# Generalizing Outside the Training Set: When Can Neural Networks Learn Identity Effects?

- Simone Brugiapaglia, Mathematics, Concordia University, Montreal, Quebec, Canada
- Paul Tupper, Dept of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada
- Matthew Liu, Mathematics and Statistics, Concordia University, Montreal, Quebec, Canada

**Abstract**Often in language and other areas of cognition, whether two components of an object are identical or not determine whether it is well formed. We call such constraints identity effects. When developing a system to learn well-formedness from examples, it is easy enough to build in an identify effect. But can identity effects be learned from the data without explicit guidance? We provide a simple framework in which we can rigorously prove that algorithms satisfying simple criteria cannot make the correct inference. We then show that a broad class of algorithms including deep neural networks with standard architecture and training with backpropagation satisfy our criteria, dependent on the encoding of inputs. Finally, we demonstrate our theory with computational experiments in which we explore the effect of different input encodings on the ability of algorithms to generalize to novel inputs.