Abstraction, Reasoning and Deep Learning: A Study of the "Look and Say" Sequence

The ability to abstract, count, and use System 2 reasoning are well-known manifestations of intelligence and understanding. In this paper, we argue, using the example of the ``Look and Say" puzzle, that although deep neural networks can exhibit high `competence' (as measured by accuracy) when trained on large data sets (2M examples in our case), they do not show any sign on the deeper understanding of the problem, or what D. Dennett calls `comprehension'. We report on two sets experiments on the ``Look and Say" puzzle data. We view the problem as building a translator from one set of tokens to another. We apply both standard LSTMs and Transformer/Attention -- based neural networks, using publicly available machine translation software. We observe that despite the amazing accuracy (on both, training and test data), the performance of the trained programs on the actual L\&S sequence is bad. We then discuss a few possible ramifications of this finding and connections to other work, experimental and theoretical. First, from the cognitive science perspective, we argue that we need better mathematical models of abstraction. Second, the classical and more recent results on the universality of neural networks should be re-examined for functions acting on discrete data sets. Mapping on discrete sets usually have no natural continuous extensions. This connects the results on a simple puzzle to more sophisticated results on modeling of mathematical functions, where algebraic functions are more difficult to model than e.g. differential equations. Third, we hypothesize that for problems such as ``Look and Say", computing the parity of bitstrings, or learning integer addition, it might be worthwhile to introduce concepts from topology, where continuity is defined without the reference to the concept of distance.