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Previous work has established that RNNs with an unbounded activation functionhave the capacity to count exactly. However, it has also been shown that RNNsare challenging to train effectively and generally do not learn exact countingbehaviour. In this paper, we focus on this problem by studying the simplestpossible RNN, a linear single-cell network. We conduct a theoretical analysisof linear RNNs and identify conditions for the models to exhibit exact countingbehaviour. We provide a formal proof that these conditions are necessary andsufficient. We also conduct an empirical analysis using tasks involving aDyck-1-like Balanced Bracket language under two different settings. We observethat linear RNNs generally do not meet the necessary and sufficient conditionsfor counting behaviour when trained with the standard approach. We investigatehow varying the length of training sequences and utilising different targetclasses impacts model behaviour during training and the ability of linear RNNmodels to effectively approximate the indicator conditions.

Theoretical Conditions and Empirical Failure of Bracket Counting on Long Sequences with Linear Recurrent Networks