In recent years, large language models (LLMs)  have had a dramatic impact on various sub-fields of AI, most notably on natural language understanding tasks. However, there is widespread agreement that the logical reasoning capabilities of contemporary LLMs are, at bestfragmentary(i.e., may work well on some problem instances but fail dramatically on others) . While traditional LLM fine-tuning approaches (e.g., those that use human feedback)  do address this problem to some degree, they suffer from many issues, including unsound black-box reward models, difficulties in collecting preference data, and sparse scalar reward values. To address these challenges, we propose a new training/fine-tuning paradigm we refer to as Reinforcement Learning via Symbolic Feedback (RLSF) , which is aimed at enhancing the reasoning capabilities of LLMs. In the RLSF setting, the LLM that is being trained/fine-tuned is considered as the RL agent, while the environment is allowed access to reasoning or domain knowledge tools (e.g., solvers, algebra systems) . Crucially, in RLSF, these reasoning tools can provide feedback to the LLMs viapoly-sized certificates(e.g., proofs) , that characterize errors in the LLM-generated object with respect to some correctness specification. The ability of RLSF-based training/fine-tuning to leverage certificate-generating symbolic tools enables sound fine-grained (token-level)  reward signals to LLMs, and thus addresses the limitations of traditional reward models mentioned above. Via extensive evaluations, we show that our RLSF-based fine-tuning of LLMs outperforms traditional approaches on two different applications, namely, program synthesis from natural language pseudo-code to programming language (C++) , and solving the Game of 24. For the former, RLSF-tuned LLMs show a significant improvement (e.g., +52.64% in compilation accuracy and +31.43% in functional correctness for Google’s CodeGemma-2b)  compared to supervised fine-tuning. Further, CodeGemma-2b gets +34.82% in compilation accuracy and +17.01% in functional correctness compared to ChatGPT (100larger) . Similarly, for the Game of 24, we observed +25% success rate using Meta’s Llama2-7b compared to traditional methods, and +7% success rate compared to ChatGPT (25larger) . A takeaway is that fine-tuning via RLSF enables relatively smaller LLMs to significantly outperform orders of magnitude larger models (e.g., ChatGPT) .