determining whether a program halts or not
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running or continue to run forever.


The Halting problem for a specific (narrow) class of programs (Turing machines) may be decidable. But in general, for arbitrary TM $M$and input $w$, the Halting problem is undecidable.