Acceptance behavior of L and NL with and without cycling
Shaull's comment answers the question:


  Do you mean that the TM repeats a configuration? If so, it can always be avoided by keeping a (logspace) counter for the maximal number of configurations, and halting when the counter resets.


This technique can even be used, without knowing (beforehand) any bound for the amount of space used. The TM can just keep track of the amount of space used till the current step (excluding the additional counter), and halt when the counter exceeds the maximal possible number of different configurations given the space used.

The additional counter will take the same amount of space as the computation already used plus the logarithm of the length of the input already scanned. Tracking of the amount of space used adds the logarithm of that used space to the total used space. So this approach to detect (avoid) repeated configurations works for any non-sublogarithmic space complexity class.



When I asked that question, I had in mind a technique for finding cycles which only works for deterministic machines: I take two copies of the machine, and always perform one step for the first machine and two steps for the second machine, and then compare the configurations of the two machines. Those configurations can only be equal if the machines cycles, and any cycle will be found before the first machine repeats a configuration.

This technique only works for deterministic machines, and only when copying the machine doesn't violate any resource restrictions. For example, it doesn't work for stack machines (because having two stacks is fundamentally different from having only one stack), and it doesn't work for machines which may only read their input once.



For machines which can read their input only in forward direction and only once, the above two techniques can be slightly modified. Each time a new symbol from the input is read, the entire machinery for detection of repeated configurations is reset, and starts again from zero. So the input position must no longer be included in the number of possible states, and the copied machines (of the second technique) no longer need any access to the input. This allows to apply those techniques also to DFAs.

Similar modifications to also make those techniques applicable to stack machines would be nice, because we know that PDAs have no problems with repeated configurations either.