Computation with Backward Feedback: Power, Observation, and Cost

How much can a computation gain by passing information back to an earlier stage, and how soon can an observer obtain the answer? We study deterministic programs whose runs are accessed at randomly selected stages. A program keeps a register, a finite block of memory. A random clock selects the current stage; each tick takes a random step, with a law fixed independently of input, program and register. When the clock moves backward, the register contents at the stage it leaves start a new run, and the clock continues where it lands. Equilibrium readout requires a distribution of clock position and register that one tick leaves unchanged, and a correct final answer under every such distribution. For bounded steps in both directions with negative average, we obtain an exact classification at each polynomial scale s. If running time and register length are both O(s), feedback gives exactly nondeterministic space O(s); without transmission the class is deterministic time O(s), which is strictly smaller. An observer that starts at stage zero with an all-zero register, reads sequentially and never accepts a nonmember sees, with fewer ticks than any exponential, only languages fixed by a bounded input prefix. Exponentially many ticks reveal every linear-time language. For a clock moving one step up or down, we bound the wait for useful backward moves and the rate of random erasures a stored counter tolerates. The results separate the power of feedback at equilibrium, observation from a fixed start, and the cost of obtaining and keeping information.