Rice's theorem

In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about the program's behavior (for instance, does the program terminate for all inputs), unlike a syntactic property (for instance, does the program contain an if-then-else statement). A property is non-trivial if it is neither true for every program, nor false for every program.

Rice's theorem can also be put in terms of functions: for any non-trivial property of partial functions, no general and effective method can decide whether an algorithm computes a partial function with that property. Here, a property of partial functions is called trivial if it holds for all partial computable functions or for none, and an effective decision method is called general if it decides correctly for every algorithm. The theorem is named after Henry Gordon Rice, who proved it in his doctoral dissertation of 1951 at Syracuse University.

In other terms, it tells us that it's impossible to create a computer program that can reliably decide certain interesting things about other programs. Specifically, when it comes to aspects like how a program behaves during its execution, there's no universal method to determine if a program has a particular, more intricate trait. If you have a property that applies to formal languages, and there's at least one example of a language with that property and one without, determining if a given Turing machine has that property becomes an undecidable problem.