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Is there a set of program synthesis problems (or a source of problems) accepted by academics as a reasonable benchmark for measuring progress these days? Or does each new approach find its own example problems to demonstrate their particular differences on?

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SyGuS competition (SyGuS-Comp)[1] is a widely-used collection of such problems, but of course there will always be problem domains that are not well-represented by standard benchmarks.

1. https://sygus.org/

Perhaps https://ml4code.github.io/tags.html#benchmark

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