Rice–Shapiro theorem

In computability theory, the Rice–Shapiro theorem is a generalization of Rice's theorem, and is named after Henry Gordon Rice and Norman Shapiro. In words, this helps in proving a set is not recursively enumerable and the only properties of the behavior of programs which can be semi-decidable are the “finitary properties” (properties which depend on the behaviour on a finite number/amount of inputs). The key idea it gives away is to leverage the existence of finite subfunctions to show undecidability.

In particular, any statement made about computable function can be proven testing the values at finitely many arguments, stating there is no general algorithm that can decide whether an arbitrary program (or function) has a specific interesting property unless that property is true for all programs or false for none.