add hw9 and hw8

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diff --git a/hw9/12-11.jl b/hw9/12-11.jl
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+# my neural net from 12.11
+
+# first, store the memory of A as a lattice of 1, -1, where 1 is the memory and -1 is the absence of memory
+A = [
+	-1 -1 -1 -1 1 1 -1 -1 -1 -1;
+	-1 -1 -1 1 1 1 1 -1 -1 -1;
+	-1 -1 1 1 -1 -1 1 1 -1 -1;
+	-1 1 1 -1 -1 -1 -1 1 1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 1 1;
+	1 1 -1 -1 -1 -1 -1 -1 1 1;
+	-1 1 1 -1 -1 -1 -1 1 1 -1;
+	-1 -1 1 1 -1 -1 1 1 -1 -1;
+	-1 -1 -1 1 1 1 1 -1 -1 -1;
+	-1 -1 -1 -1 1 1 -1 -1 -1 -1
+]
+
+B = [
+	1 1 1 1 1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 1 1 1 -1 -1 -1 -1 -1;
+	1 1 1 1 1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 1 1 1 -1 -1 -1 -1 -1
+]
+
+C = [
+	-1 1 1 1 1 1 1 -1 -1 -1;
+	1 1 1 1 1 1 1 -1 -1 -1;
+	1 1 1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 1 1 1 1 1 -1 -1 -1;
+	-1 1 1 1 1 1 1 -1 -1 -1
+]
+
+D = [
+	1 1 1 1 1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 -1 -1 1 1 -1 -1 -1 -1;
+	1 1 1 1 1 -1 -1 -1 -1 -1
+]
+
+E_letter = [
+	1 1 1 1 1 1 1 1 1 1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 1 1 1 1 1 1 1 1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 1 1 1 1 1 1 1 1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 1 1 1 1 1 1 1 1
+]
+
+F = [
+	1 1 1 1 1 1 1 1 1 1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 1 1 1 1 1 1 1 1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1
+]
+
+G = [
+	1 1 1 1 1 1 1 1 1 1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 -1 -1 -1;
+	1 1 -1 -1 -1 -1 -1 1 1 1;
+	1 1 -1 -1 -1 -1 -1 -1 1 1;
+	1 1 -1 -1 -1 -1 -1 -1 1 1;
+	1 1 -1 -1 -1 -1 -1 -1 1 1;
+	-1 1 1 1 1 1 1 1 1 1;
+	-1 -1 1 1 1 1 1 1 1 1
+]
+
+encodings = [A, B, F, G]
+
+spin_numbers(row, col) = (row - 1) * 10 + col
+
+# create strength of interactions between ith and jth spinds in Ji,j
+J = zeros(100, 100)
+for m in 1:100 # row
+	for n in 1:100 # col
+		i = (m - 1) % 10 + 1
+		j = (n - 1) % 10 + 1
+		k = (m - 1) ÷ 10 + 1
+		l = (n - 1) ÷ 10 + 1
+
+		J[m, n] = 0
+		for encoding in encodings
+			J[m, n] += encoding[i, k] * encoding[j, l]
+		end
+		J[m, n] /= length(encodings)
+	end
+end
+
+# the energy function
+function energy(s, J)
+	E = 0
+	for i in 1:100
+		for j in 1:100
+			E += J[i, j] * s[i] * s[j]
+		end
+	end
+	return -E
+end
+
+# the update function (uses monte carlo steps)
+function monte_carlo(s, J)
+	for i in 1:100 # systematically go through each point in the lattice
+		# calculate the energy of the system
+		E = energy(s, J)
+		# randomly flip a spin
+		s[i] = -s[i]
+		# calculate the new energy of the system
+		E_new = energy(s, J)
+		# calculate the change in energy
+		dE = E_new - E
+		# if the change in energy is positive, flip the spin back
+		if dE > 0
+			s[i] = -s[i]
+		end
+	end
+	return s
+end
+
+# create the main function
+function main(s, J, nsteps)
+	E = energy(s, J)
+	for i in 1:nsteps
+		s = monte_carlo(s, J)
+		E = energy(s, J)
+	end
+	return s, E
+end
+
+# run the main function
+
+# randomly change some values in A to see if NN works
+function produce_test_arr(enc, prob_change = 0.1)
+	tmp = zeros(10, 10)
+	for i in 1:10
+		for j in 1:10
+			if rand() < prob_change
+				tmp[i, j] = rand([-1, 1]) # set some random values
+			else
+				tmp[i, j] = enc[i, j]
+			end
+		end
+	end
+	return tmp
+end
+
+function check_if_same(s, enc)
+	for i in 1:10
+		for j in 1:10
+			if s[i, j] != enc[i, j]
+				println("The neural net did not work")
+				return false
+			end
+		end
+	end
+	# println("The neural net worked")
+	return true
+end
+
+function apply_damage_to_J(J, prob_damage = 0.8)
+	println("Applying damage to J with probability ", prob_damage)
+	ret = copy(J)
+	for i in 1:100
+		for j in 1:100
+			if rand() < prob_damage
+				ret[i, j] = 0
+			else
+				ret[i, j] = J[i, j]
+			end
+		end
+	end
+	return ret
+end
+
+function run_tests(num_times, Js, MC_steps = 2)
+	num_correct = 0
+	num_total = 0
+
+	for i in 1:num_times
+		# randomly select a memory
+		enc = encodings[rand(1:length(encodings))]
+		# produce a test array
+		s = produce_test_arr(enc)
+		# run theNN 
+		s, E = main(s, Js, MC_steps)
+		# check if the NN worked
+		if check_if_same(s, enc)
+			num_correct += 1
+		end
+		num_total += 1
+	end
+
+	println("Number of correct tests: ", num_correct)
+
+	return num_correct / num_total
+end
+
+function run_tests_for_damage_range(damages, num_times, MC_steps = 2, init_J = J)
+	res = []
+
+	for damage in damages
+		damaged_J = apply_damage_to_J(init_J, damage)
+		p = run_tests(num_times, damaged_J, MC_steps)
+
+		push!(res, p)
+	end
+
+	return res
+end
+
+
+# J = apply_damage_to_J(J, 0.8)
+
+# s = produce_test_arr(F)
+# s, E = main(s, J, 2)
+# check_if_same(s, F)
+
+s = produce_test_arr(A)
+s, E = main(s, J, 2)
+check_if_same(s, A)
+
+# s = produce_test_arr(C, 0.0)
+# s, E = main(s, J, 2)
+# check_if_same(s, C)
+
+damage_range = collect(0.1:0.05:1.0)
+res = run_tests_for_damage_range(damage_range, 100)
+
+#plot the results
+using Plots
+plot(damage_range, res, xlabel = "Damage", ylabel = "Success rate", title = "NN success rate vs damage probability", marker = :circle, label = "MC steps = 2")
+
+# do the same for different number of MC steps
+res = run_tests_for_damage_range(damage_range, 100, 5)
+plot!(damage_range, res, xlabel = "Damage", ylabel = "Success rate", title = "NN success rate vs damage probability", marker = :circle, label = "MC steps = 5")
+
+res = run_tests_for_damage_range(damage_range, 100, 10)
+plot!(damage_range, res, xlabel = "Damage", ylabel = "Success rate", title = "NN success rate vs damage probability", marker = :circle, label = "MC steps = 10")
+
+res = run_tests_for_damage_range(damage_range, 100, 20)
+plot!(damage_range, res, xlabel = "Damage", ylabel = "Success rate", title = "NN success rate vs damage probability", marker = :circle, label = "MC steps = 20")
+
+savefig("12-11-all.png")
+
+
+