How to use SimpleRNN to model a modulus function & predict many to many sequence - keras

Question

I have designed this toy problem to understand the working of SimpleRNN in Keras.
My input sequence is:
[x1,x2,x3,x4,x5]
and the corresponding output is:
[(0+x1)%2,(x1+x2)%2,(x2+x3)%2,(x3+x4)%2,(x4+x5)%2)]
My code is:
import numpy as np
import random
from scipy.ndimage.interpolation import shift
def generate_sequence():
max_len = 5
x = np.random.randint(1,100,max_len)
shifted_x = shift(x, 1, cval=0)
y = (x + shifted_x) % 2
return x.reshape(max_len,1),y.reshape(max_len,1),shifted_x.reshape(max_len,1)
X_train = np.zeros((100,5,1))
y_train = np.zeros((100,5,1))
for i in range(100):
x,y,z = generate_sequence()
X_train[i] = x
y_train[i] = y
X_test = np.zeros((100,5,1))
y_test = np.zeros((100,5,1))
for i in range(100):
x,y,z = generate_sequence()
X_test[i] = x
y_test[i] = y
from keras.layers import SimpleRNN
model = Sequential()
model.add(SimpleRNN(3,input_shape=(5,1),return_sequences=True,name='rnn'))
model.add(Dense(1,activation='sigmoid'))
# try using different optimizers and different optimizer configs
model.compile(loss='binary_crossentropy',
optimizer='sgd',
metrics=['accuracy'])
print('Train...')
model.fit(X_train, y_train,
batch_size=70,
epochs=200,verbose=0,validation_split=0.3)
score, acc = model.evaluate(X_test, y_test,
batch_size=batch_size)
print('Test score:', score)
print('Test accuracy:', acc)
When I train this SimpleRNN I only get an accuracy of 50%, each item in the sequence only depends on the previous item. Why is the RNN struggling to learn this?
100/100 [==============================] - 0s 37us/step
Test score: 0.6975522041320801
Test accuracy: 0.5120000243186951
UPDATE:
It turns out mod function is very hard to model, I switched to simple data generation strategy like y[t] = x[t] < x[t-1], then I could see the model performing with 80% binary accuracy.
def generate_rnn_sequence():
max_len = 5
x = np.random.randint(1,100,max_len)
shifted_x = shift(x, 1, cval=0)
y = (x < shifted_x).astype(float)
return x.reshape(5,1),y.reshape(5,1)
So How do i model a mod function using a RNN?