Training and testing are done in alternating steps: In each epoch, for training, we first present to an LSTM network 1000 samples in a given language, which are generated according to a certain discrete probability distribution supported on a closed finite interval. We then freeze all the weights in our model, exhaustively enumerate all the sequences in the language by their lengths, and determine the first $k$ shortest sequences whose outputs the model produces inaccurately.  experimented with 1, 2, 3, and 36 hidden units for $a^n b^n$ ; 2, 3, 4, and 36 hidden units for $a^n b^n c^n$ ; and 3, 4, 5, and 36 hidden units for $a^n b^n c^n d^n$ .  Following the traditional approach adopted by BIBREF7 , BIBREF12 , BIBREF9 and many other studies, we train our neural network as follows. At each time step, we present one input character to our model and then ask it to predict the set of next possible characters, based on the current character and the prior hidden states. Given a vocabulary $\mathcal {V}^{(i)}$ of size $d$ , we use a one-hot representation to encode the input values; therefore, all the input vectors are $d$ -dimensional binary vectors. The output values are $(d+1)$ -dimensional though, since they may further contain the termination symbol $\dashv $ , in addition to the symbols in $\mathcal {V}^{(i)}$ . The output values are not always one-hot encoded, because there can be multiple possibilities for the next character in the sequence, therefore we instead use a $k$ -hot representation to encode the output values. Our objective is to minimize the mean-squared error (MSE) of the sequence predictions. 
 Question: What training settings did they try?
Output: Training and testing are done in alternating steps: In each epoch, for training, we first present to an LSTM network 1000 samples in a given language, which are generated according to a certain discrete probability distribution supported on a closed finite interval. We then freeze all the weights in our model, exhaustively enumerate all the sequences in the language by their lengths, and determine the first $k$ shortest sequences whose outputs the model produces inaccurately.  experimented with 1, 2, 3, and 36 hidden units for $a^n b^n$ ; 2, 3, 4, and 36 hidden units for $a^n b^n c^n$ ; and 3, 4, 5, and 36 hidden units for $a^n b^n c^n d^n$ .  Following the traditional approach adopted by BIBREF7 , BIBREF12 , BIBREF9 and many other studies, we train our neural network as follows. At each time step, we present one input character to our model and then ask it to predict the set of next possible characters, based on the current character and the prior hidden states. Given a vocabulary $\mathcal {V}^{(i)}$ of size $d$ , we use a one-hot representation to encode the input values; therefore, all the input vectors are $d$ -dimensional binary vectors. The output values are $(d+1)$ -dimensional though, since they may further contain the termination symbol $\dashv $ , in addition to the symbols in $\mathcal {V}^{(i)}$ . The output values are not always one-hot encoded, because there can be multiple possibilities for the next character in the sequence, therefore we instead use a $k$ -hot representation to encode the output values. Our objective is to minimize the mean-squared error (MSE) of the sequence predictions.