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While forward reasoning (i.e. find the answer given the question) has beenexplored extensively in the recent literature, backward reasoning is relativelyunexplored. We examine the backward reasoning capabilities of LLMs on Math WordProblems (MWPs): given a mathematical question and its answer, with somedetails omitted from the question, can LLMs effectively retrieve the missinginformation?  In this paper, we formally define the backward reasoning task on math wordproblems and modify three datasets to evaluate this task: GSM8k, SVAMP andMultiArith. Our findings show a significant drop in the accuracy of models onbackward reasoning compared to forward reasoning across four SOTA LLMs (GPT4,GPT3.5, PaLM-2, and LLaMa-2). Utilizing the specific format of this task, wepropose three novel techniques that improve performance: Rephrase reformulatesthe given problem into a forward reasoning problem, PAL-Tools combines the ideaof Program-Aided LLMs to produce a set of equations that can be solved by anexternal solver, and Check your Work exploits the availability of naturalverifier of high accuracy in the forward direction, interleaving solving andverification steps. Finally, realizing that each of our base methods correctlysolves a different set of problems, we propose a novel Bayesian formulation forcreating an ensemble over these base methods aided by a verifier to furtherboost the accuracy by a significant margin. Extensive experimentationdemonstrates that our techniques successively improve the performance of LLMson the backward reasoning task, with the final ensemble-based method resultingin a substantial performance gain compared to the raw LLMs with standardprompting techniques such as chain-of-thought.

Fill in the Blank: Exploring and Enhancing LLM Capabilities for Backward Reasoning in Math Word Problems