> The halting problem is decidable for any deterministic system with finite memory. Either you halt, or you repeat a state. This covers most real-world computer programs.
Problem being that you cannot decide whether a system with such finite memory halts without using another different system that has even more memory. And if that larger system doesn't "real-world" exist, can you truly say that the halting problem is decidable?
You can determine if a program halts by running two copies in lockstep, one at half the speed of the other. If their states are ever the same after the first step, they're in an infinite loop.
That was actually used in an early batch system for running student jobs in an interpreter. A successful student job ran for under a second; one in a loop ran for 30 seconds until it was killed for taking too long. So detecting simple loops, even with substantial extra overhead, was worth it.
Yes, but the point is, if you have an amount of memory M, there exists an amount of memory N for which there is no program that fits into M that can successfully predict whether an arbitrary that fits into N will halt. No infinites required.
(since your example is using two copies, you're essentially using 2M memory)
Problem being that you cannot decide whether a system with such finite memory halts without using another different system that has even more memory. And if that larger system doesn't "real-world" exist, can you truly say that the halting problem is decidable?