Learning and logic are distinct and remarkable approaches to prediction. Machine learning has experienced a surge in popularity because it is robust to noise and achieves high performance; however, ML experiences many issues with knowledge transfer and extrapolation. In contrast, logic is easily intepreted, and logical rules are easy to chain and transfer between systems; however, inductive logic is brittle to noise. We then explore the premise of combining learning with inductive logic into AI Reasoning Systems. Specifically, we summarize findings from PAC learning (conceptual graphs, robust logics, knowledge infusion) and deep learning (DSRL, $\partial$ILP, DeepLogic) by reproducing proofs of tractability, presenting algorithms in pseudocode, highlighting results, and synthesizing between fields. We conclude with suggestions for integrated models by combining the modules listed above and with a list of unsolved (likely intractable) problems. AI Reasoning Systems: PAC and Applied Methods