Many problems in programming language theory and formal methods are undecidable, so they cannot be solved precisely. Practical techniques for dealing with undecidable problems are often based on decidable approximations. Undecidability implies that those approximations are always imprecise. Typically, practitioners use heuristics and \emph{ad hoc} reasoning to identify imprecision issues and improve approximations, but there is a lack of computability-theoretic foundations about whether those efforts can succeed.

This paper shows a surprising interplay between undecidability and decidable approximations: there exists a class of undecidable problems, such that it is computable to transform any decidable approximation to a witness input demonstrating its imprecision. We call those undecidable problems \emph{witnessable problems}. For example, if a program property $P$ is witnessable, then there exists a computable function $f_P$, such that $f_P$ takes as input the code of any program analyzer targeting $P$ and produces an input program $w$ on which the program analyzer is imprecise. An even more surprising fact is that the class of witnessable problems includes almost all undecidable problems in programming language theory and formal methods. Specifically, we prove the diagonal halting problem $K$ is witnessable, and the class of witnessable problems is closed under complements and many-one reductions. In particular, all ``non-trivial semantic properties of programs'' mentioned in Rice's theorem are witnessable. We also explicitly construct a problem in the non-witnessable (and undecidable) class and show that both classes have cardinality $2^{\aleph_0}$.

Our results offer a new perspective on the understanding of undecidability: for witnessable problems, although it is impossible to solve them precisely, it is always possible to improve any decidable approximation to make it closer to the precise solution. This fact formally demonstrates that research efforts on such approximations are promising and shows there exist universal ways to identify precision issues of program analyzers, program verifiers, SMT solvers, etc., because their essences are decidable approximations of witnessable problems.