If at each processor instruction, you dump the 4 Gb of memory (as well as caches, disk and loggers), any programs that are running will eventually repeat a state and you can prove this with the dump. After all, if memory has a finite size, then the amount of existing states is finite. The halting problem applies to cases where memory is infinite, but this case is the one with finite memory.
The halting problem is (theoretically) solvable in cases where memory is finite doing exactly what you propose. However, it is necessary to look at this "theoretically" with a lot of attention.
In the worst case, you would have to go through all 2n memory states until it repeats, where n is the number of bits existing in the memory as a whole. If n is a sufficiently small value (for some device with a few bytes of total memory), until it is feasible. For a typical computer of a few gigabytes of memory, this is totally unfeasible. If we only consider the 4 Gb of memory and disregard the disks, the number of existing states would be 234.359.738.368 (4 Gb = 34,359,738,368 bits). This results in a number with more than 10 billion decimal digits!
Each bit existing memory bending the amount of states possible and consequently doubles the time and energy consumption needed to traverse all these states. With each byte of memory added, the number of states increases by 256 times. Now, imagine having 34,359,738,368 bits, each doubling the amount of possible states (that’s 4 Gb, and look at this is just the main memory of a computer already considered limited to the present day, plus we’re disregarding the disks).
You just can’t walk all these states because it will take a long time and take a lot of energy. Thermodynamic physics will impose barriers before you can finish this. Soon, one of these things happens before:
The computer is destroyed by forces outside of it.
The universe ends before you go through all the states. I don’t know if it would be with Big Crunch, Big Rip, Big Freeze, Big Bounce, Heat Death or anything else. But whatever the case, it would not be good for you. This implies in hypothesis 1 above.
You consume all the energy in the universe in order to create and store enough memory dumps (or even if it’s just to keep the computer running through all these states). This implies in hypothesis 2 above which in turn implies hypothesis 1.
Thus, it is not difficult to see that with a sufficient amount of memory, one of the three things enumerated above would occur (and occur quite early!) in this process.
Related: https://answall.com/q/276648/64969
– Jefferson Quesado