Large language models (LLMs) excel at implementing code from functionality descriptions, but struggle with algorithmic problems that require not only implementation but also identification of the suitable algorithm. Moreover, LLM-generated programs lack guaranteed correctness and require human verification. To address these challenges, we propose ALGO, a framework that synthesizes Algorithmic programs with LLM-Generated Oracles to guide the creation and verify their correctness. ALGO first generates a probably correct but possibly slow reference oracle by prompting an LLM to exhaustively enumerate all the combinations of relevant variables. This oracle is then utilized to guide an arbitrary search strategy in exploring the algorithm space and to verify the algorithms synthesized. Our study shows that the LLM-generated oracles are correct for 88% of the cases. With the oracles as verifiers, ALGO can be integrated with any existing code generation model in a model-agnostic manner to enhance its performance. Experiments show that when equipped with ALGO, we achieve an 8x better one-submission pass rate over the Codex model and a 2.6x better one-submission pass rate over CodeT, the current state-of-the-art model on CodeContests. We can also get 1.3x better pass rate over the ChatGPT Code Interpreter on unseen problems.
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