We argue that symmetry is an important consideration in addressing the problem of systematicity and investigate two forms of symmetry relevant to symbolic processes. We implement this approach in terms of convolution and show that it can be used to achieve effective generalisation in three toy problems: rule learning, composition and grammar learning.
======================================== Update after rebuttal =============================================
I have now read the author rebuttal, but my concerns about the paper remain. The training details are not described in anywhere near sufficient detail (optimizer?, batch size?, learning rate?, initialization?, etc). The baseline architectures “recurrent net” or the “multi-layer perceptron” are not described at all, despite my explicit request to that effect. I had also requested to see the source code for the experiments as this would perhaps have illuminated a lot of the details left out in the paper, but the authors have not provided it. I understand that the authors are not required to provide their code, but this should have been a relatively straightforward request in this case given the simplicity of the experiments and as I mentioned in my initial review, it would have been very useful in evaluating the paper. 
In their rebuttal, the authors also claimed that the results in Fig. 1 and Table 1 are training results (that even though the architecture is "innate", the weights are learned), but I'm concerned about this claim. I happen to be doing some experiments along these lines at the moment, and it is not trivial at all to get such crisp results as those shown in Fig. 1 & Table 1 in these kinds of experiments (even when the architecture is correctly specified). Again, it would have been very helpful if the authors had either provided their source code or had described their experimental setup in sufficient detail to allow the reproduction of these results. 
Given these concerns, I have decided to keep my score as it is.
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This paper addresses an important problem: systematic generalization in neural networks. However, the paper is very confusing and I have some serious concerns about the models and the results presented in sections 3 and 4. Here are the main issues:
1) In section 3, there are only 10x10=100 possible combinations in this composition task. Yet, the paper says “we randomly sample 1000 such translation pairs, choose three combinations and remove all instances of them from the training data and then exclusively test on unseen pairings of command and modifier.” How can you sample 1000 pairs out of a possible 100 combinations? Also being able to generalize to 3 held-out combinations out of 100 is not very impressive. On the contrary, it is almost trivial.
2) No details are given about the “recurrent net” or the “multi-layer perceptron” baselines in section 3. What are these models? The fact that they have exactly zero accuracy is a bit suspicious, especially given the almost trivial nature of the task in section 3. Previous works reported perfect or near perfect accuracy with similar baselines in similar tasks (see e.g., Lake & Baroni, 2018).
3) I'm afraid the proposed model in section 3 also doesn’t make sense to me. It is explicitly acknowledged (Appendix B) that y is an M+1-dimensional vector, g is an Nx(M+1) matrix. Then by all accounts, the convolution of these should be an N-dimensional vector. Yet, somehow, h_t+1 in Equation 2 manages to be an NxM matrix. How? Please clarify this. If possible, making the source code available would be very helpful.
4) What is the semantics of x and y in section 3? What exactly are they supposed to be doing? This is not explained in the paper beyond a vague description.
5) Similar problems arise in section 4. The task is not explicitly described in the text. We only learn from Appendix C that it is actually to predict the next symbol. The task description mentions “strings of length 15, 17, 19, 21, 23 and 24,...” (p. 5). But, the grammar in Fig. 3 can only generate odd length strings, it cannot generate a string of length 24. Is this a typo?
6) Again in section 4, I have no idea how the proposed model is actually supposed to work. The motivation for the model and its description are not clear at all.
7) The model in section 2 is hand-coded. It is not shown that it can actually learn this solution from data. What happens if the sequences are longer or if the rules are different, for example? Then you have to hand-code a completely different architecture.
8) Which brings me to another important issue I have with this paper (and with similar papers): this whole set-up is very misguided in my mind. I think the real problem is not to come up with an architecture that would generalize systematically in a very specific (and usually toy) problem. It is to come up with an architecture that would learn to generalize systematically in a much broader set of problems. The learning aspect in sections 3-4 is a step in the right direction, but there’s no evidence that the models proposed there can learn to generalize in any task other than the very specific tasks they were designed for (if they can actually do that).