Incorporating syntax into neural approaches in NLP has a multitude of practical and scientific benefits. For instance, a language model that is syntax-aware is likely to be able to produce better samples; even a discriminative model like BERT with a syntax module could be used for core NLP tasks like unsupervised syntactic parsing. Rapid progress in recent years was arguably spurred on by the empirical success of the Parsing-Reading-Predict architecture of (Shen et al., 2018a), later simplified by the Order Neuron LSTM of (Shen et al., 2019). Most notably, this is the first time neural approaches were able to successfully perform unsupervised syntactic parsing (evaluated by various metrics like F-1 score). However, even heuristic (much less fully mathematical) understanding of why and when these architectures work is lagging severely behind. In this work, we answer representational questions raised by the architectures in (Shen et al., 2018a, 2019), as well as some transition-based syntax-aware language models (Dyer et al., 2016): what kind of syntactic structure can current neural approaches to syntax represent? Concretely, we ground this question in the sandbox of probabilistic context-free-grammars (PCFGs), and identify a key aspect of the representational power of these approaches: the amount and directionality of context that the predictor has access to when forced to make parsing decision. We show that with limited context (either bounded, or unidirectional), there are PCFGs, for which these approaches cannot represent the max-likelihood parse; conversely, if the context is unlimited, they can represent the max-likelihood parse of any PCFG.