A Virtual Machine that uses only one type of instruction to be turing complete (by combining assignment+decision) and implemented some programs in this bytecode / program representation.
Yes, usually the Busy Beaver problem is undecidable. I solved the Busy Beaver problem for a variant: Turing machines with finite tape, instead of Turing machines with infinite tape. This conduces to a further termination condition, so there are 3 termination conditions:
* Halting State is reached (of course)
* End Of Tape is reached (because of finite tape)
* Infinite Looping is detected (see note below)
Infinite Looping is detected -> (This is implemented by an idea of mine: tracking the previous states and when a previous state reappears this is a sign that the programs infinitely repeats (see note below about Euclid division algorithm). This is also because there are no inputs in busy beavers programs, only output they produce, so there is no input that can make the program run on a different path. When the same point is reached, considering snapshots of states, it is assured that the program repeats, thus solving the halting problem for Turing machines that have no input and have finite tape)
)
note about Euclid division algorithm) This works very similarly to an Euclid division algorithm, where this possible infinite repetition of sequences of digits can occur when a previously occurred dividend appears. The divisor does not change, only the dividend changes. And when that changing part is "unchanging" (reoccurs) the behavior repeats, and this is shown in repeating digits, when this happen.
Conclusion: the infinite repetition can be detected (and the Euclid division algorithm showed me how). This relates to the "halting problem".
Dario Cangialosi