AbstractStatic type errors are a common stumbling block for newcomers to typed functional languages. We present a dynamic approach to explaining type errors by generating counterexample witness inputs that illustrate how an ill-typed program goes wrong. First, given an ill-typed function, we symbolically execute the body to synthesize witness values that make the program go wrong. We prove that our procedure synthesizes general witnesses in that if a witness is found, then for all inhabited input types, there exist values that can make the function go wrong. Second, we show how to extend this procedure to produce a reduction graph that can be used to interactively visualize and debug witness executions. Third, we evaluate the coverage of our approach on two data sets comprising over 4,500 ill-typed student programs. Our technique is able to generate witnesses for around 85% of the programs, our reduction graph yields small counterexamples for over 80% of the witnesses, and a simple heuristic allows us to use witnesses to locate the source of type errors with around 70% accuracy. Finally, we evaluate whether our witnesses help students understand and fix type errors, and find that students presented with our witnesses show a greater understanding of type errors than those presented with a standard error message.
Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System