We introduce a framework in which production-rule based computational cognitive modeling and Reinforcement Learning can systematically interact and inform each other. We focus on linguistic applications because the sophisticated rule-based cognitive models needed to capture linguistic behavioral data promise to provide a stringent test suite for RL algorithms, connecting RL algorithms to both accuracy and reaction-time experimental data. Thus, we open a path towards assembling an experimentally rigorous and cognitively realistic benchmark for RL algorithms. We extend our previous work on lexical decision tasks and tabular RL algorithms (Brasoveanu and Dotlačil, 2020b) with a discussion of neural-network based approaches, and a discussion of how parsing can be formalized as an RL problem.