So an idea I've had bouncing around for a while goes like this: suppose we have some TM that runs in possibly exponential time, and thus can use possibly exponential space. But let's say that it doesn't use its available space very often: say, it's oblivious and makes a constant number of passes back and forth across the input.
It seems to me that such a machine can have its space usage improved, but I'm not sure of any relevant results. Can we simulate such a machine in some smaller space bound (ideally polynomial or close to it...), if we allow the simulator to use that smaller amount of space however it wants? Anyone know of such a result?