Does the Halting Problem have practical relevance? I can calculate all outputs for a finite number of states and inputs Coming from a digital functional hardware verification background, I don’t really understand the Halting Problem. I can represent the program as a state machine and show whether all inputs in all states generate a valid output or lead to an infinite loop.

The only requirement is that the inputs and states are finite. E.g. an infinite counter would lead to an infinite number of states.

Since infinite counters or infinite inputs are impossible in reality, does the Halting Problem really have a practical relevance?
Indeed, if states and inputs are finite (and the tape is finite, which is quite reasonable), then you can easily reduce the Turing machine to a Finite State Machine. However, this is completely impractical, since it doesn't yield any reasonably effective algorithms. I.e. your solution for the halting problem will be to explode the states of the TM with all possible input combinations of size N, and then find a path that reaches a stop state, right? Probably don't want to implement such a check in your compiler.