13.2874 -1.747 TD
(8)Tj
/F4 1 Tf
.552 0 TD
(k)Tj
/F1 1 Tf
.4666 0 TD
()Tj
/F4 1 Tf
.5008 0 TD
(t)Tj
/F10 1 Tf
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(<)Tj
/F4 1 Tf
1.0471 0 TD
(T)Tj
/F8 1 Tf
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(:)Tj
/F1 1 Tf
5.1044 0 TD
(\()Tj
/F4 1 Tf
.3357 0 TD
(w)Tj
6.9738 0 0 6.9738 277.3416 283.2377 Tm
(k)Tj
/F1 1 Tf
.4715 0 TD
()Tj
/F4 1 Tf
.4959 0 TD
(t)Tj
/F7 1 Tf
9.9626 0 0 9.9626 288.907 284.7117 Tm
(^)Tj
/F4 1 Tf
.882 0 TD
(w)Tj
6.9738 0 0 6.9738 304.3275 283.2377 Tm
(k)Tj
/F1 1 Tf
.4715 0 TD
()Tj
/F4 1 Tf
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(t)Tj
/F6 1 Tf
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(+)Tj
/F1 1 Tf
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(1)Tj
9.9626 0 0 9.9626 323.3763 284.7117 Tm
(\))Tj
/F7 1 Tf
.6146 0 TD
(!)Tj
/F4 1 Tf
1.2747 0 TD
(a)Tj
6.9738 0 0 6.9738 347.1873 283.2377 Tm
(k)Tj
/F1 1 Tf
.4715 0 TD
()Tj
/F4 1 Tf
.504 0 TD
(t)Tj
/F6 1 Tf
.4634 0 TD
(+)Tj
/F1 1 Tf
.7235 0 TD
(1)Tj
/F6 1 Tf
9.9626 0 0 9.9626 367.9369 284.7117 Tm
(=)Tj
/F4 1 Tf
.717 0 TD
(F)Tj
/F1 1 Tf
.6544 0 TD
(\()Tj
/F4 1 Tf
.3357 0 TD
(a)Tj
6.9738 0 0 6.9738 389.9338 283.2377 Tm
(k)Tj
/F1 1 Tf
.4715 0 TD
()Tj
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[(4,)-336.9(we)-337.3(focus)-338.8(the)-336.9(descrip)-9.1(tion)-332.9(on)]TJ
/F4 1 Tf
12.7924 0 TD
(k)Tj
/F6 1 Tf
.6373 0 TD
(=)Tj
/F1 1 Tf
.717 0 TD
(1.)Tj
-13.1452 -1.2007 TD
[(We)-472.5(benchmar)-11.3(ked)-470.7(the)-467.8(sensit)-8.6(ivity)-473(of)-464.4(the)-473.5(different)-477.1(models)-474.8(against)-470.9(rando)-7.6(mized)-474.6(models)-474.8(in)-465.2(which)-473.2(both)]TJ
-1.0015 -1.2007 TD
[(topol)-9.3(ogy)-303.9(and)-305.7(logic)-307(function)-9.7(s)-297.9(are)-309.4(rando)-7.6(mized.)-310(Specically)-10.9(,)-302(the)-308.4(topology)-307.5(was)-305.1(rando)-7.6(mized)-309.6(while)-305.3(pres)-8(erving)]TJ
0 -1.195 TD
[(in-)-416.7(and)-413.8(out-degr)-8.9(ees)-416.8(using)-411.8(the)-416.6(switchi)-9.1(ng)-411.2(method)-416.7(wi)-8.2(th)-408.3(10)]TJ
/F4 1 Tf
23.6499 0 TD
(n)Tj
/F1 1 Tf
.9162 0 TD
[(switch)-415.3(attempt)-9.3(s;)-415.2(the)-410.9(Boo)-7(lean)-413.4(functi)-8.2(ons)-413.2(were)]TJ
-24.5661 -1.2007 TD
[(rando)-7.6(mized)-286.8(while)-288.2(preserving)-292.2(the)-280(numb)-6.6(er)-284.1(of)-282.3(1s)-281.6(in)-277.4(their)-286.2(tru)-7(th)-277.4(tables)-288(by)-280.4(perm)-10.3(uting)-282.5(table)-286.1(rows.)-283.7(We)-284.7(gener)-9.4(ated)]TJ
T*
[(40)-337.3(rando)-7.6(m)-334.3(models)-338.2(for)-336.3(each)-338.5(real)-341.4(model.)]TJ
1.0015 -1.2007 TD
[(We)-261.9(split)-263.7(the)-262.9(processi)-11.1(ng)-257.6(jobs)-263.1(among)-265.2(ve)-259.4(serv)-8(ers.)-258(Two)-267.8(with)-262.9(72)-257.6(CPUs)-262.5(\(2300)-170.8(MH)-7.4(z\))-261.4(and)-260.2(258)-264.1(GB)-260.6(RAM,)-263.6(and)]TJ
-1.0015 -1.2007 TD
[(the)-257.2(remainin)-9.3(g)-245.5(thr)-7(ee)-249.8(with)-251.5(40)-251.9(CPUs)-256.8(\(2800)-170.8(MHz\))-257.4(and)-248.8(258)-252.7(GB)-254.9(RAM.)-252.3(When)-257.8(running)-254.8(the)-251.5(stochasti)-9.7(c)-247.2(algo)-7.6(rithms,)]TJ
T*
[(we)-269(ran)-262.1(itera)-9.2(tions)-266.6(in)-260.4(para)-8.6(llel,)-263.9(and)-265.8(let)-265.7(the)-262.9(ILP)-265.1(solv)-7(er)-261.4(\(G)-7.4(urobi\))-267.4(utilize)-264.6(all)-265.7(core)-8.6(s.)-264.1(Table)-266(1)-262.5(provid)-9.3(es)-260.6(runni)-8.6(ng)-263.3(time)]TJ
T*
[(sta)-8(tistics.)-259.9(Running)-257.5(time)-8.2(s)-252.3(varied)-257.6(consi)-9.6(derably)-258.3(betwee)-10.9(n)-251.2(mode)-8.4(ls,)-250.6(and)-260.2(thus,)-257.8(we)-251.9(set)-258.4(a)-252.9(time)-8.2(-out)-252.5(of)-259.6(90)-251.9(minut)-7.2(es)-254.9(\(in)]TJ
T*
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0 -1.195 TD
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0 -1.2007 TD
[(we)-451.1(collecte)-10.1(d)-438.9(resu)-8(lts)-441.5(for)-450.1(30)-439.7(of)-447.4(the)-445(70)-445.4(models)-8.1(.)-438.6(T)0(h)-7(e)-440.7(sourc)-8.8(e)-440.7(code)-450.5(used)-444.2(for)-444.4(the)-450.7(analysis)-450.1(can)-444(be)-447.2(found)-449.7(at)]TJ
T*
[(git)-7.8(hub.com/)-8.1(arielbro/at)-11.7(tractor_lear)-13.4(ning,)-336.2(com)-7.6(mit)-335.7(hash)-336.1(cd0874b)-7.1(6103f1)-6.6(42ae3b)-8.1(b828e56)-7.1(5e6162e)-8.9(7ff2f0e)]TJ
/F4 1 Tf
10.959 0 0 10.959 74.0976 464.4282 Tm
[(4.2.)-502(Real)-330.8(models)-329.8(are)-333.8(more)-334.9(robust)-330.5(than)-329.7(their)-335.7(randomized)-327.9(counterparts)]TJ
/F1 1 Tf
9.9626 0 0 9.9626 84.0755 446.5133 Tm
[(We)-375.7(applied)-378.6(our)-374.2(robustne)-8.1(ss)-368.1(com)-7.6(putation)-380.4(pipeline)-376.5(to)-374.2(the)-371(30)-371.4(mode)-8.4(ls)-369.7(describ)-9.1(ed)-373.2(above)-377.3(and)-374(compared)-381.2(the)]TJ
-1.0015 -1.2007 TD
[(resu)-8(lts)-373.2(with)-376.7(thos)-7(e)-372.4(obtaine)-8.7(d)-370.7(o)0(n)-371.4(rando)-7.6(mized)-377.9(mode)-8.4(ls.)-375.7(When)-377.3(applying)-377.6(the)-376.7(ILP-ba)-9.3(sed)-375.2(sensitivity)-379.2(anal)-9.4(ysis)-372.4(of)]TJ
T*
[(mode)-8.4(l)-276.7(change)-9.9(s,)-275.5(we)-280.4(obser)-8.8(ved)-277.2(that)-283.5(in)-277.4(almost)-284.1(all)-282.7(cases)-284.7(\(for)-276.4(both)-284.7(real)-278.8(and)-282.9(random)-279.3(mode)-8.4(ls\))-281.4(a)-275.7(sing)-7(le)-279.2(bit)-275.3(change)]TJ
T*
[(was)-401.8(enough)-393.3(to)-396.9(elimina)-9.3(te)-393(the)-393.8(entire)-396.9(attra)-9.2(ctor)-396.5(space)-397.3(of)-390.5(a)-395.2(mode)-8.4(l.)-390.9(Thus,)-397.1(we)-394.2(com)-7.6(puted)-395.3(an)-396(alternative)-404.1(score)]TJ
T*
[(whe)-8(re)-318.3(the)-319.8(change)-328.6(is)-318.5(const)-9.6(rained)-320.2(to)-323(a)-315.5(speci)-11.3(c)-315.5(nod)-8(e)-315.5(a)0(t)-319.1(a)-321.2(time)-8.2(,)-313.4(produc)-8.3(ing)-318(a)-321.2(sensit)-8.6(ivity)-319.4(scor)-8(e)-315.5(for)-324.9(each)-321.4(of)-322.2(the)]TJ
T*
[(mode)-8.4(ls)-338.3(nodes,)-337.2(and)-334.1(modi)-9.3(ed)-333.4(the)-336.9(stochast)-11.9(ic)-330.4(v)-6.5(ersion)-336.7(similar)-10.7(ly.)]TJ
1.0015 -1.2007 TD
[(The)-288.4(d)-6.5(istribution)-8.3(s)-286.5(o)0(f)-288(the)-285.7(four)-291.5(scor)-8(e)-287.1(variants)-291.5(on)-291.8(real)-290.2(and)-288.6(random)-290.7(mode)-8.4(ls)-284.3(are)-292.4(given)-292.9(in)-283.1(Figur)-7.8(e)-287.1(1)0(.)-285.7(Evide)-8.2(ntly,)]TJ
-1.0015 -1.195 TD
[(real)-375.6(models)-372.3(have)-370.8(lower)-372.8(sensit)-8.6(ivity)-370.6(scores)-371(than)-371.8(rando)-7.6(m)-368.5(models)-372.3(across)-371(all)-368.1(scor)-8(e)-366.7(variant)-10.7(s.)-366.5(To)-365.5(qu)-7.2(antify)-372.4(the)]TJ
0 -1.2007 TD
[(devi)-7.6(ation)-335.5(of)-333.6(the)-331.2(real)-335.7(models)-338.2(from)-334.9(the)-331.2(correspo)-9.1(nding)-336.6(distribut)-9.7(ions)-331.4(of)-333.6(random)-336.2(scores,)-337.2(we)-331.6(used)-336.1(a)-332.6(Wilcox)-8.7(on)]TJ
T*
[(sign)-7(ed-rank)-479.5(tes)-8(t,)-470.5(whe)-8(re)-477.6(the)-479.2(score)-477.2(of)-475.8(each)-480.7(real)-478(mode)-8.4(l)-475.8(was)-475.8(paired)-485.2(with)-473.4(the)-479.2(mea)-9.4(n)-473.1(score)-477.2(o)-6.5(f)-469.4(it)-7.1(s)-474.3(random)]TJ
T*
[(count)-8.4(erpart)-9.2(s.)-292.6(For)-293.7(the)-297.1(node-)-7.6(based)-298.8(model)-296.4(perturb)-8.1(ation)-295.7(scor)-8(es,)-295.1(we)-297.5(paired)-297.4(the)-297.1(mean)-294.7(scor)-8(e)-292.8(o)0(f)-293.7(vertic)-10(es)-294.7(in)-294.5(a)-292.8(real)]TJ
T*
[(mode)-8.4(l)-333.6(with)-336.9(the)-336.9(mean)-340.2(scores)-342.6(of)-333.6(nodes)-336.8(in)-334.3(its)-339.1(random)-336.2(count)-8.4(erpart)-9.2(s.)]TJ
1.0015 -1.2007 TD
[(The)-311.2(com)-7.6(parison)-314.7(results)-312.9(are)-315.1(summa)-8.1(rized)-313.7(in)-311.6(Table)-311.6(2,)-308.4(reve)-8.6(aling)-312.7(that)-312(in)-305.9(all)-311.2(case)-9.6(s)-309.2(the)-308.4(real)-313(mode)-8.4(ls)-307.1(are)-315.1(more)]TJ
-1.0015 -1.2007 TD
[(robus)-7(t)-356.3(t)0(o)-357.1(change)-9.9(s)-360.5(than)-360.4(their)-360.2(rando)-7.6(mized)-360.8(instanc)-9.9(es.)-357.7(These)-366.9(difference)-12.1(s)-354.8(were)-365.3(signica)-9.9(nt)-357.1(for)-359.1(all)-362.4(score)-363.4(types)]TJ
T*
(\()Tj
/F4 1 Tf
.3357 0 TD
(p)Tj
/F10 1 Tf
.7796 0 TD
(<)Tj
/F1 1 Tf
1.0471 0 TD
(0)Tj
/F10 1 Tf
.5065 0 TD
(:)Tj
/F1 1 Tf
.2675 0 TD
[(05\),)-266.4(and)-271.5(especia)-10.7(lly)-263.9(for)-268(the)-268.6(state)-270.2(changes)-268(\()]TJ
/F4 1 Tf
16.7302 0 TD
(p)Tj
/F10 1 Tf
.7796 0 TD
(<)Tj
/F1 1 Tf
1.0528 0 TD
(0)Tj
/F10 1 Tf
.5008 0 TD
(:)Tj
/F1 1 Tf
.2675 0 TD
[(005\),)-267.2(as)-272(also)-264.9(evid)-7.6(ent)-262.9(from)-272.3(the)-268.6(box)-264.1(plots)-266.6(in)-266.1(Figur)-7.8(e)-264.3(2)0(.)]TJ
/F4 1 Tf
10.959 0 0 10.959 74.0976 255.2315 Tm
[(4.3.)-502(Stochastic)-327(versus)-332.5(optimization-based)-330(sensitivity)-334.7(analysis)]TJ
/F1 1 Tf
9.9626 0 0 9.9626 84.0755 237.2598 Tm
[(It)-262.3(is)-267.2(interesting)-270.9(to)-260.4(com)-7.6(pare)-264.7(the)-262.9(stoc)-8.8(hastic)-265.2(scor)-8(es)-260.6(that)-266.4(mea)-9.4(sure)-264.1(the)-262.9(mean)-266.2(sensi)-10.7(tivity)-266(to)-260.4(a)-264.3(change)-266(vers)-8(us)-264.5(the)]TJ
-1.0015 -1.195 TD
[(ILP)-9(-based)-318.6(optimizati)-11.5(on)-308.8(scor)-8(es)-311.8(that)-317.6(measur)-9.1(e)-309.8(the)-314.1(extrem)-10.7(es.)-312.2(As)-313.9(an)-316.3(exampl)-8.7(e,)-310.2(Th)-7(e)-309.8(major)-8.2(ity)-315.1(of)-310.8(models)-315.4(h)-6.5(a)0(d)-310.6(a)0(n)]TJ
0 -1.2007 TD
[(op)-7.2(timization)-387(score)-380.4(of)-379.1(1)-382(for)-376.1(attra)-9.2(ctor)-379.5(elim)-8.2(ination,)-380(bu)-7.2(t)-373.4(les)-8(s)-377.5(than)-377.5(5)]TJ
/F4 1 Tf
26.7114 0 TD
(%)Tj
/F1 1 Tf
1.2121 0 TD
[(of)-379.1(the)-382.4(mode)-8.4(ls)-375.4(had)-385.3(the)-376.7(correspo)-9.1(nding)]TJ
-27.9235 -1.2007 TD
[(stoc)-8.8(hastic)-362(score)-363.4(of)-356.3(1,)-359.7(sugges)-8.7(ting)-355.7(that)-363.2(there)-359.2(is)-358.3(a)-361.1(marked)-366.5(difference)-364.9(betwee)-10.9(n)-359.3(a)-355.4(rando)-7.6(m)-357.1(perturb)-8.1(ation)-358.2(to)-357.1(the)]TJ
ET
173.764 150.746 231.137 .51024 re
f
173.764 81.071 231.137 .51025 re
f
173.764 132.321 231.137 .51022 re
f
BT
/F3 1 Tf
8.9663 0 0 8.9663 180.2267 167.9244 Tm
(Table)Tj
/F1 1 Tf
3.0034 0 TD
(1.)Tj
/F3 1 Tf
1.2456 0 TD
[(Running)-338.3(Times)-336.1(in)-331.6(Seconds)-334.8(of)-335.1(the)-333.5(Different)]TJ
3.9265 -1.2266 TD
[(Scoring)-331.3(Variants)]TJ
/F4 1 Tf
-8.8963 -2.0233 TD
[(Variant)-9970.5(Median)-2699.2(Mean)-2816.3(Max)]TJ
/F1 1 Tf
0 -2.0739 TD
[(Model)-333.5(change,)-336(ILP)-6689.2(0)-5121.1(1)-4096.8(11)]TJ
0 -1.2266 TD
[(Model)-333.5(change,)-336(stochastic)-3734.7(27)-4621.1(83)-3090.9(1917)]TJ
0 -1.2203 TD
[(State)-331.5(change,)-336(ILP)-6796.7(14)-4121.6(322)-3090.4(6728)]TJ
T*
[(State)-331.5(change,)-336(stochastic)-4841.2(6)-4621.6(3)0(8)-3090.9(1371)]TJ
7.9702 0 0 7.9702 181.7007 70.1291 Tm
[(ILP,)-328(integer)-333.7(linear)-328.6(programm)13.8(ing.)]TJ
/F2 1 Tf
9.9626 0 0 9.9626 74.0976 737.4045 Tm
[(13)-7.2(8)-30433.8(BRU)-9.2(NER)-339.2(AND)-338.4(SHAR)-8(AN)]TJ
ET
endstream
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BT
/F1 1 Tf
9.9626 0 0 9.9626 107.6031 357.5621 Tm
0 0 0 rg
0 Tc
0 Tw
[(logic)-386.7(of)-390.5(a)-383.8(model)-387.5(and)-391(a)-383.8(target)-10(ed)-384.6(one.)-385.7(Fo)-6.4(r)-384(individu)-8.7(al)-381.7(nodes,)-394.1(there)-387.7(was)-390.5(a)-383.8(correl)-9.2(ation)-386.7(of)-390.5(0)]TJ
/F10 1 Tf
36.5789 0 TD
(:)Tj
/F1 1 Tf
.2675 0 TD
[(20)-388.5(between)-393(the)]TJ
-36.8463 -1.2007 TD
[(stochasti)-9.7(c)-252.9(and)-254.5(ILP-ba)-9.3(sed)-255.7(variant)-10.7(s.)-252.7(For)-253.9(state)-258.8(perturb)-8.1(ations,)-258.1(there)-256.8(was)-259.6(a)-252.9(strong)-260.9(correlation)-265(\(0)]TJ
/F10 1 Tf
36.5049 0 TD
(:)Tj
/F1 1 Tf
.2731 0 TD
[(74\))-254.7(betwee)-10.9(n)-251.2(the)]TJ
-36.778 -1.2007 TD
[(stochasti)-9.7(c)-275.7(and)-271.5(ILP-ba)-9.3(sed)-272.7(variant)-10.7(s.)-269.8(Th)-7(ese)-274.5(relations)-10.5(hips)-274.5(are)-275.3(visu)-7(alized)-277.2(in)-271.8(Figur)-7.8(e)-270(3)0(.)-274.3(The)-277(ILP-ba)-9.3(sed)-272.7(scor)-8(es)-272(are)]TJ
T*
[(not)-318(a)-315.5(perf)-8.8(ect)-315.9(upper)-320.6(bound)-322.5(for)-319.2(the)-319.8(stochasti)-9.7(c)-315.5(ones)-319(sinc)-8.8(e)-315.5(they)-320.6(assume)-321.3(a)-321.2(restricted)-328(path)-314.9(leng)-7.6(th)-317.3(to)-317.3(a)-315.5(modi)-9.3(ed)]TJ
T*
[(attract)-9.6(or,)-333.9(but)-335.1(are)-337.9(almost)-341(always)-340.4(highe)-8.4(r)-332.8(for)-336.3(the)-336.9(same)-335.7(mode)-8.4(l.)]TJ
/F2 1 Tf
10.959 0 0 10.959 275.9243 273.8834 Tm
[(5.)-496.7(CONCLUSIONS)]TJ
/F1 1 Tf
9.9626 0 0 9.9626 117.5811 249.959 Tm
[(We)-244.9(have)-245.6(pres)-8(ented)-243.5(a)-247.2(rst)-241.5(anal)-9.4(ysis)-241.5(of)-242.5(the)-245.8(attra)-9.2(ctor)-242.9(land)-7.6(scape)-249.4(of)-242.5(Boolea)-9.9(n)-239.8(mode)-8.4(ls)-238.8(of)-248.2(cellular)-249.3(circuits)-248.7(and)-248.8(its)]TJ
-1.0015 -1.2007 TD
[(sensit)-8.6(ivity)-473(to)-470.9(state)-475(and)-476.4(logic)-472.1(change)-9.9(s.)-469(Our)-474.8(analysi)-9.9(s)-468.6(com)-7.6(bined)-475(novel)-475(ILP)-469.9(formul)-9.1(ations)-474(with)-473.4(stoc)-8.8(hastic)]TJ
T*
[(simulat)-8.7(ions)-411(to)-408.3(overc)-9.4(ome)-411.6(the)-410.9(inhe)-7.6(rent)-413.6(comple)-8.7(xity)-406.9(of)-413.2(the)-410.9(probl)-8.6(em.)-411.2(We)-409.9(found)-415.5(that)-414.4(real)-409.7(mode)-8.4(ls)-409.5(are)-411.9(mor)-7.8(e)]TJ
T*
[(robus)-7(t)-401.9(than)-405.9(their)-411.4(randomi)-9.7(zed)-404.2(count)-8.4(erparts)-409.5(with)-410.8(respect)-409.3(to)-408.3(both)-404.2(state)-412.4(and)-408.1(logic)-403.8(change)-9.9(s.)-406.4(The)-407.9(signica)-9.9(nt)]TJ
T*
[(difference)-290.9(betwee)-10.9(n)-279.6(real)-284.5(and)-282.9(rando)-7.6(m)-277.4(mode)-8.4(ls,)-279(acro)-8.6(ss)-282.7(the)-280(four)-285.8(measur)-9.1(es,)-283.7(supports)-286.4(the)-280(motiv)-7.2(ating)-284.3(assumpt)-10.1(ion)]TJ
T*
[(that)-334.7(the)-336.9(abst)-8.8(ract)-335.7(notion)-340.2(of)-333.6(robus)-7(tness)-335.1(is)-335.5(capture)-9.7(d)-336.5(i)0(n)-334.3(the)-336.9(dynamics)-341.5(of)-333.6(Boo)-7(lean)-333.7(mode)-8.4(ls.)]TJ
ET
q
430.485 0 0 296.107 107.66 426.16 cm
/Im1 Do
Q
BT
/F2 1 Tf
8.9663 0 0 8.9663 107.6031 410.1731 Tm
[(FIG.)-353.7(1.)]TJ
/F1 1 Tf
4.1352 0 TD
[(Distribution)-356.5(of)-355.7(model)-355.3(scores)-361(for)-351.5(each)-359.4(score)-360.3(variant,)-356.4(comparing)-358.4(real)-357.3(and)-355(random)-353.4(models.)]TJ
/F2 1 Tf
36.9258 0 TD
(\(a\))Tj
/F1 1 Tf
1.5238 0 TD
[(ILP)-353.6(per-node)]TJ
-42.5848 -1.2203 TD
[(scores)-348.3(for)-351.5(model)-349(perturbation.)]TJ
/F2 1 Tf
12.728 0 TD
(\(b\))Tj
/F1 1 Tf
1.5681 0 TD
[(Stochastic)-349.3(per-node)-347.6(scores)-354.6(for)-351.5(model)-349(perturbation.)]TJ
/F2 1 Tf
21.1059 0 TD
(\(c\))Tj
/F1 1 Tf
1.4669 0 TD
[(ILP)-347.3(scores)-348.3(for)-351.5(state)-353.1(pertur-)]TJ
-36.8689 -1.2266 TD
(bation.)Tj
/F2 1 Tf
3.0919 0 TD
(\(d\))Tj
/F1 1 Tf
1.5554 0 TD
[(Stochastic)-336.7(scores)-335.7(for)-332.5(state)-334.1(perturbation.)-335.3(ILP,)-331.2(integer)-340.9(linear)-335(programming.)]TJ
ET
207.213 140.315 231.137 .51024 re
f
207.213 70.639 231.137 .45358 re
f
207.213 121.89 231.137 .51022 re
f
BT
/F3 1 Tf
8.9663 0 0 8.9663 214.4125 157.4362 Tm
(Table)Tj
/F1 1 Tf
3.0034 0 TD
(2.)Tj
/F3 1 Tf
1.2456 0 TD
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-6.9299 -2.0233 TD
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(p)Tj
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T*
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/F2 1 Tf
9.9626 0 0 9.9626 107.6031 737.4045 Tm
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