Decidability of the halting problem on real-life finite computers given unlimited input
The halting problem can be stated as follows:


  Given a program P and an input x, does P halt when run on x?


Here both P and x are finite. In some cases (for example, in program verification) we might also be interested in infinite input which is eventually periodic, or at the very least, has an effective finite description (i.e., it is given by some algorithm). Assuming that this algorithm is also executed on a real machine (otherwise it is "cheating"), the entire setup can be folded to a program running on a real machine, and so the property of having a finite number of states is recovered.